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AEnglish Español

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absolute value (28) A number’s distance from zeroon the number line, represented by x .

absolute value function (90) A function written asf(x) � x , where f(x) � 0 for all values of x.

absolute value inequalities (42) For all real numbersa and b, b � 0, the following statements are true.1. If a � b, then �b � a � b2. If a � b, then a � b or a � �b.

algebraic expression (7) An expression thatcontains at least one variable.

amplitude (763) For functions in the form y � a sin b� or y � a cos b�, the amplitude is a .

angle of depression (705) The angle between ahorizontal line and the line of sight from theobserver to an object at a lower level.

angle of elevation (705) The angle between ahorizontal line and the line of sight from theobserver to an object at a higher level.

arccosine (747) The inverse of y � cos x, written as x � arccos y.

arcsine (747) The inverse of y � sin x, written as x � arcsin y.

arctangent (747) The inverse of y � tan x written asx � arctan y.

arithmetic mean (580) The terms between any twononconsecutive terms of an arithmetic sequence.

arithmetic sequence (578) A sequence in whicheach term after the first is found by adding aconstant, the common difference d, to theprevious term.

arithmetic series (583) The indicated sum of theterms of an arithmetic sequence.

asymptote (442, 485) A line that a graph approachesbut never crosses.

augmented matrix (208) A coefficient matrix withan extra column containing the constant terms.

axis of symmetry (287) A line about which a figureis symmetric.

f (x)

xO

axis of symmetry

valor absoluto Distancia entre un número y cero enuna recta numérica; se denota con x .

función del valor absoluto Una función que seescribe f(x) � x , donde f(x) � 0, para todos losvalores de x.

desigualdades con valor absoluto Para todonúmero real a y b, b � 0, se cumple lo siguiente.1. Si a � b, entonces �b � a � b2. Si a � b, entonces a � b o a � �b.

expresión algebraica Expresión que contiene almenos una variable.

amplitud Para funciones de la forma y � a sen b� oy � a cos b�, la amplitud es a .

ángulo de depresión Ángulo entre una rectahorizontal y la línea visual de un observador auna figura en un nivel inferior.

ángulo de elevación Ángulo entre una rectahorizontal y la línea visual de un observador auna figura en un nivel superior.

arcocoseno La inversa de y � cos x, que se escribecomo x � arccos y.

arcoseno La inversa de y � sen x, que se escribecomo x � arcsen y.

arcotangente La inversa de y � tan x que se escribecomo x � arctan y.

media aritmética Cualquier término entre dos térmi-nos no consecutivos de una sucesión aritmética.

sucesión aritmética Sucesión en que cualquiertérmino después del primero puede hallarsesumando una constante, la diferencia común d, altérmino anterior.

serie aritmética Suma específica de los términos deuna sucesión aritmética.

asíntota Recta a la que se aproxima una gráfica, sinjamás cruzarla.

matriz ampliada Matriz coeficiente con una colum-na extra que contiene los términos constantes.

eje de simetría Recta respecto a la cual una figuraes simétrica.

f (x)

xO

eje de simetría

Cómo usar el glosario en español:1. Busca el término en inglés que desees encontrar.2. El término en español, junto con la definición,

se encuentran en la columna de la derecha.

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R2 Glossary/Glosario

Bb

�n1

�(257) For any real number b and for any positive

integer n, b�n1

�� n

b�, except when b � 0 and n iseven.

binomial (229) A polynomial that has two unliketerms.

binomial experiment (677) An experiment in whichthere are exactly two possible outcomes for eachtrial, a fixed number of independent trials, andthe probabilities for each trial are the same.

Binomial Theorem (613) If n is a nonnegativeinteger, then (a � b)n �

1anb0 � �n1

�an � 1b1 � �n(n

1 ��

21)

� an � 2b2 �…� 1a0bn.

boundary (96) A line or curve that separates thecoordinate plane into two regions.

bounded (129) A region is bounded when the graphof a system of constraints is a polygonal region.

Cartesian coordinate plane (56) A plane dividedinto four quadrants by the intersection of the x-axis and the y-axis at the origin.

center of a circle (426) The point from which allpoints on a circle are equidistant.

center of an ellipse (434) The point at which themajor axis and minor axis of an ellipse intersect.

center of a hyperbola (442) The midpoint of thesegment whose endpoints are the foci.

change of base formula (548) For all positivenumbers a, b, and n, where a � 1 and b � 1,

loga n � �lloogg

b

b

na

�.

circle (426) The set of all points in a plane that areequidistant from a givenpoint in the plane, calledthe center.

circular functions (740) Functions defined using aunit circle.

y

xO

(x, y)

(h, k)

r

radius

center

O

(3, 2)

x-axis

y-axisQuadrant II

Quadrant III Quadrant IV

Quadrant I

x-coordinate

y-coordinateorigin

b�n1

�Para cualquier número real b y para cualquier

entero positivo n, b�n1

�� n

b�, excepto cuando b � 0y n es par.

binomio Polinomio con dos términos diferentes.

experimento binomial Experimento con exactamen-te dos resultados posibles para cada prueba, unnúmero fijo de pruebas independientes y en elcual cada prueba tiene igual probabilidad.

Teorema del binomio Si n es un entero no negativo,entonces (a � b)n �

1anb0 � �n1

�an � 1b1 � �n(n

1 ��

21)

� an � 2b2 �…� 1a0bn.

frontera Recta o curva que divide un plano decoordenadas en dos regiones.

acotada Una región está acotada cuando la gráfica deun sistema de restricciones es una región poligonal.

plano de coordenadas cartesiano Plano dividido encuatro cuadrantes mediante la intersección en elorigen de los ejes x y y.

centro de un círculo El punto desde el cual todoslos puntos de un círculo están equidistantes.

centro de una elipse Punto de intersección de losejes mayor y menor de una elipse.

centro de una hipérbola Punto medio del segmentocuyos extremos son los focos.

fórmula del cambio de base Para todo númeropositivo a, b y n, donde a � 1 y b � 1,

logb n � �lloogg

b

b

na

�.

círculo Conjunto de todos los puntos en un planoque equidistan de unpunto dado del planollamado centro.

funciones circulares Funciones definidas en uncírculo unitario.

y

xO

(x, y)

(h, k)

r

radio

centro

O

(3, 2)

eje x

eje yCuadrante II

Cuadrante III Cuadrante IV

Cuadrante I

coordenada x

coordenada yorigen

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coeficiente Factor numérico de un monomio.

matriz columna Matriz que sólo tiene una columna.

combinación Arreglo de elementos en que el ordenno es importante.

diferencia común Diferencia entre términosconsecutivos de una sucesión aritmética.

logaritmos comunes El logaritmo de base 10.

razón común Razón entre términos consecutivos deuna sucesión geométrica.

Propiedad conmutativa de la adición Paracualquier número real a y b, a � b � b � a.

Propiedad conmutativa de la multiplicación Paracualquier número real a y b, a � b � b � a.

completar el cuadrado Proceso mediante el cualuna expresión cuadrática se transforma en untrinomio cuadrado perfecto.

conjugados complejos Dos números complejos dela forma a � bi y a � bi.

fracción compleja Expresión racional cuyonumerador o denominador contiene unaexpresión racional.

número complejo Cualquier número que puedeescribirse de la forma a � bi, donde a y b sonnúmeros reales e i es la unidad imaginaria.

composición de funciones Se evalúa una funcióny luego se evalúa una segunda función en elresultado de la primera función. La composiciónde f y g se define con f � g y [f � g](x) � f[g(x)].

evento compuesto Dos o más eventos simples.

desigualdad compuesta Dos desigualdades unidaspor las palabras y u o.

sección cónica Cualquier figura obtenida medianteel corte de un cono doble.

eje conjugado El segmento de 2b unidades delongitud que es perpendicular al eje transversalen el centro.

conjugados Binomios de la forma a�b� � c�d� ya�b� � c�d�, donde a, b, c y d son númerosracionales.

consistente Sistema de ecuaciones que posee por lomenos una solución.

constante Monomios que carecen de variables.

función constante Función lineal de la forma f(x) � b.

constante de variación La constante k que se usa envariación directa o inversa.

coefficient (222) The numerical factor of amonomial.

column matrix (155) A matrix that has only onecolumn.

combination (640) An arrangement of objects inwhich order is not important.

common difference (578) The difference betweenthe successive terms of an arithmetic sequence.

common logarithms (547) Logarithms that use 10 asthe base.

common ratio (588) The ratio of successive terms ofa geometric sequence.

Commutative Property of Addition (12) For anyreal numbers a and b, a � b � b � a.

Commutative Property of Multiplication (12) Forany real numbers a and b, a � b � b � a.

completing the square (307) A process used tomake a quadratic expression into a perfect squaretrinomial.

complex conjugates (273) Two complex numbers ofthe form a � bi and a � bi.

complex fraction (475) A rational expression whosenumerator and/or denominator contains arational expression.

complex number (271) Any number that can bewritten in the form a � bi, where a and b are realnumbers and i is the imaginary unit.

composition of functions (384) A function isperformed, and then a second function isperformed on the result of the first function.The composition of f and g is denoted by f � g,and [f � g](x) � f[g(x)].

compound event (658) Two or more simple events.

compound inequality (40) Two inequalities joinedby the word and or or.

conic section (419) Any figure that can be obtainedby slicing a double cone.

conjugate axis (442) The segment of length 2b unitsthat is perpendicular to the transverse axis at thecenter.

conjugates (253) Binomials of the form a�b� � c�d�and a�b� � c�d�, where a, b, c, and d are rationalnumbers.

consistent (111) A system of equations that has atleast one solution.

constant (222) Monomials that contain no variables.

constant function (90) A linear function of the formf(x) � b.

constant of variation (492) The constant k used withdirect or inverse variation.

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término constante En f(x) � ax2 � bx � c, c es eltérmino constante.

restricciones Condiciones a que están sujetas lasvariables, a menudo escritas como desigualdadeslineales.

continuidad La gráfica de una función que sepuede calcar sin levantar nunca el lápiz del papel.

distribución de probabilidad continua El resultadopuede ser cualquier valor de un intervalo denúmeros reales, representados por curvas.

cosecante Para cualquier ángulo de medida �, un punto P(x, y) en su lado terminal, r � �x2 � y�2�, csc � � �

yr

�.

coseno Para cualquier ángulo de medida �, un punto P(x, y) en su lado terminal, r � �x2 � y�2�, cos � � �

xr�.

cotangente Para cualquier ángulo de medida �, un punto P(x, y) en su lado terminal, r � �x2 � y�2�, cot � � �

xy

�.

ángulos coterminales Dos ángulos en posiciónestándar que tienen el mismo lado terminal.

Regla de Crámer Método que usa determinantespara resolver un sistema de ecuaciones lineales.

grado Suma de los exponentes de las variables deun monomio.

grado de un polinomio de una variable Elexponente máximo de la variable del polinomio.

eventos dependientes El resultado de un eventoafecta el resultado de otro evento.

sistema dependiente Sistema de ecuaciones queposee un número infinito de soluciones.

variable dependiente La otra variable de una fun-ción, por lo general y, cuyo valor depende de x.

polinomio reducido El cociente cuando se divideun polinomio entre uno de sus factoresbinomiales.

determinante Arreglo cuadrado de números ovariables encerrados entre dos rectas paralelas.

dilatación Transformación en que se amplía oreduce una figura geométrica.

análisis dimensional Realizar operaciones conunidades.

tamaño de una matriz El número de filas, m, ycolumnas, n, de una matriz, lo que se escribe m n.

constant term (286) In f(x) � ax2 � bx � c, c is theconstant term.

constraints (129) Conditions given to variables,often expressed as linear inequalities.

continuity (485) A graph of a function that can betraced with a pencil that never leaves the paper.

continuous probability distribution (671) Theoutcome can be any value in an interval of realnumbers, represented by curves.

cosecant (701) For any angle, with measure �, a point P(x, y) on its terminal side, r � �x2 � y�2�, csc � � �

yr

�.

cosine (701) For any angle, with measure �, a point P(x, y) on its terminal side, r � �x2 � y�2�, cos � � �

xr�.

cotangent (701) For any angle, with measure �, a point P(x, y) on its terminal side, r � �x2 � y�2�, cot � � �

xy

�.

coterminal angles (711) Two angles in standardposition that have the same terminal side.

Cramer’s Rule (189) A method that uses determinantsto solve a system of linear equations.

degree (222) The sum of the exponents of thevariables of a monomial.

degree of a polynomial in one variable (346) Thegreatest exponent of the variable of thepolynomial.

dependent events (633) The outcome of one eventdoes affect the outcome of another event.

dependent system (111) A consistent system ofequations that has an infinite number of solutions.

dependent variable (59) The other variable in afunction, usually y, whose values depend on x.

depressed polynomial (366) The quotient when apolynomial is divided by one of its binomialfactors.

determinant (182) A square array of numbers orvariables enclosed between two parallel lines.

dilation (176) A transformation in which ageometric figure is enlarged or reduced.

dimensional analysis (225) Performing operationswith units.

dimensions of a matrix (155) The number of rows,m, and the number of columns, n, of the matrixwritten as m n.

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directriz Véase parábola.

variación directa y varía directamente con x si hay una constante no nula k tal que y � kx. k se llama la constante de variación.

distribución de probabilidad discreta Probabilidadesque tienen un número finito de valores posibles.

discriminante En la fórmula cuadrática, laexpresión b2 � 4ac.

Fórmula de la distancia La distancia entre dospuntos (x1, y1) y (x2, y2) viene dada por

d � �(x2 ��x1)2 �� (y2 �� y1)2�.

dominio El conjunto de todas las coordenadas x delos pares ordenados de una relación.

e El número irracional 2.71828.... e es la base de loslogaritmos naturales.

elemento Cada valor de una matriz.

método de eliminación Eliminar una de lasvariables de un sistema de ecuaciones sumando orestando las ecuaciones.

elipse Conjunto de todos los puntos de un plano enlos que la suma de sus distancias a dos puntosdados del plano, llamados focos, es constante.

conjunto vacío Conjunto solución de una ecuaciónque no tiene solución, denotado por { } o �.

comportamiento final El comportamiento de unagráfica a medida que x tiende a más infinito (+�)o menos infinito (��).

matrices iguales Dos matrices que tienen lasmismas dimensiones y en las que cada elementode una de ellas es igual al elementocorrespondiente en la otra matriz.

ecuación Enunciado matemático que afirma laigualdad de dos expresiones matemáticas.

expansión por determinantes menores Un métodode calcular el determinante de tercer orden omayor mediante el uso de determinantes deorden más bajo.

x

y

O

(a, 0)

F1 (�c, 0)

F2 (c, 0)

(�a, 0)a a

c

b

eje menorcentro

eje mayor

directrix (419) See parabola.

direct variation (492) y varies directly as x if there is some nonzero constant k such that y � kx. k is called the constant of variation.

discrete probability distributions (671) Probabilitiesthat have a finite number of possible values.

discriminant (316) In the Quadratic Formula, theexpression b2 � 4ac.

Distance Formula (413) The distance between twopoints with coordinates (x1, y1) and (x2, y2) is

given by d � �(x2 ��x1)2 �� (y2 �� y1)2�.

domain (56) The set of all x-coordinates of theordered pairs of a relation.

e (554) The irrational number 2.71828.... e is the baseof the natural logarithms.

element (155) Each value in a matrix.

elimination method (118) Eliminate one of thevariables in a system of equations by adding orsubtracting the equations.

ellipse (433) The set of all points in a plane suchthat the sum of the distances from two givenpoints in the plane, called foci, is constant.

empty set (29) The solution set for an equation thathas no solution, symbolized by { } or �.

end behavior (349) The behavior of the graph as xapproaches positive infinity (+�) or negativeinfinity (��).

equal matrices (155) Two matrices that have thesame dimensions and each element of one matrixis equal to the corresponding element of the othermatrix.

equation (20) A mathematical sentence stating thattwo mathematical expressions are equal.

expansion by minors (183) A method of evaluatinga third or high order determinant by usingdeterminants of lower order.

x

y

O

(a, 0)

F1 (�c, 0)

F2 (c, 0)

(�a, 0)a a

c

b

Minor axisCenter

Major axis

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desintegración exponencial Ocurre cuando una can-tidad disminuye exponencialmente con el tiempo.

ecuación exponencial Ecuación en que las variablesaparecen en los exponentes.

función exponencial Una función de la formay � abx, donde a � 0, b � 0, y b � 1.

crecimiento exponencial El que ocurre cuando unacantidad aumenta exponencialmente con eltiempo.

solución extraña Número que no satisface laecuación original.

extrapolación Predicción para un valor de x mayorque cualquiera de los de un conjunto de datos.

factorial Si n es un entero positivo, entonces n! � n(n � 1)(n � 2) … 2 � 1.

fracaso Cualquier resultado distinto del deseado.

familia de gráficas Grupo de gráficas que presentanuna o más características similares.

región viable Intersección de las gráficas de unsistema de restricciones.

sucesión de Fibonacci Sucesión en que los dosprimeros términos son iguales a 1 y cada términoque sigue es igual a la suma de los dos anteriores.

foco Véase parábola, elipse, hipérbola.

método FOIL El producto de dos binomios es lasuma de los productos de los primeros (First) tér-minos, los términos exteriores (Outer), los términosinteriores (Inner) y los últimos (Last) términos.

fórmula Enunciado matemático que describe larelación entre ciertas cantidades.

f (x)

xO

1

1�1�2 2

2

3

crecimientoexponencial

f (x)

xO

1

1�1�2 2

2

3

desintegraciónexponencial

exponential decay (524) Exponential decay occurswhen a quantity decreases exponentially over time.

exponential equation (526) An equation in whichthe variables occur as exponents.

exponential function (524) A function of the formy � abx, where a � 0, b � 0, and b � 1.

exponential growth (524) Exponential growthoccurs when a quantity increases exponentiallyover time.

extraneous solution (263) A number that does notsatisfy the original equation.

extrapolation (82) Predicting for an x-value greaterthan any in the data set.

factorial (613) If n is a positive integer, then n! � n(n � 1)(n � 2) … 2 � 1.

failure (644) Any outcome other than the desiredoutcome.

family of graphs (70) A group of graphs thatdisplays one or more similar characteristics.

feasible region (129) The intersection of the graphsin a system of constraints.

Fibonacci sequence (606) A sequence in which thefirst two terms are 1 and each of the additionalterms is the sum of the two previous terms.

focus (419, 433, 441) See parabola, ellipse, hyperbola.

FOIL method (230) The product of two binomialsis the sum of the products of F the first terms, O the outer terms, I the inner terms, and L thelast terms.

formula (8) A mathematical sentence that expressesthe relationship between certain quantities.

f (x)

xO

1

1�1�2 2

2

3

ExponentialGrowth

f (x)

xO

1

1�1�2 2

2

3

ExponentialDecay

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function (57) A relation in which each element ofthe domain is paired with exactly one element inthe range.

function notation (59) An equation of y in terms ofx can be rewritten so that y � f(x). For example,y � 2x � 1 can be written as f(x) � 2x � 1.

geometric mean (590) The terms between any twononsuccessive terms of a geometric sequence.

geometric sequence (588) A sequence in which eachterm after the first is found by multiplying theprevious term by a constant r, called the commonratio.

geometric series (594) The sum of the terms of ageometric sequence.

greatest integer function (89) A step function,written as f(x) � x�, where f(x) is the greatestinteger less than or equal to x.

hyperbola (441) The set of all points in the planesuch that the absolute value of the difference ofthe distances from two given points in the plane,called foci, is constant.

hypothesis (686) A statement to be tested.

identity function (90, 391) The function I(x) � x.

identity matrix (195) A square matrix that, whenmultiplied by another matrix, equals that samematrix. If A is any n n matrix and I is the n nidentity matrix, then A � I � A and I � A � A.

image (175) The graph of an object after atransformation.

imaginary unit (270) i, or the principal square rootof �1.

inclusive (659) Two events whose outcomes may bethe same.

y

xO

asymptote

conjugate axis

transverse axis

center

asymptote

vertex vertex F1 F2

b c

a

función Relación en que a cada elemento deldominio le corresponde un solo elemento delrango.

notación funcional Una ecuación de y en términosde x puede escribirse en la forma y � f(x). Porejemplo, y � 2x � 1 puede escribirse como f(x) � 2x � 1.

media geométrica Cualquier término entre dos tér-minos no consecutivos de una sucesión geométrica.

sucesión geométrica Sucesión en que cualquiertérmino después del primero puede hallarsemultiplicando el término anterior por unaconstante r, llamada razón común .

serie geométrica La suma de los términos de unasucesión geométrica.

función del máximo entero Una función etapa quese escribe f(x) � [x], donde f(x) es el meaximoentero que es menor que o igual a x.

hipérbola Conjunto de todos los puntos de unplano en los que el valor absoluto de la diferenciade sus distancias a dos puntos dados del plano,llamados focos, es constante.

hipótesis Proposición que debe ser verificada.

función identidad La función I(x) � x.

matriz identidad Matriz cuadrada que al multipli-carse por otra matriz, es igual a la misma matriz.Si A es una matriz de n n e I es la matriz identi-dad de n n, entonces A � I � A y I � A � A.

imagen Gráfica de una figura después de unatransformación.

unidad imaginaria i, o la raíz cuadrada principal de�1.

inclusivo Dos eventos que pueden tener los mismosresultados.

y

xO

asíntota

eje conjugado

eje transversal

centro

asíntota

vértice vértice F1 F2

b c

a


G

H

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inconsistente Sistema de ecuaciones que no tienesolución alguna.

independiente Sistema de ecuaciones que sólo tieneuna solución.

eventos independientes Eventos que no se afectanmutuamente.

variable independiente En una función, la variable,por lo general x, cuyos valores forman el dominio.

índice de suma Variable que se usa con el símbolode suma. En la siguiente expresión, el índice desuma es n.

�3

n�14n

hipótesis inductiva El suponer que un enunciado esverdadero para algún entero positivo k, donde k� n.

serie geométrica infinita Serie geométrica con unnúmero infinito de términos.

lado inicial de un ángulo El rayo fijo de un ángulo.

interpolación Predecir un valor de x entre losvalores máximo y mínimo del conjunto de datos.

intersección Gráfica de una desigualdad compuestaque contiene la palabra y.

notación de intervalo Uso de los símbolos deinfinito, �� y ��, para indicar que el conjuntosolución de una desigualdad no es acotado en ladirección positiva o negativa, respectivamente.

inversa Dos matrices de n n son inversas mutuassi su producto es la matriz identidad.

función inversa Dos funciones f y g son inversasmutuas si y sólo si las composiciones de ambasson la función identidad.

inversa de una función trigonométrica Lasrelaciones arcocoseno, arcoseno y arcotangente.

relaciones inversas Dos relaciones son relacionesinversas mutuas si y sólo si cada vez que una delas relaciones contiene el elemento (a, b), la otracontiene el elemento (b, a).

variación inversa y varía inversamente con x si hayuna constante no nula k tal que xy � k oy � �

xk

�.

y

xO

ladoterminal

lado inicial

vértice

180˚

270˚

90˚

inconsistent (111) A system of equations that has nosolutions.

independent (111) A system of equations that hasexactly one solution.

independent events (632) Events that do not affecteach other.

independent variable (59) In a function, the variable,usually x, whose values make up the domain.

index of summation (585) The variable used withthe summation symbol. In the expression below,the index of summation is n.

�3

n�14n

inductive hypothesis (618) The assumption that astatement is true for some positive integer k,where k � n.

infinite geometric series (599) A geometric serieswith an infinite number of terms.

initial side of an angle (709) The fixed ray of anangle.

interpolation (82) Predicting for an x-value betweenthe least and greatest values of the set.

intersection (40) The graph of a compoundinequality containing and.

interval notation (35) Using the infinity symbols,�� and ��, to indicate that the solution set of aninequality is unbounded in the positive ornegative direction, respectively.

inverse (195) Two n n matrices are inverses ofeach other if their product is the identity matrix.

inverse function (391) Two functions f and g areinverse functions if and only if both of theircompositions are the identity function.

inverse of a trigonometric function (746) Thearccosine, arcsine, and arctangent relations.

inverse relations (390) Two relations are inverserelations if and only if whenever one relationcontains the element (a, b) the other relationcontains the element (b, a).

inverse variation (493) y varies inversely as x ifthere is some nonzero constant k such that xy � kor y � �

xk

�.

y

xO

terminalside

initial side

vertex

180˚

270˚

90˚

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irrational number (11) A real number that is notrational. The decimal form neither terminates norrepeats.

isometry (175) A transformation in which theimage and preimage are congruent figures.

iteration (608) The process of composing a functionwith itself repeatedly.

joint variation (493) y varies jointly as x and z ifthere is some nonzero constant k such thaty � kxz, where x � 0 and z � 0.

latus rectum (421) The line segment through thefocus of a parabola and perpendicular to the axisof symmetry.

Law of Cosines (733–734) Let �ABC be any trianglewith a, b, and c representing the measures ofsides, and opposite angles with measures A, B,and C, respectively. Then the following equationsare true.a2 � b2 � c2 � 2bc cos Ab2 � a2 � c2 � 2ac cos Bc2 � a2 � b2 � 2ab cos C

Law of Sines (726) Let �ABC be any triangle witha, b, and c representing the measures of sidesopposite angles with measurements A, B, and C, respectively. Then �sin

aA

� � �sin

bB

� � �sin

cC

�.

leading coefficient (346) The coefficient of the termwith the highest degree.

like radical expressions (252) Two radicalexpressions in which both the radicands andindices are alike.

like terms (229) Monomials that can be combined.

limit (593) The value that the terms of a sequenceapproach.

linear equation (63) An equation that has nooperations other than addition, subtraction, andmultiplication of a variable by a constant.

linear function (63) A function whose ordered pairssatisfy a linear equation.

linear permutation (638) The arrangement ofobjects or people in a line.

linear programming (130) The process of findingthe maximum or minimum values of a functionfor a region defined by inequalities.

linear term (286) In the equation f(x) � ax2 � bx � c,bx is the linear term.

L

J

número irracional Número que no es racional. Suexpansión decimal no es ni terminal ni periódica.

isometría Transformación en que la imagen y lapreimagen son figuras congruentes.

iteración Proceso de componer una función consigomisma repetidamente.

variación conjunta y varía conjuntamente con x y zsi hay una constante no nula k tal quey � kxz, donde x � 0 y z � 0.

latus rectum El segmento de recta que pasa por elfoco de una parábola y que es perpendicular a sueje de simetría.

Ley de los cosenos Sea �ABC un triángulocualquiera, con a, b y c las longitudes de los ladosy con ángulos opuestos de medidas A, B y C,respectivamente. Entonces se cumplen lassiguientes ecuaciones.a2 � b2 � c2 � 2bc cos Ab2 � a2 � c2 � 2ac cos Bc2 � a2 � b2 � 2ab cos C

Ley de los senos Sea �ABC cualquier triángulo cona, b y c las longitudes de los lados y con ángulosopuestos de medidas A, B y C, respectivamente.Entonces �sin

aA

� � �sin

bB

� � �sin

cC

�.

coeficiente líder Coeficiente del término de mayorgrado.

expresiones radicales semejantes Dos expresionesradicales en que tanto los radicandos como losíndices son semejantes.

términos semejantes Monomios que puedencombinarse.

límite El valor al que tienden los términos de unasucesión.

ecuación lineal Ecuación sin otras operaciones quelas de adición, sustracción y multiplicación deuna variable por una constante.

función lineal Función cuyos pares ordenadossatisfacen una ecuación lineal.

permutación lineal Arreglo de personas o figurasen una línea.

programación lineal Proceso de hallar los valoresmáximo o mínimo de una función lineal en unaregión definida por las desigualdades.

término lineal En la ecuación f(x) � ax2 � bx � c, eltérmino lineal es bx.


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line of fit (81) A line that closely approximates a setof data.

logarithm (531) In the function x � by, y is called thelogarithm, base b, of x. Usually written as y �logb x and is read “y equals log base b of x.”

logarithmic equation (533) An equation thatcontains one or more logarithms.

logarithmic function (532) The function y � logb x,where b � 0 and b � 1, which is the inverse of theexponential function y � bx.

m � n matrix (155) A matrix with m rows and ncolumns.

major axis (434) The longer of the two line segmentsthat form the axes of symmetry of an ellipse.

mapping (57) How each member of the domain ispaired with each member of the range.

margin of sampling error (ME) (682) The limit onthe difference between how a sample respondsand how the total population would respond.

mathematical induction (618) A method of proofused to prove statements about positive integers.

matrix (154) Any rectangular array of variables orconstants in horizontal rows and vertical columns.

maximum value (288) The y-coordinate of the vertexof the quadratic function f(x) � ax2 � bx � c,where a � 0.

measure of central tendency (665) A number thatrepresents the center or middle of a set of data.

measure of variation (664) A representation of howspread out or scattered a set of data is.

midline (771) A horizontal axis used as thereference line about which the graph of a periodicfunction oscillates.

minimum value (288) The y-coordinate of thevertex of the quadratic function f(x) � ax2 � bx �c, where a � 0.

minor (183) The determinant formed when the rowand column containing that element are deleted.

minor axis (434) The shorter of the two line segmentsthat form the axes of symmetry of an ellipse.

monomial (222) An expression that is a number, avariable, or the product of a number and one ormore variables.

recta de ajuste Recta que se aproxima estrecha-mente a un conjunto de datos.

logaritmo En la función x � by, y es el logaritmo enbase b, de x. Generalmente escrito como y � logbx y se lee “y es igual al logaritmo en base b de x.”

ecuación logarítmica Ecuación que contiene uno omás logaritmos.

función logarítmica La función y � logb x, donde b � 0 y b � 1, inversa de la función exponencial y � bx.

matriz de m � n Matriz de m filas y n columnas.

eje mayor El más largo de dos segmentos de rectaque forman los ejes de simetría de una elipse.

transformaciones La correspondencia entre cadamiembro del dominio con cada miembro del rango.

margen de error muestral (EM) Límite en la diferen-cia entre las respuestas obtenidas con una muestray cómo pudiera responder la población entera.

inducción matemática Método de demostrarenunciados sobre los enteros positivos.

matriz Arreglo rectangular de variables o constan-tes en filas horizontales y columnas verticales.

valor máximo La coordenada y del vértice de lafunción cuadrática f(x) � ax2 � bx � c, dondea � 0.

medida de tendencia central Número que represen-ta el centro o medio de un conjunto de datos.

medida de variación Número que representa ladispersión de un conjunto de datos.

recta central Eje horizontal que se usa como rectade referencia alrededor de la cual oscila la gráficade una función periódica.

valor mínimo La coordenada y del vértice de lafunción cuadrática f(x) � ax2 � bx � c, dondea � 0.

determinante menor El que se forma cuando sedescartan la fila y columna que contienen dichoelemento.

eje menor El más corto de los dos segmentos derecta de los ejes de simetría de una elipse.

monomio Expresión que es un número, una varia-ble o el producto de un número por una o másvariables.


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mutually exclusive (658) Two events that cannotoccur at the same time.

nth root (245) For any real numbers a and b, and anypositive integer n, if an � b, then a is an nth root of b.

natural base exponential function (554) Anexponential function with base e, y � ex.

natural logarithm (554) Logarithms with base e,written ln x.

natural logarithmic function (554) y � ln x, theinverse of the natural base exponential functiony � ex.

negative exponent (222) For any real number a � 0

and any integer n, a�n � �a1n� and �

a�1

n� � an.

normal distribution (671) A frequency distributionthat often occurs when there is a large number ofvalues in a set of data: about 68% of the valuesare within one standard deviation of the mean,95% of the values are within two standarddeviations from the mean, and 99% of the valuesare within three standard deviations.

octants (136) The eight regions of three-dimensionalspace.

odds (645) The ratio of the number of the successes ofan event to the number of failures.

one-to-one function (57, 392) 1. A function whereeach element of the range is paired with exactlyone element of the domain 2. A function whoseinverse is a function.

open sentence (20) A mathematical sentencecontaining one or more variables.

ordered pair (56) A pair of coordinates, written inthe form (x, y), used to locate any point on acoordinate plane.

ordered triple (136, 139) 1. The coordinates of apoint in space 2. The solution of a system ofequations in three variables x, y, and z.

Normal Distribution

mutuamente exclusivos Dos eventos que nopueden ocurrir simultáneamente.

raíz enésima Para cualquier número real a y b ycualquier entero positivo n, si an � b, entonces ase llama una raíz enésima de b.

función exponencial natural La funciónexponencial de base e, y � ex.

logaritmo natural Logaritmo de base e, el que seescribe ln x.

función logarítmica natural y � ln x, la inversa de la función exponencial natural y � ex.

exponente negativo Para cualquier número real a � 0

cualquier entero positivo n, a�n � �a1n� y �

a�1

n� � an.

distribución normal Distribución de frecuencia queaparece a menudo cuando hay un número grandede datos: cerca del 68% de los datos están dentrode una desviación estándar de la media, 95%están dentro de dos desviaciones estándar de lamedia y 99% están dentro de tres desviacionesestándar de la media.

octantes Las ocho regiones del espaciotridimensional.

posibilidades Razón del número de éxitos de unevento a su número de fracasos.

función biunívoca 1. Función en la que a cadaelemento del rango le corresponde sólo unelemento del dominio. 2. Función cuya inversaes una función.

enunciado abierto Enunciado matemático quecontiene una o más variables.

par ordenado Un par de números, escrito en laforma (x, y), que se usa para ubicar cualquierpunto en un plano de coordenadas.

triple ordenado 1. Las coordenadas de un punto enel espacio 2. Solución de un sistema deecuaciones en tres variables x, y y z.

Distribución normal


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P

Order of Operations (6)Step 1 Evaluate expressions inside groupingsymbols.Step 2 Evaluate all powers.Step 3 Do all multiplications and/or divisionsfrom left to right.Step 4 Do all additions and subtractions fromleft to right.

outcomes (632) The results of a probabilityexperiment/an event.

outlier (826) A data point that does not appear tobelong to the rest of the set.

parabola (286, 419) The set of all points in a planethat are the same distance from a given point,called the focus, and a given line, called thedirectrix.

parallel lines (70) Nonvertical coplanar lines withthe same slope.

parent graph (70) The simplest of graphs in afamily.

partial sum (599) The sum of the first n terms of aseries.

Pascal’s triangle (612) A triangular array ofnumbers such that the (n � 1)th row is thecoefficient of the terms of the expansion (x � y)n

for n � 0, 1, 2 ...

period (741) The least possible value of a for whichf(x) � f(x � a).

periodic function (741) A function is called periodicif there is a number a such that f(x) � f(x � a) forall x in the domain of the function.

permutation (638) An arrangement of objects inwhich order is important.

perpendicular lines (71) In a plane, any two obliquelines the product of whose slopes is �1.

phase shift (769) A horizontal translation of atrigonometric function.

piecewise function (91) A function that is writtenusing two or more expressions.

y

xO

vertex

axis ofsymmetry

(h, k)

x � h

Orden de las operacionesPaso 1 Evalúa las expresiones dentro desímbolos de agrupamiento.Paso 2 Evalúa todas las potencias.Paso 3 Ejecuta todas las multiplicaciones ydivisiones de izquierda a derecha.Paso 4 Ejecuta todas las adiciones ysustracciones de izquierda a derecha.

resultados Lo que produce un experimento oevento probabilístico.

valor atípico Dato que no parece pertenecer al restoel conjunto.

parábola Conjunto de todos los puntos de un planoque están a la misma distancia de un punto dado,llamado foco, y de una recta dada, llamadadirectriz.

rectas paralelas Rectas coplanares no verticales conla misma pendiente.

gráfica madre La gráfica más sencilla en una familiade gráficas.

suma parcial La suma de los primeros n términosde una serie.

Triángulo de Pascal Arreglo triangular de númerosen el que la fila (n � 1)n proporciona loscoeficientes de los términos de la expansión de (x � y)n para n � 0, 1, 2 ...

período El menor valor positivo posible para a, parael cual f(x) � f(x � a).

función periódica Función para la cual hay unnúmero a tal que f(x) � f(x � a) para todo x en eldominio de la función .

permutación Arreglo de elementos en que el ordenes importante.

rectas perpendiculares En un plano, dos rectasoblicuas cualesquiera cuyas pendientes tienen unproducto igual a �1.

desvío de fase Traslación horizontal de una funcióntrigonométrica.

función a intervalos Función que se escribe usandodos o más expresiones.

y

xO

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(h, k)

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point discontinuity (485) If the original function isundefined for x � a but the related rationalexpression of the function in simplest form isdefined for x � a, then there is a hole in the graphat x � a.

point-slope form (76) An equation in the formy � y1 � m(x � x1) where (x1, y1) are thecoordinates of a point on the line and m is theslope of the line.

polynomial (229) A monomial or a sum ofmonomials.

polynomial function (347) A function that isrepresented by a polynomial equation.

polynomial in one variable (346) a0xn � a1xn � 1 �… � an�2x2 � an � 1x � an, where the coefficientsa0, a1, …, an represent real numbers, and a0 is notzero and n is a nonnegative integer.

power (222) An expression of the form xn.

power function (704) An equation in the form f(x) � axb, where a and b are real numbers.

prediction equation (81) An equation suggested bythe points of a scatter plot that is used to predictother points.

preimage (175) The graph of an object before atransformation.

principal root (246) The nonnegative root.

principal values (746) The values in the restricteddomains of trigonometric functions.

probability (644) A ratio that measures the chancesof an event occurring.

probability distribution (646) A function that mapsthe sample space to the probabilities of theoutcomes in the sample space for a particularrandom variable.

pure imaginary number (270) The square roots ofnegative real numbers. For any positive real

number b, ��b2� � �b2� � ��1�, or bi.

quadrantal angle (718) An angle in standard positionwhose terminal side coincides with one of the axes.

quadrants (56) The four areas of a Cartesiancoordinate plane.

f (x)

xO

pointdiscontinuity

discontinuidad evitable Si la función original noestá definida en x � a pero la expresión racionalreducida correspondiente de la función estádefinida en x � a, entonces la gráfica tiene unaruptura o corte en x � a.

forma punto-pendiente Ecuación de la formay � y1 � m(x � x1) donde (x1, y1) es un punto enla recta y m es la pendiente de la recta.

polinomio Monomio o suma de monomios.

función polinomial Función representada por unaecuación polinomial.

polinomio de una variable a0xn � a1xn � 1 �… � an�2x2 � an � 1x � an, donde los coeficientesa0, a1, …, an son números reales, a0 no es nulo y nes un entero no negativo.

potencia Expresión de la forma xn.

función potencia Ecuación de la forma f(x) � axb, donde a y b son números reales.

ecuación de predicción Ecuación sugerida por lospuntos de una gráfica de dispersión y que se usapara predecir otros puntos.

preimagen Gráfica de una figura antes de unatransformación.

raíz principal La raíz no negativa.

valores principales Valores en los dominiosrestringidos de las funciones trigonométricas.

probabilidad Razón que mide la posibilidad de queocurra un evento.

distribución de probabilidad Función que aplica elespacio muestral a las probabilidades de losresultados en el espacio muestral obtenidos parauna variable aleatoria particular.

número imaginario puro Raíz cuadrada de unnúmero real negativo. Para cualquier número

real positivo b, ��b2� � �b2� � ��1� ó bi.

ángulo de cuadrante Ángulo en posición estándarcuyo lado terminal coincide con uno de los ejes.

cuadrantes Las cuatro regiones de un plano decoordenadas cartesiano.

f (x)

xO

discontinuidadevitable

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quadratic equation (294) A quadratic function setequal to a value, in the form ax2 � bx � c, where a � 0.

quadratic form (360) For any numbers a, b, and c,except for a � 0, an equation that can be writtenin the form a[f(x)2] � b[f(x)] � c � 0, where f(x) issome expression in x.

Quadratic Formula (313) The solutions of a quadraticequation of the form ax2 � bx � c � 0, where a � 0, are given by the Quadratic Formula, which

is x � .

quadratic function (286) A function described bythe equation f(x) � ax2 � bx � c, where a � 0.

quadratic term (286) In the equation f(x) � ax2 �bx � c, ax2 is the quadratic term.

radian (710) The measure of an angle � in standardposition whose rays intercept an arc of length 1unit on the unit circle.

radical equation (263) An equation with radicalsthat have variables in the radicands.

radical inequality (264) An inequality that has avariable in the radicand.

random (645) All outcomes have an equally likelychance of happening.

random variable (646) The outcome of a randomprocess that has a numerical value.

range (56) The set of all y-coordinates of a relation.

rate of change (69) How much a quantity changeson average, relative to the change in anotherquantity, often time.

rate of decay (560) The percent decrease r in theequation y � a(1 � r)t.

rate of growth (562) The percent increase r in theequation y � a(1 � r)t.

rational equation (505) Any equation that containsone or more rational expressions.

rational exponent (258) For any nonzero realnumber b, and any integers m and n, with n � 1,

b�mn�

� �n bm� � ��n b�m, except when b � 0 and n iseven.

rational expression (472) A ratio of two polynomialexpressions.

rational function (472) An equation of the

form f(x) � �pq((xx))

�, where p(x) and q(x) are

polynomial functions, and q(x) � 0.

�b � �b2 � 4�ac��

ecuación cuadrática Función cuadrática igual a unvalor, de la forma ax2 � bx � c, donde a � 0.

forma de ecuación cuadrática Para cualquiernúmero a, b y c, excepto a � 0, una ecuación quepuede escribirse de la forma a[f(x)2] � b[f(x)] � c� 0, donde f(x) es una expresión en x.

Fórmula cuadrática Las soluciones de una ecuacióncuadrática de la forma ax2 � bx � c � 0, donde a � 0, se dan por la fórmula cuadrática, que es

x � .

función cuadrática Función descrita por la ecuaciónf(x) � ax2 � bx � c, donde a � 0.

término cuadrático En la ecuación f(x) � ax2 �bx � c, el término cuadrático es ax2.

radián Medida de un ángulo � en posición normalcuyos rayos intersecan un arco de 1 unidad delongitud en el círculo unitario.

ecuación radical Ecuación con radicales que tienenvariables en el radicando.

desigualdad radical Desigualdad que tiene unavariable en el radicando.

aleatorio Todos los resultados son equiprobables.

variable aleatoria El resultado de un procesoaleatorio que tiene un valor numérico.

rango Conjunto de todas las coordenadas y de unarelación.

tasa de cambio Lo que cambia una cantidad enpromedio, respecto al cambio en otra cantidad,por lo general el tiempo.

tasa de desintegración Disminución porcentual r enla ecuación y � a(1 � r)t.

tasa de crecimiento Aumento porcentual r en laecuación y � a(1 � r)t.

ecuación racional Cualquier ecuación que contieneuna o más expresiones racionales.

exponente racional Para cualquier número real nonulo b y cualquier entero m y n, con n � 1,

b�mn�

� �nbm� � ��n

b�m, excepto cuando b � 0 y n espar.

expresión racional Razón de dos expresionespolinomiales.

función racional Ecuación de la forma

f(x) � �pq((xx))

�, donde p(x) y q(x) son funciones

polinomiales y q(x) � 0.

�b � �b2 � 4�ac��2a


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desigualdad racional Cualquier desigualdad quecontiene una o más expresiones racionales.

racionalizar el denominador La eliminación deradicales de un denominador o de fracciones deun radicando.

número racional Cualquier número �mn

�, donde m y n

son enteros y n no es cero. Su expansión decimales o terminal o periódica.

números reales Todos los números que se usan enla vida cotidiana; el conjunto de los todos losnúmeros racionales e irracionales.

fórmula recursiva Cada término proviene de uno omás términos anteriores.

ángulo de referencia El ángulo agudo formado porel lado terminal de un ángulo en posiciónestándar y el eje x.

reflexión Transformación en que cada punto de unafigura se aplica a través de una recta de simetría asu imagen correspondiente.

matriz de reflexión Matriz que se usa para reflejaruna figura sobre una recta o plano.

recta de regresión Una recta de óptimo ajuste.

relación Conjunto de pares ordenados.

histograma de frecuencia relativa Tabla de probabi-lidades o gráfica para asistir en la visualizaciónde una distribución de probabilidad.

máximo relativo Punto en la gráfica de una funciónen donde ningún otro punto cercano tiene unacoordenada y mayor.

mínimo relativo Punto en la gráfica de una funciónen donde ningún otro punto cercano tiene unacoordenada y menor.

raíz Las soluciones de una ecuación cuadrática.

rotación Transformación en que una figura se hacegirar alrededor de un punto central, generalmenteel origen.

matriz de rotación Matriz que se usa para hacergirar un objeto.

matriz fila Matriz que sólo tiene una fila.

f (x)

xO

máximo relativo

mínimo relativo

rational inequality (508) Any inequality thatcontains one or more rational expressions.

rationalizing the denominator (251) To eliminateradicals from a denominator or fractions from aradicand.

rational number (11) Any number �mn

�, where m and

n are integers and n is not zero. The decimal formis either a terminating or repeating decimal.

real numbers (11) All numbers used in everydaylife; the set of all rational and irrational numbers.

recursive formula (606) Each term is formulatedfrom one or more previous terms.

reference angle (718) The acute angle formed by theterminal side of an angle in standard position andthe x-axis.

reflection (177) A transformation in which everypoint of a figure is mapped to a correspondingimage across a line of symmetry.

reflection matrix (177) A matrix used to reflect anobject over a line or plane.

regression line (87) A line of best fit.

relation (56) A set of ordered pairs.

relative frequency histogram (646) A table ofprobabilities or a graph to help visualize aprobability distribution.

relative maximum (354) A point on the graph of afunction where no other nearby points have agreater y-coordinate.

relative minimum (354) A point on the graph of afunction where no other nearby points have alesser y-coordinate.

root (294) The solutions of a quadratic equation.

rotation (178) A transformation in which an object ismoved around a center point, usually the origin.

rotation matrix (178) A matrix used to rotate anobject.

row matrix (155) A matrix that has only one row.

f (x)

xO

relative maximum

relative minimum

Glo

ssary

/G

losa

rio

espacio muestral Conjunto de todos los resultadosposibles de un experimento probabilístico.

escalar Una constante.

multiplicación por escalares Multiplicación de unamatriz por una constante llamada escalar;producto de un escalar k y una matriz de m n.

gráfica de dispersión Conjuntos de datos grafica-dos como pares ordenados en un plano decoordenadas.

notación científica Escritura de un número en la forma a 10n, donde 1 � a � 10 y n es unentero.

secante Para cualquier ángulo de medida �, un punto P(x, y) en su lado terminal, r � �x2 � y�2�, sec � � �

xr

�.

determinante de segundo orden El determinantede una matriz de 2 2.

sucesión Lista de números en un orden particular.

serie Suma específica de los términos de una sucesión.

notación de construcción de conjuntos Escrituradel conjunto solución de una desigualdad, porejemplo, {x x � 9}.

notación de suma Para cualquier sucesión a1, a2,a3,…, la suma de los k primeros términos puede

escribirse �k

n�1an, lo que se lee “la suma de n � 1 a

k de los an.” Así, �k

n�1an � a1 � a2 � a3 � … � ak,

donde k es un valor entero.

evento simple Un solo evento.

reducir Escribir una expresión sin paréntesis oexponentes negativos.

simulación Uso de un experimento probabilísticopara imitar una situación de la vida real.

seno Para cualquier ángulo, de medida �, un puntoP(x, y) en su lado terminal, r � �x2 + y�2�, sin

� � �yr

�.

distribución asimétrica Curva o histograma que noes simétrico.

Negativamente AlabeadaPositivamente Alabeada


Ssample space (632) The set of all possible outcomes

of an event.

scalar (162) A constant.

scalar multiplication (162) Multiplying any matrixby a constant called a scalar; the product of ascalar k and an m n matrix.

scatter plot (81) A set of data graphed as orderedpairs in a coordinate plane.

scientific notation (225) The expression of anumber in the form a 10n, where 1 � a � 10and n is an integer.

secant (701) For any angle, with measure �, a point P(x, y) on its terminal side, r � �x2 � y�2�, sec � � �

xr

�.

second-order determinant (182) The determinant ofa 2 2 matrix.

sequence (578) A list of numbers in a particular order.

series (583) The sum of the terms of a sequence.

set-builder notation (34) The expression of thesolution set of an inequality, for example {x x � 9}.

sigma notation (585) For any sequence a1, a2, a3,…,

the sum of the first k terms may be written �k

n�1an,

which is read “the summation from n � 1 to k of

an.” Thus, �k

n�1an � a1 � a2 � a3 � … � ak, where k

is an integer value.

simple event (658) One event.

simplify (222) To rewrite an expression withoutparentheses or negative exponents.

simulation (681) The use of a probabilityexperiment to mimic a real-life situation.

sine (701) For any angle, with measure �, a pointP(x, y) on its terminal side, r � �x2 + y�2�, sin

� � �yr

�.

skewed distribution (671) A curve or histogramthat is not symmetric.

Negatively SkewedPositively Skewed

Glo

ssary

/G

losa

rio

slope (68) The ratio of the change in y-coordinatesto the change in x-coordinates.

slope-intercept form (75) The equation of a line inthe form y � mx � b, where m is the slope and bis the y-intercept.

solution (20) A replacement for the variable in anopen sentence that results in a true sentence.

solving a right triangle (704) The process of findingthe measures of all of the sides and angles of aright triangle.

square matrix (155) A matrix with the same numberof rows and columns.

square root (245) For any real numbers a and b, if a2 � b, then a is a square root of b.

square root function (395) A function that containsa square root of a variable.

Square Root Property (306) For any real number n,if x2 � n, then x � � �n�.

standard deviation (665) The square root of thevariance, represented by �.

standard form (64) A linear equation written in theform Ax � By � C, where A, B, and C are realnumbers and A and B are not both zero.

standard position (709) An angle positioned so thatits vertex is at the origin and its initial side isalong the positive x-axis.

step function (89) A function whose graph is a seriesof line segments.

substitution method (116) A method of solving asystem of equations in which one equation issolved for one variable in terms of the other.

success (644) The desired outcome of an event.

synthetic division (234) A method used to divide apolynomial by a binomial.

synthetic substitution (365) The use of syntheticdivision to evaluate a function.

system of equations (110) A set of equations withthe same variables.

system of inequalities (123) A set of inequalitieswith the same variables.

tangent (427, 701) 1. A line that intersects a circle atexactly one point. 2. For any angle, withmeasure �, a point P(x, y) on its terminal side,

r � �x2 � y�2�, tan � � �yx

�.

pendiente La razón del cambio en coordenadas y alcambio en coordenadas x.

forma pendiente-intersección Ecuación de unarecta de la forma y � mx � b, donde m es lapendiente y b la intersección.

solución Sustitución de la variable de un enunciadoabierto que resulta en un enunciado verdadero.

resolver un triángulo rectángulo Proceso de hallarlas medidas de todos los lados y ángulos de untriángulo rectángulo.

matriz cuadrada Matriz con el mismo número defilas y columnas.

raíz cuadrada Para cualquier número real a y b, si a2 � b, entonces a es una raíz cuadrada de b.

función radical Función que contiene la raízcuadrada de una variable.

Propiedad de la raíz cuadrada Para cualquiernúmero real n, si x2 � n, entonces x � � �n�.

desviación estándar La raíz cuadrada de lavarianza, la que se escribe �.

forma estándar Ecuación lineal escrita de la formaAx � By � C, donde A, B, y C son números realesy A y B no son cero simultáneamente.

posición estándar Ángulo en posición tal que suvértice está en el origen y su lado inicial está a lolargo del eje x positivo.

función etapa Función cuya gráfica es una serie desegmentos de recta.

método de sustitución Método para resolver unsistema de ecuaciones en que una de lasecuaciones se resuelve en una de las variables entérminos de la otra.

éxito El resultado deseado de un evento.

división sintética Método que se usa para dividirun polinomio entre un binomio.

sustitución sintética Uso de la división sintéticapara evaluar una función polinomial.

sistema de ecuaciones Conjunto de ecuaciones conlas mismas variables.

sistema de desigualdades Conjunto dedesigualdades con las mismas variables.

tangente 1. Recta que interseca un círculo en un solopunto. 2. Para cualquier ángulo, de medida �,un punto P(x, y) en su lado terminal,

r � �x2 � y�2�, tan � � �yx

�.


T

Glo

ssary

/G

losa

rio

term (229, 578) 1. The monomials that make up apolynomial. 2. Each number in a sequence orseries.

terminal side of an angle (709) A ray of an anglethat rotates about the center.

third-order determinant (183) Determinants of a 3 3 matrix.

transformation (175) Functions that map points of apre-image onto its image.

translation (175) A figure is moved from onelocation to another on the coordinate planewithout changing its size, shape, or orientation.

translation matrix (175) A matrix that represents atranslated figure.

transverse axis (442) The segment of length 2awhose endpoints are the vertices of a hyperbola.

trigonometric equation (799) An equationcontaining at least one trigonometric functionthat is true for some but not all values of thevariable.

trigonometric functions (701, 717) For any angle,with measure �, a point P(x, y) on its terminal side, r � �x2 � y�2�, the trigonometric functions of� are as follows.

sin � � �yr

� cos � � �xr

� tan � � �yx

�

csc � � �yr

� sec � � �xr

� cot � � �xy

�

trigonometric identity (777) An equation involvinga trigonometric function that is true for all valuesof the variable.

trigonometry (701) The study of the relationshipsbetween the angles and sides of a right triangle.

trinomial (229) A polynomial with three unlike terms.

unbiased sample (682) A sample in which everypossible sample has an equal chance of beingselected.

unbounded (130) A system of inequalities thatforms a region that is open.

union (41) The graph of a compound inequalitycontaining or.

y

xO

terminalside

initial side

vertex

180˚

270˚

90˚

término 1. Los monomios que constituyen unpolinomio. 2. Cada número de una sucesión oserie.

lado terminal de un ángulo Rayo de un ángulo quegira alrededor de un centro.

determinante de tercer orden Determinante de unamatriz de 3 3.

transformación Funciones que aplican puntos deuna preimagen en su imagen.

traslación Se mueve una figura de un lugar a otroen un plano de coordenadas sin cambiar sutamaño, forma u orientación.

matriz de traslación Matriz que representa unafigura trasladada.

eje transversal El segmento de longitud 2a cuyosextremos son los vértices de una hipérbola.

ecuación trigonométrica Ecuación que contiene porlo menos una función trigonométrica y que sólose cumple para algunos valores de la variable.

funciones trigonométricas Para cualquier ángulo,de medida �, un punto P(x, y) en su lado terminal, r � �x2 � y�2�, las funcionestrigonométricas de � son las siguientes.

sen � � �yr

� cos � � �xr

� tan � � �yx

�

csc � � �yr

� sec � � �xr

� cot � � �xy

�

identidad trigonométrica Ecuación que involucrauna o más funciones trigonométricas y que secumple para todos los valores de la variable.

trigonometría Estudio de las relaciones entre loslados y ángulos de un triángulo rectángulo.

trinomio Polinomio con tres términos diferentes.

muestra no sesgada Muestra en que cualquiermuestra posible tiene la misma posibilidad deseleccionarse.

no acotado Sistema de desigualdades que formauna región abierta.

unión Gráfica de una desigualdad compuesta quecontiene la palabra o.

y

xO

ladoterminal

lado inicial

vértice

180˚

270˚

90˚


U

Glo

ssary

/G

losa

rio

unit circle (710) A circle of radius 1 unit whosecenter is at the origin of a coordinate system.

variables (7) Symbols, usually letters, used torepresent unknown quantities.

variance (665) The mean of the squares of thedeviations from the arithmetic mean.

vertex (287, 442) 1. The point at which the axis ofsymmetry intersects a parabola. 2. The point oneach branch nearest the center of a hyperbola.

vertex form (322) A quadratic function in the formy � a(x � h)2 � k, where (h, k) is the vertex of theparabola and x � h is its axis of symmetry.

vertex matrix (175) A matrix used to represent thecoordinates of the vertices of a polygon.

vertical asymptote (485) If the related rationalexpression of a function is written in simplestform and is undefined for x � a, then x � a is avertical asymptote.

vertical line test (57) If no vertical line intersects agraph in more than one point, then the graphrepresents a function.

vertices (129) The maximum or minimum valuethat a linear function has for the points in afeasible region.

x-intercept (65) The x-coordinate of the point atwhich a graph crosses the x-axis.

y-intercept (65) The y-coordinate of the point atwhich a graph crosses the y-axis.

zeros (294) The x-intercepts of the graph of aquadratic equation; the points for which f(x) � 0.

zero matrix (155) A matrix in which every elementis zero.

1 unit

� measures 1 radian.

x

y

O

(�1, 0)

(0, �1)

(1, 0)

(0, 1)

1

�

círculo unitario Círculo de radio 1 cuyo centro es elorigen de un sistema de coordenadas.

variables Símbolos, por lo general letras, que seusan para representar cantidades desconocidas.

varianza Media de los cuadrados de lasdesviaciones de la media aritmética.

vértice 1. Punto en el que el eje de simetríainterseca una parábola. 2. El punto en cadarama más cercano al centro de una hipérbola.

forma de vértice Función cuadrática de la formay � a(x � h)2 � k, donde (h, k) es el vértice de laparábola y x � h es su eje de simetría.

matriz de vértice Matriz que se usa para escribir lascoordenadas de los vértices de un polígono.

asíntota vertical Si la expresión racional quecorresponde a una función racional se reduce yestá no definida en x � a, entonces x � a es unaasíntota vertical.

prueba de la recta vertical Si ninguna recta verticalinterseca una gráfica en más de un punto,entonces la gráfica representa una función.

vértices El valor máximo o mínimo que unafunción lineal tiene para los puntos en unaregión viable.

intersección x La coordenada x del punto o puntosen que una gráfica interseca o cruza el eje x.

intersección y La coordenada y del punto o puntosen que una gráfica interseca o cruza el eje y.

ceros Las intersecciones x de la gráfica de una ecua-ción cuadrática; los puntos x para los que f(x) � 0.

matriz nula Matriz cuyos elementos son todos iguala cero.

1 unidad

� mide 1 radián.

x

y

O

(�1, 0)

(0, �1)

(1, 0)

(0, 1)

1

�


V

X

Y

Z

Selected AnswersSele

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Chapter 1 Solving Equations andInequalities

Page 5 Chapter 1 Getting Started

1. 19.84 3. �17.51 5. ��152� 7. �2�

16

� 9. 0.48 11. 1.1

13. �2�23

� 15. 8�45

� 17. 8 19. 49 21. 0.64 23. �49

� 25. false

27. true 29. false 31. true

Pages 8–10 Lesson 1-11. First, find the sum of c and d. Divide this sum by e.Multiply the quotient by b. Finally, add a. 3. b; The sum ofthe cost of adult and children tickets should be subtractedfrom 50. Therefore parentheses need to be inserted aroundthis sum to insure that this addition is done beforesubtraction. 5. 6 7. 1 9. 119 11. �23 13. $432

15. $1162.50 17. 3 19. 25 21. �34 23. 5 25. �31

27. 14 29. �3 31. 162 33. 2.56 35. 25�13

� 37. 31.25

drops per min 39. 2 41. �4.2 43. �4 45. 1.4 47. �8

49. 2�16

� 51. �16 53. $8266.03 55. Sample answer:

4 � 4 � 4 4 � 1; 4 4 � 4 4 � 2; (4 � 4 � 4) 4 � 3;4 (4 � 4) � 4 � 4; (4 4 � 4) 4 � 5; (4 � 4) 4 � 4 � 6;44 4 � 4 � 7; (4 � 4) (4 4) � 8; 4 � 4 � 4 4 � 9;

(44 � 4) 4 � 10 57. C 59. 3 61. 10 63. �2 65. �23

�

Pages 14–17 Lesson 1-21a. Sample answer: 2 1b. Sample answer: 5 1c. Sampleanswer: �11 1d. Sample answer: 1.3 1e. Sampleanswer: �2� 1f. Sample answer: �1.3 3. 0; Zero does

not have a multiplicative inverse since �10

� is undefined.

5. N, W, Z, Q, R 7. Multiplicative Inverse 9. Additive

Identity 11. ��13

�, 3 13. �2x � 4y 15. 3c � 18d

17. 1.5(10 � 15 � 12 � 8 � 19 � 22 � 31) or 1.5(10) �1.5(15) � 1.5(12) � 1.5(8) � 1.5(19) � 1.5(22) � 1.5(31)19. W, Z, Q, R 21. N, W, Z, Q, R 23. I, R25. N, W, Z, Q, R 27. Q, R; 2.4, 2.49, 2.4�9�, 2.49�, 2.9�29. Associative ( ) 31. Associative (�) 33. MultiplicativeInverse 35. Multiplicative Identity 37. �m; AdditiveInverse 39. 1 41. �2� units 43. 10; ��

110� 45. 0.125; �8

47. ��43

�, �34

� 49. 3a � 2b 51. 40x � 7y 53. �12r � 4t

55. �3.4m � 1.8n 57. �8 � 9y 59. true 61. false; 663. 6.5(4.5 � 4.25 � 5.25 � 6.5 � 5) or 6.5(4.5) � 6.5(4.25) �(6.5)5.25 � 6.5(6.5) � 6.5(5)

65. 3�2�14

� � 2�1�18

�� 3�2 � �

14

� � 2�1 � �18

� Definition of a mixed number

� 3(2) � 3��14

� � 2(1) � 2��18

� Distributive Property

� 6 � �34

� � 2 � �14

� Multiply.

� 6 � 2 � �34

� � �14

� Commutative Property (�)

� 8 � �34

� � �14

� Add.

� 8 � ��34

� � �14

� Associative Property (�)

� 8 � 1 or 9 Add.67. 4700 ft2 69. $62.15

71. Answers should include the following.• Instead of doubling each coupon value and then adding

these values together, the Distributive Property could beapplied allowing you to add the coupon values first andthen double the sum.

• If a store had a 25% off sale on all merchandise, theDistributive Property could be used to calculate thesesavings. For example, the savings on a $15 shirt, $40 pairof jeans, and $25 pair of slacks could be calculated as0.25(15) � 0.25(40) � 0.25(25) or as 0.25(15 � 40 � 25)using the Distributive Property.

73. C 75. False; 0 � 1 � �1, which is not a whole number.

77. False; 2 3 � �23

�, which is not a whole number. 79. 6

81. �2.75 83. �11 85. �4.3

Page 17 Practice Quiz 11. 14 3. 6 5. 2 amperes 7. N, W, Z, Q, R 9. ��

67

�, �76

�

Pages 24–27 Lesson 1-31. Sample answer: 2x � �143. Jamal; his method can be confirmed by solving theequation using an alternative method.

C � �59

�(F � 32)

C � �59

�F � �59

�(32)

C � �59

�(32) � �59

�F

�95

��C � �59

�(32) � F

�95

�C � 32 � F

5. 2n � n3 7. Sample answer: 5 plus 3 times the square ofa number is twice that number. 9. Addition (�) 11. 14

13. �4.8 15. 16 17. p � �rIt� 19. 5 � 3n 21. n2 � 4

23. 5(9 � n) 25. ��n4

�2 27. 2rh � 2r2 29. Sample answer:

5 less than a number is 12. 31. Sample answer: A numbersquared is equal to 4 times the number. 33. Sampleanswer: A number divided by 4 is equal to twice the sum ofthat number and 1. 35. Substitution (�) 37. Transitive (�)

39. Symmetric (�) 41. 7 43. 3.2 45. �112� 47. �8 49. �7

51. 1 53. �14

� 55. ��525� 57. �

dt� � r 59. �

3V

r2� � h

61. b � �x(c

a� 3)� � 2 63. n � number of games;

2(1.50) � n(2.50) � 16.75; 5 65. x � cost of gasoline permile; 972 � 114 � 105 � 7600x � 1837; 8.5¢/mi67. a � Chun-Wei's age; a � (2a � 8) � (2a � 8 � 3) � 94;Chun-Wei: 15 yrs old, mother: 38 yrs old, father: 41 yrs old69. n � number of lamps broken; 12(125) � 45n � 1365; 3 lamps 71. 15.1 mi/month 73. The Central Pacific hadto lay their track through the Rocky Mountains, while theUnion Pacific mainly built track over flat prairie. 75. theproduct of 3 and the difference of a number and 5 added tothe product of four times the number and the sum of thenumber and 1 77. B 79. �6x � 8y � 4z 81. 6.6

83. 105 cm2 85. 3 87. ��14

� 89. �5 � 6y

R20 Selected Answers

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Pages 30–32 Lesson 1-4

1. a � �a when a is a negative number and the negativeof a negative number is positive. 3. Always; since theopposite of 0 is still 0, this equation has only one case,

ax � b � 0. The solution is ��ab

�. 5. 8 7. �17 9. {�18, �12}

11. {�32, 36} 13. {8} 15. least: 158°F; greatest: 162°F17. 15 19. 0 21. 3 23. �4 25. �9.4 27. 55 29. {8, 42}

31. {�45, 21} 33. {�2, 16} 35. ��32

�� 37. �2, �92

�� 39. �

41. {�5, 11} 43. ��131�, �3� 45. {8} 47. x � 200 � 5;

maximum: 205°F; minimum: 195°F 49. x � 13 � 5;maximum: 18 km, minimum: 8 km 51. sometimes; true only if c � 0 53. B 55. x � 1 � 2 � x � 4;

x � 1 � 2 � �(x � 4) 57. {�1.5} 59. 2(n � 11) 61. �136�

63. 14 65. Distributive 67. Additive Identity 69. true

71. false; 1.2 73. 364 ft2 75. 8 77. �23

� 79. ��34

�

Pages 37–39 Lesson 1-51. Dividing by a number is the same as multiplying by itsinverse. 3. Sample answer: x � 2 � x � 1

5. �x x � �53

�� or ��, �53

�

7. {y y � 6} or (6, ��)

9. {p p � 15} or (15, ��)

11. all real numbers or (��, ��)

13. 2n � 3 � 5; n � 415. {n n � �11} or [�11, ��)

17. {x x � 7} or (��, 7)

19. {g g � 27} or (��, 27]

21. {k k � �3.5} or [�3.5, ��)

23. {m m � �4} or (�4, ��)

25. {t t � 0} or (��, 0]

27. {n n � 1.75} or [1.75, ��)

29. {x x � �279} or (��, �279)

31. {d d � �5} or [�5, ��)

33. {g g � 2} or (��, 2)

35. �y y � �15

�� or ��, �15

�

37. �

39. at least 25 h 41. n � 8 � 2; n � �6 43. �12

�n � 7 � 5;

n � 24 45. 2(n � 5) � 3n � 11; n � �1 47. 2(7m) � 17;

m � �1174�; at least 2 child care staff members

49. n � 34.97; She must sell at least 35 cars. 51. s � 91;Ahmik must score at least 91 on her next test to have an Atest average. 53. Answers should include the following.• 150 � 400• Let n equal the number of minutes used. Write an

expression representing the cost of Plan 1 and for Plan 2for n minutes. The cost for Plan 1 would include a monthlyaccess fee of $35 plus 40¢ for each minute over 150 minutesor 35 � 0.4(n � 150). The cost for Plan 2 for 400 minutesor less would be $55. To find where Plan 2 would costless than Plan 1 solve 55 � 35 � 0.4(n � 150) for n. Thesolution set is {n n � 200}, which means that for morethan 200 minutes of calls, Plan 2 is cheaper.

55. D 57. x � �2 59. {�14, 20} 61. � 63. N, W, Z, Q, R

65. I, R 67. {�7, 7} 69. �4, ��45

�� 71. {�11, �1}

Page 39 Practice Quiz 2

1. 0.5 3. 14 5. �m m � �49

�� or ��49

�, ��

Pages 43–46 Lesson 1-61. 5 � c � 15 3. Sabrina; an absolute value inequality ofthe form a � b should be rewritten as an or compoundinequality, a � b or a �� b. 5. n � 3

42�6 �4 �2 0

10 89

23

29� 4

929

2 4�6 �4 �2 0

�1 135

15

35� 1

5�

�6 �4 �2 0 2 4

�6 �4 �2�8 0 2

�286 �284 �282 �280 �278 �276

0 0.5 1 1.5 2 2.5

�4 �2 0 2 4 6

2 4�6 �4 �2 0

�6 �5 �4 �3 �2�7

24 2620 22 28 30

3 4 6 72 5 8 9 10�1 10

�14 �12 �10 �6�8 �4

�6 �4 �2 20 4

16 191514 181710 1398 1211

�1 0 1 3 4 6 72 5 8 9 10

0 1 2 3

Selected Answers R21

Sele

cted A

nsw

ers

7. n � 29. {d �2 � d � 3}

11. {g �13 � g � 5}

13. all real numbers

15. n � 5

17. n � 4

19. n � 8

21. n � 1 23. n � 1.5 25. n � 1 � 127. {p p � 2 or p � 8}

29. {x �2 � x � 4}

31. {f �7 � f � �5}

33. {g �9 � g � 9}

35. �

37. {b b � 10 or b � �2}

39. �w ��73

� � w � 1�

41. all real numbers

43. �n n � �72

��45. 6.8 � x � 7.4 47. 45 � s � 5549. 108 in. � L � D � 130 in.51. a � b � c, a � c � b, b � c � a

53a.

53b.

53c.

53d. 3 � x � 2 � 8 can be rewritten as x � 2 � 3 and x � 2 � 8. The solution of x � 2 � 3 is x � 1 or x � �5. The solution of x � 2 � 8 is �10 � x � 6.Therefore, the union of these two sets is (x � 1 or x � �5)and (�10 � x � 6). The union of the graph of x � 1 or x � �5 and the graph of �10 � x � 6 is shown below.From this we can see that solution can be rewritten as (�10 � x � �5) or (1 � x � 6).

55. x � �5 or x � �6

57.

59. (5x � 2 � 3) or (5x � 2 � �3); {x x � 0.2 or x � �1}61. {d d � �6} or [�6, ��)

63. {n n � �1} or (��, �1)

65. {�10, 16} 67. � 69. Symmetric (�) 71. 3a � 7b 73.2 75. �7

Pages 47–50 Chapter 1 Study Guide and Review1. compound inequality 3. Commutative ( )5. Reflexive (�) 7. Multiplicative Inverse 9. absolute value11. 22 13. �49 15. �23 17. 37.5 19. Q, R 21. I, R

23. �5a � 24b 25. �14 27. �13 29. �4 31. x � �C �

ABy

�

33. p � �1 �

Art

� 35. {6, �18} 37. {6} 39. ��32

�, �1�41. {x x � 5} or [5, ��)

43. {a a � 2} or (2, � �)

45. {x x � �1.8} or (�1.8, ��)

47. �y �53

� � y � 5�

1 2 3 4 5 6

�2.2 �2.0 �1.8 �1.6 �1.4 �1.2

�2 20 6 84

�1 321 654 9 10870

4 62�4 �2 0

2�4�6�8 �2 0

�4�12 �8 0 4 8

4 62�4 �2 0

4 62�4 �2 0

4 62�4 �2 0

32 540 1

4 62�4 �2 0

10�2 �1

8 12 164�4 0

4 62�4 �2 0

8 124�8 �4 0

�10 �8 �6 �4 �2 0

4 62�4 �2 0

8 124�8 �4 0

8 124�8 �4 0

4 62�4 �2 0

8 124�8 �4 0

4 62�4 �2 0

4�8�12�16 �4 0

4 62�4 �2 0


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49. {y �9 � y � 18}

51. �b b � �4 or b � ��130��

Chapter 2 Linear Relations andFunctions

Page 55 Chapter 2 Getting Started1. (�3, 3) 3. (�3, �1) 5. (0, �4) 7. �2 9. 9 11. 2

13. x � 1 15. 2x � 6 17. �12

�x � 2 19. 3 21. 15 23. 2.5

Pages 60–62 Lesson 2-11. Sample answer: {(�4, 3), (�2, 3), (1, 5), (�2, 1)}3. Molly; to find g(2a), replace x with 2a. Teisha found 2g(a),not g(2a). 5. yes7. D � {7}, R � {�1, 2, 5, 8}, 9. D � all reals, R � all no reals, yes

11. 10 13. D � {70, 72, 88}, R � {95, 97, 105, 114}

15.

17. yes 19. no 21. yes 23. D � {�3, 1, 2}, R � 25. D � {�2, 3}, R � {5, 7, 8}; {0, 1, 5}; yes no

27. D � {�3.6, 0, 1.4, 2}, 29. D � all reals, R � all R � {�3, �1.1, 2, 8}; yes reals; yes

31. D � all reals, R � all 33. D � all reals, R � {y y � 0}; reals; yes yes

35.

37. No; the domain value 56 is paired with two differentrange values.

39.

41. Yes; each domain value is paired with only one rangevalue.

Pri

ce (

$)

70

60

50

40

30

20

10

0

Year19981996 2000 2002 2004

Stock Price

RB

I

170

165

160

155

150

145

140

HR480 50 52 54 56

American League Leaders

y

O x

y � x 2

y

O x

y � 3x � 4

y

O x

y � �5x

y

O x

(1.4, 2)

(2, �3)(0, �1.1)

(�3.6, 8)

y

O x

(3, 7)

(�2, 5)

(�2, 8)y

O x

(2, 1)

(1, 5)

(�3, 0)

July

95

100

105

110

115

January700 80 90

Record High Temperatures

y

O x

y � �2x � 1

(7, 2)

(7, �1)

(7, 5)

(7, 8)y

O x

�3 �2 �1�4

�12 �6 181260


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43.

45. Yes; no; each domain value is paired with only onerange value so the relation is a function, but the rangevalue 12 is paired with two domain values so the functionis not one-to-one. 47. 6 49. �3 51. 25n2 � 5n 53. 1155. f(x) � 4x � 3 57. B 59. discrete 61. discrete63. {y �8 � y � 6} 65. {x x � 5.1} 67. $29.8269. 31a � 10b 71. 2 73. 15

Pages 65–67 Lesson 2-21. The function can be written as f(x) � �

12

�x � 1, so it is of

the form f(x) � mx � b, where m � �12

� and b � 1. 3. Sample

answer: x � y � 2 5. yes 7. 2x � 5y � 3; 2, �5, 3

9. ��53

�, �5 11. 2, 3

13. $177.62 15. yes 17. No; y is inside a square root.19. No; x appears in a denominator. 21. No; x has anexponent other than 1. 23. x2 � 5y � 0 25. 7200 m27. 3x � y � 4; 3, 1, 4 29. x � 4y � �5; 1, �4, �531. 2x � y � 5; 2, �1, 5 33. x � y � 12; 1, 1, 12 35. x � 6;1, 0, 6 37. 25x � 2y � 9; 25, 2, 9

39. 3, 5 41. �130�, ��

52

�

43. 0, 0 45. none, �2

47. 8, none 49. �14

�, �1

51. The lines are parallelbut have different y-intercepts.53. 90°C

55. 57.

59. no 61. A linear equation can be used to relate theamounts of time that a student spends on each of twosubjects if the total amount of time is fixed. Answers shouldinclude the following.• x and y must be nonnegative because Lolita cannot

spend a negative amount of time studying a subject.• The intercepts represent Lolita spending all of her time

on one subject. The x-intercept represents her spendingall of her time on math, and the y-intercept representsher spending all of her time on chemistry.

c

100 200 400

35030025020015010050

b0

1.75b � 1.5c � 525

T(d )

O1 2�2�3�4 3 4

1601208040

�40�80

�120�160

d

T(d ) � 35d � 20

y

Ox

x � y � 5

x � y � �5

x � y � 0

f (x)

O x

f (x) � 4x � 1

x � 8

2 4�4�6�8 6

8642

�2�4�6�8

�2x

y

O

y

O x

y � �2

y

Ox

y � x

y

O x

3x � 4y � 10 � 0

y

Ox

5x � 3y � 15

y

O x

3x � 2y � 6

y

O x

y � �3x � 5

Rep

rese

nta

tives

14

12

10

8

6

4

2

0

Year’87 ’91 ’95 ’99

30+ Years of Service


Sele

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ers

63. B 65. D � {0, 1, 2}, R � {�1, 0, 2, 3}; no67. {x x � �6 or x � �2}

69. 3s � 14 71. �13

� 73. 275. �5 77. 0.4

Pages 71–74 Lesson 2-31. Sample answer: y � 1 3. Luisa; Mark did not subtract ina consistent manner when using the slope formula. If y2 � 5and y1 � 4, then x2 must be �1 and x1 must be 2, not

vice versa. 5. ��12

�

7. 9.

11. 13. 1.25°/hr 15. ��52

� 17. �35

�

19. 0 21. 8 23. �425. undefined 27. 129. about 0.6

31. 33.

35. 37. about 68 million per year39. The number of cassettetapes shipped has beendecreasing. 41. 45 mph

43. 45.

47. 49.

51. Yes; slopes show that adjacent sides are perpendicular.53. The grade or steepness of a road can be interpretedmathematically as a slope. Answers should include thefollowing.• Think of the diagram at the beginning of the lesson as

being in a coordinate plane. Then the rise is a change iny-coordinates and the horizontal distance is a change inx-coordinates. Thus, the grade is a slope expressed as apercent.

•

55. D 57. The graphs have the same y-intercept. As theslopes become more negative, the lines get steeper.

59. �2, �83

�

61. �7 63. ��52

� 65. {x �1 � x � 3} 67. at least 8 69. 9

71. y � �4x � 2 73. y � �52

�x � �12

� 75. y � ��23

�x � �131�

Page 74 Practice Quiz 11. D � {�7, �3, 0, 2}, R � {�2, 1, 2, 4, 5} 3. 6x � y � 4

y

O x

4x � 3y � 8 � 0

y

O

x

y � 0.08x

y

Ox

y

O x

y

O x

y

O x

y

O x

y

O x

y

O x

y

O x

y

O x

y

O x

y

O x(1, 0)

(1, 3)(0, 2)

(2, �1)


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5.

Pages 78–80 Lesson 2-41. Sample answer: y � 3x � 2 3. Solve the equation for y

to get y � �35

�x � �25

�. The slope of this line is �35

�. The slope of a

parallel line is the same. 5. ��32

�, 5 7. y � ��34

�x � 2

9. y � ��35

�x � �156� 11. y � �

54

�x � 7 13. ��23

�, �4 15. �12

�, ��52

�

17. undefined, none 19. y � 0.8x 21. y � �4

23. y � 3x � 6 25. y � ��12

�x � �72

� 27. y � �0.5x � 2

29. y � ��45

�x � �157� 31. y � 0 33. y � x � 4

35. y � �23

�x � �130� 37. y � ��

115�x � �

253� 39. y � 3x � 2

41. d � 180c � 360 43. 540° 45. 10 mi 47. 68°F

49. y � 0.35x � 1.25 51. y � 2x � 4 53. C 55. � �y5

� � 1

57. �2 59. 0 61. � 63. {r r � 6} 65. 6.5 67. 5.85

Pages 83–86 Lesson 2-51. d 3. Sample answer using (4, 130.0) and (6, 140.0): y � 5x � 110

5a.

5b. Sample answer using (1992, 57) and (1998, 67): y � 1.67x � 3269.64 5c. Sample answer: about 87 million

7a.

7b. Sample answer using (4, 5) and (32, 37): y � 1.14x � 0.447c. Sample answer: about 13

9a.

9b. Sample answer using (1, 499) and (3, 588): y � 44.5x � 454.5, where x is the number of seasons since1995–1996 9c. Sample answer: about $1078 million or $1.1billion 11. Sample answer: $1091 13. Sample answer:Using the data for August and November, a predictionequation for Company 1 is y � �0.86x � 25.13, where x isthe number of months since August. The negative slopesuggests that the value of Company 1’s stock is goingdown. Using the data for October and November, aprediction equation for Company 2 is y � 0.38x � 31.3,where x is the number of months since August. Thepositive slope suggests that the value of Company 2’s stockis going up. Since the value of Company 1’s stock appearsto be going down, and the value of Company 2’s stockappears to be going up, Della should buy Company 2.15.

17. Sample answer: about 23 in. 19. Sample answer: Using(1975, 62.5) and (1995, 81.7): 96.1% 23. D 25. 1988, 1993,1998; 247, 360.5, 461 27. 354 29. y � 21.4x � 42,294.03

31. y � 4x � 6 33. 3 35. �239� 37. {x x � �7 or x � �1}

39. 11 41. �23

�

Pages 92–95 Lesson 2-61. Sample answer: [[1.9]] � 1 3. Sample answer: f(x) � x � 1 5. S7. D � all reals, R � all 9. D � all reals, R � all integers nonnegative reals

xO

f (x) � |3x � 2|

f(x)

xO

g(x) � 2x�

g(x)

World Cities

5

0

10

15

20

Prec

ipit

atio

n (

in.)

25

30

3540

Elevation (ft)200 400 600

BroadwayPlay Revenue

100

0

200

300

400

Rev

enu

e ($

mill

ion

s)

500600700

Seasons Since ’95–’961 2 3 4

2000–2001Detroit Red Wings

10

0

20

30

40

Ass

ists

5060

Goals10 20 30 40

Cable Television

100

20

30

40

Ho

use

ho

lds

(mill

ion

s)

50

60

7080

Year’88 ’90 ’92 ’94 ’96 ’98 ’00

x�

y

O x


Sele

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11. D � all reals, R � all 13.reals

15. C 17. S 19. A 21. 23. $1.00

25. D � all reals, R � all 27. D � all reals, R � {3a a is integers an integer.}

29. D � all reals, R � all 31. D � all reals, R � all integers nonnegative reals

33. D � all reals, R � 35. D � all reals, R � all {y y � �4} nonnegative reals

37. D � all reals, R � all 39. D � {x x � �2 or x � 2}, nonnegative reals R � {�1, 1}

41. D � all reals, 43. D � all reals, R � all R � {y y � 2} nonnegative whole numbers

45. f(x) � x � 2

47. 49.

51. B 53.

55. Sample answer: 78.7 yr 57. y � x � 2

59. �y y � �56

��

61. no 63. yes 65. yes

3210�1�3 �2

Life Expectancy

66

0

68

70

72

Exp

ecta

ncy

(yr

)

74

76

78

Years Since 195010 20 30 40 50

xO

y|x| � |y| � 3

xO

f(x)

xO

g(x) � |x�|

g(x)

xO

g(x)

xO

h(x)

xOf(x) � |x � |1

2

f(x)

xO

f (x) � |x � 2|

f(x)

xO

g(x) � |x | � 4

g(x)

xO

h (x) � |�x |

h(x)

xO

f (x) � x� � 1

f(x)

x

h (x) � �3x�6

1�3�6�9

�12

�2�3�4 2 3 4

912

O

h(x)

�1xO

g(x) � x � 2�

g(x)

1

60 180 300

2345

xO

y

Co

st (

$)

Time (hr)0

xO

h(x)


Sele

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nsw

ers


1. y � ��23

�x � �131� 3. Sample answer using (66, 138) and

(74, 178): y � 5x � 192 5. D � all reals, R � nonnegativereals

Pages 98–99 Lesson 2-71. y � �3x � 4 3. Sample answer: y � x5. 7.

9. 11.

13. 15.

17. 19.

21. 23.

25. 27.

29. 31. x � �2

33.

35. 4a � 3s � 2000 37. yes 39. yes 41. Linearinequalities can be used to track the performance of playersin fantasy football leagues. Answers should include thefollowing.• Let x be the number of receiving yards and let y be the

number of touchdowns. The number of points Dana getsfrom receiving yards is 5x and the number of points hegets from touchdowns is 100y. His total number of pointsis 5x � 100y. He wants at least 1000 points, so theinequality 5x � 100y � 1000 represents the situation.

50 150 250 350 xO

y

50

150

250

350

0.4x � 0.6y � 90

x

y

O

x � �2

xO

y

x � y � 1

x � y � �1

xO

y

y � |x| � 3

xO

y

y � |x|

xO

y

y � x � 513

xO

y

4x � 5y � 10 � 0

xO

y

y � 1

xO

y

y � �4x � 3

xO

y

y � 6x � 2

xO

y

x � y � �5

cO

d

10c � 13d � 40

x

O

y

y � 3|x| � 1

xO

y

x � 2y � 5xO

y

y � 2x � 3

xO

f (x) � |x � 1|

f(x)


Sele

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nsw

ers

• • the first one

43. B 45.

47.

49. D � all reals, R � {y y � �1}

51.

53. Sample answer: $10,000 55. 3

Pages 100–104 Chapter 2 Study Guide and Review1. identity 3. standard 5. domain 7. slope

9. D � {�2, 2, 6}, R � {1, 3}; 11. D � all reals, R � all reals; yes yes

13. 21 15. 5y � 9 17. No; x has an exponent other than 1.19. No; x is inside a square root. 21. 5x � 2y � �4; 5, 2, �4

23. �4, �20 25. 9, �9

27. ��131�

29. 31.

33. 35.

37. y � ��53

�x � 3 39. y � ��34

�x � �147� 41. Sample answer

using (1980, 29.3) and (1990, 33.6): y � 0.43x � 822.1

43. D � all reals, R � all 45. D � all reals, R � {y y � 4}integers

xO

g(x) � |x | � 4

g(x)

xO

f (x)

f(x) � x� � 2

xO

y

xO

y

y

xO

y

xO

yxO

2 4 6

2

8 10 12�2

�4�6�8

�10�12�14

14

y � x � �9

yxO

4

4

8 12�4�8

�8�12

�12�16�20�24�28

�16 16

� y � x � 415

�4

y

xO

y � 0.5x

y

xO

(2 , 1)(�2, 3)

(6, 3)

Sales vs. Experience

2000

0

4000

6000

8000

Sale

s ($

)

10,000

Years1 2 3 4 5 6 7

xO

g(x) � |x | � 1

g(x)

[�10, 10] scl: 1 by [�10, 10] scl: 1

[�10, 10] scl: 1 by [�10, 10] scl: 1

100�50 200 300 xO

y

2

4

6

8

10

12

5x � 100y � 1000


Sele

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ers

47. D � all reals, 49.R � {y y � 0 or y � 2}

51. 53.

Chapter 3 Systems of Equations andInequalities


1. 3.

5. 7. y � �2x 9. y � 6 � 3x11. y � 2 � 6x

13. 15.

17. 19. �9 21. 0 23. �22

Pages 112–115 Lesson 3-11. Two lines cannot intersect in exactly two points.3. A graph is used to estimate the solution. To determinethat the point lies on both lines, you must check that itsatisfies both equations.

5. 7. consistent and independent

9. consistent and dependent 11. The cost is $5.60 for both stores to develop 30 prints.

13. 15.

17. 19.

x

y

O

(3.5, 0)2x � 3y � 7

2x � 3y � 7

x

y

O

(5, 3)3x � 7y � �6

x � 2y � 11

x

y

O(4, 1)

x � 2y � 6

2x � y � 9

x

y

O

(1, �2)y � 2x � 4

y � �3x � 1

x

y

O

x � 2y � 8

x � y � 412

x

y

O

(1, 5)y � x � 4

y � 6 � x

x

y

O

(2, 2)

2x � 3y � 10

3x � 2y � 10

y

xO

2x � y � 6

y

xO

y � 2x � 2

y

xOy � �2

y

xO

2x � 3y � �12

y

xO

y � 2x � 3

y

xO

2y � x

xO

y

y � |x | � 2

xO

y

y � 0.5x � 4

xO

y

y � 3x � 5

xO

f (x)


Sele

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21. 23.

25. inconsistent 27. consistent and independent

29. inconsistent 31. consistent and independent

33. consistent and 35. inconsistentindependent

37. (�3, 1) 39. y � 52 � 0.23x, y � 80 41. Deluxe Plan43. Supply, 300,000; demand, 200,000; prices will tend tofall. 45. y � 304x � 15,982, y � 98.6x � 18,976 47. FL willprobably be ranked third by 2020. The graphs intersect inthe year 2015, so NY will still have a higher population in2010, but FL will have a higher population in 2020.

49. You can use a system of equations to track sales andmake predictions about future growth based on pastperformance and trends in the graphs. Answers shouldinclude the following.• The coordinates (6, 54) represent that 6 years after 1999

both the in-store sales and online sales will be $54,000.• The in-store sales and the online sales will never be equal

and in-store sales will continue to be higher than onlinesales.

51. C 53. (�5.56, 12) 55. no solution 57. (2.64, 42.43)

59. 61. A 63. P 65. {�15, 9}67. {�2, 3} 69. {9}

71. x2 � 6 73. �3z

� � 1

75. 9y � 177. 12x � 18y � 679. x � 4y

Pages 119–122 Lesson 3-2 3. Vincent; Juanita subtracted the two equations incorrectly;�y � y � �2y, not 0. 5. (1, 3) 7. (5, 2) 9. (6, �20)

11. �3�13

�, 2�23

� 13. (9, 5) 15. (3, �2) 17. no solution

19. (4, 3) 21. (2, 0) 23. (10, �1) 25. (4, �3) 27. (�8, �3)

29. no solution 31. ��12

�, �32

� 33. (�6, 11) 35. (1.5, 0.5)37. 8, 6 39. x � y � 28, 16x � 19y � 478 41. 4 2-bedroom,2 3-bedroom 43. x � y � 30, 700x � 200y � 15,00045. 2x � 4y � 100, y � 2x 47. Yes; they should finish thetest within 40 minutes. 49. 25 min of step aerobics, 15 minof stretching 51. You can use a system of equations to findthe monthly fee and rate per minute charged during themonths of January and February. Answers should includethe following.• The coordinates of the point of intersection are (0.08, 3.5).• Currently, Yolanda is paying a monthly fee of $3.50 and

an additional 8¢ per minute. If she graphs y = 0.08x +3.5 (to represent what she is paying currently) and y = 0.10x + 3 (to represent the other long-distance plan)and finds the intersection, she can identify which plan would be better for a person with her level of usage.

53. A 55. consistent and dependent

57. 59.

61. x � y � 0; 1, �1, 0 63. 2x � y � �3; 2, �1, �365. 3x � 2y � 21; 3, 2, 21 67. yes 69. no

y

xO

3x � 9y � �15

y

xO

x � y � 3

x

y

O

4y � 2x � 4

y � x � 112

y

xO

2x � y � �4

x

y

O

3y � x � �2

y � x � 213

x

y

O

1.2x � 2.5y � 4

0.8x � 1.5y � �10

(�5, 4)

x

y

O

( , )12

14

2y � x

8y � 2x � 1x

y

O

2y � 2x � 8

y � x � 5

x

y

O

(�1, 5)

�4x � y � 9

x � y � 4

x

y

O

y � x � 4

y � x � 4

x

y

O

(�4, �2)

x � y � �214

12

x � y � 012

x

y

O

(4, 2)

2x � y � 6

x � 2y � 514


Sele

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1. 3. (2, 7) 5. Hartsfield,78 million; O’Hare,72.5 million

Pages 125–127 Lesson 3-3 1. Sample answer: y � x � 3, y � x � 2 3a. 4 3b. 2 3c. 13d. 3

5. 7.

9. (�4, 3), (1, �2), (2, 9), (7, 4) 11. Sample answer: 3 packages of bagels, 4 packages of muffins; 4 packages of bagels, 4 packages of muffins; 3 packages of bagels, 5 packages of muffins13. 15.

17. 19. no solution

21. 23.

25. (�3, �4), (5, �4), (1, 4) 27. (�6, �9), (2, 7), (10, �1)

29. (�4, 3), (�2, 7), (4, �1), �7�13

�, 2�13

� 31. 64 units2

33. s � 111, s � 130, h � 9, h � 12

35.

37. 6 pumpkin, 8 soda

39. The range for normal blood pressure satisfies fourinequalities that can be graphed to find their intersection.Answers should include the following.• Graph the blood pressure as an ordered pair; if the point

lies in the shaded region, it is in the normal range.• High systolic pressure is represented by the region to the

right of x � 140 and high diastolic pressure isrepresented by the region above y � 90.

41. Sample answer: y � 6, y � 2, x � 5, x � 1 43. (6, 5)

45. 47.

49. �5 51. 8 53. 5

x

y

O

�x � 8y � 12

2x � y � 6

(4, 2)

y

xO

y � 2x � 1

y � � x � 412

(�2, �3)

Swed

ish

So

da

2

0

4

6

8

10

12

14

Pumpkin2 4 6 8 10 12 14

x � 2.5y � 26

2x � 1.5y � 24y

x

Sto

rm S

urg

e (f

t)

0

8

10

12

14

16

Wind Speed (mph)80 100 120 140 160

s � 111

s � 130

h � 12

h � 9

h

s

y

xO

2x � 4y � �7

x � 3y � 2

2x � y � 4y

xO

x � 3y � 6

x � �4

x � 2

y

xO

2y � x � �6

4x � 3y � 7

y

xO y � �2

y � 2

y � x � 3

y

xO

y � �4x � �1

y

xO x � 2y � �3

y � 2x � 1x � 1

y

xO

y � x � 2

y � �2x � 4

x

y

O

(�1, 7)

y � �x � 6

y � 3x � 10


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ers

Pages 132–135 Lesson 3-41. sometimes 3. vertices: (1, 2), (1, 4), (5, 2);

max: f(5, 2) � 4, min: f(1, 4) � �10

5. vertices: (0, 1), (1, 3),(6, 3), (10, 1); max:f(10, 1) � 31, min: f(0, 1) � 1

7. vertices: (�2, 4), (�2, �3),(2, �3), (4, 1); max: f(2, �3) � 5; min: f(�2, 4) � �6

9. c � 0, � � 0, c � 3� � 56, 4c � 2� � 104 11. (0, 0),

(26, 0), (20, 12), �0, 18�23

�13. 20 canvas tote bags and12 leather tote bags

15. vertices: (0, 1), (6, 1), (6, 13);max: f(6, 13) � 19; min: f(0, 1) � 1

17. vertices: (1, 4), (5, 8), (5, 2),(1, 2); max: f(5, 2) � 11, min: f(1, 4) � �5

19. vertices: (�3, �1), (3, 5);min: f(�3, �1) � �9; no maximum

21. vertices: (0, 0), (0, 2), (2, 1),(3, 0); max: f(0, 2) � 6; min: f(3, 0) � �12

23. vertices: (3, 0), (0, �3); min: f(0, �3) � �12; no maximum

25. vertices: (0, 2), (4, 3),

��73

�, ��13

�; max: f(4, 3) � 25,

min: f(0, 2) � 6

27. vertices: (2, 5), (3, 0); no maximum;no minimum

29. vertices: (2, 1), (2, 3), (4, 1),(4, 4), (5, 3); max: f(4, 1) � 0, min: f(4, 4) � �12

x

y

O

(2, 3)

(2, 1) (4, 1)

(4, 4)

(5, 3)

x

y

O

(3, 0)

(2, 5)

x

y

O

( , )13�7

3

(4, 3)(0, 2)

x

y

O

(3, 0)

(0, �3)

x

y

O(3, 0)(0, 0)

(0, 2) (2, 1)

x

y

O(�3, �1)

(3, 5)

x

y

O

(1, 2)

(1, 4)

(5, 2)

(5, 8)

x

y

O

(6, 13)

(6, 1)(0, 1)

x

y

O

(�2, 4)

(4, 1)

(2, �3)

(�2, �3)

x

y

O

(1, 3) (6, 3)

(0, 1) (10, 1)

x

y

O

(1, 4)

(1, 2)

(5, 2)


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ers

31. g � 0, c � 0, 1.5g � c � 85, 2g � 0.5c � 40 33. (0, 0), (0, 20), (80, 0) 35. 0 graphing calculators, 80 CAS calculators39. (0, 0), (0, 4000),

(2500, 2000), (4500, 0)

41. 4500 acres corn, 0 acres soybeans; $130,50043. There are many variables in scheduling tasks. Linearprogramming can help make sure that all the requirementsare met. Answers should include the following.• Let x � the number of buoy replacements and let

y � the number of buoy repairs. Then, x � 0, y � 0, x � 8 and x � 2.5y � 24.

• The captain would want to maximize the number ofbuoys that a crew could repair and replace so f(x, y) � x � y.

• Graph the inequalities and find the vertices of theintersection of the graphs. The coordinate (0, 24)maximizes the function. So the crew can service themaximum number of buoys if they replace 0 and repair24 buoys.

45. C 47.

49. (2, 3) 51. c � average cost each year; 15c � 3479 � 748953. Additive Inverse 55. Multiplicative Inverse 57. 959. 16 61. 8Page 135 Practice Quiz 2

1. 3.

5. vertices: (1, �3), (�1, 3), (5, 6), (5, 1); max: f(5, 1) � 17, min: f(�1, 3) � �13

Pages 142–144 Lesson 3-51. You can use elimination or substitution to eliminate oneof the variables. Then you can solve two equations in twovariables.3. Sample answer: x � y � z � 4, 2x � y � z � �9, x � 2y � z � 5; �3 � 5 � 2 � 4, 2(�3) � 5 � 2 � �9,�3 � 2(5) � 2 � 5 5. (�1, �3, 7) 7. (5, 2, �1) 9. (4, 0, 8)

11. 4�12

� lb chicken, 3 lb sausage, 6 lb rice 13. (�2, 1, 5)

15. (4, 0, �1) 17. (1, 5, 7) 19. infinitely many

21. ��13

�, ��12

�, �14

� 23. (�5, 9, 4) 25. 8, 1, 3 27. enchilada,

$2.50; taco, $1.95; burrito, $2.65 29. x � y � z � 355,

x � 2y � 3z � 646, y � z � 27 31. a � �32�, b � 0, c � 3;

y � �32�x2 � 0x � 3 or y � �

32�x2 � 3 33. D 35. 120 units of

notebook paper and 80 units of newsprint37. 39. Sample answer using

(7, 15) and (14, 22): y � x � 841. x � 3y 43. 9s � 4t

Pages 145–148 Study Guide and Review1. c 3. f 5. a 7. h 9. d11.

13. 15. (3, 2) 17. (9, 4)19. (�1, 2)

21. 23.

25. 160 My Real Babies, 320 My First Babies 27. (4, �2, 1)

y

xO

y � x � 1

x � 5

y

xO

y � 4

y � �3

y

xO

y � 2x � 8

(�8, �8)

y � x � 4 12

y

xO

x � 2y � 4

3x � 2y � 12

(4, 0)

x

y

O

4y � 2x � 4

3x � y � 3

x

y

O

(5, 1)

(5, 6)

(�1, 3)

(1, �3)

x

y

O

x � 3y � 15

4x � y � 16

x

y

O

y � x � 4

y � x � 0

x

y

O

3x � 2y � �6

y � x � 132

S

c0

1000

2000

2000 4000

3000

4000

(0, 0)

(2500, 2000)(0, 4000)

(4500, 0)


�

Sele

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Chapter 4 Matrices


1. 6 3. 4�34

� 5. �13 7. �3; �13

� 9. �8; �18

� 11. �1.25; 0.8

13. �83

�; ��38

�

15. 17.

19. (6, 1) 21. (8, �5) 23. (2, �2)

Pages 156–158 Lesson 4-11. The matrices must have the same dimensions and eachelement of one matrix must be equal to the correspondingelement of the other matrix. 3. Corresponding elementsare elements in the same row and column positions.5. 3 4 7. (3, 3) 9. 2 5 11. 3 1 13. 3 3

15. 3 2 17. �3, ��13

� 19. (3, �5, 6) 21. (4, �3)

23. (14, 15) 25. (5, 3, 2) 27. 3 3 29. Sample answer:Mason’s Steakhouse; it was given the highest ratingpossible for service and atmosphere, location was given oneof the highest ratings, and it is moderately priced.

Single Double Suite

31. � 33. row 6, column 9 35. B 37. (7, 5, 4) 39. ��

45

�, �35

�, �1141. vertices: (3, 1), ��

125�, �

52

�, ��32

�, �127�; max: f��

125�, �

52

� �

35, min: f��32

�, �127� � �1

43. 45. $4.50 47. 2 49. 2051. �10 53. �18 55. �3

57. �32

�

Pages 163–166 Lesson 4-21. They must have the same dimensions.

3. � 5. � 7. � 9. �

11. Males � � , Females � � 13. No; many schools offer the same sport for males andfemales, so those schools would be counted twice.

15. impossible 17. � 19. � 21. � 23. � 25. � 27. � 29. � 31. �

33. � 35. 1996, floods; 1997, floods; 1998, floods; 1999, tornadoes; 2000, lightning

37. � 39. � 41. You can use matrices to track dietary requirements andadd them to find the total each day or each week. Answersshould include the following.

• Breakfast � � , Lunch � � ,

Dinner � � • Add the three matrices: � .

43. A 45. 1 4 47. 3 3 49. 4 3 51. (5, 3, 7)53. (2, 5) 55. (6, �1)57. 59. Multiplicative Inverse

61. Distributive


1. Sample answer: � � � 3. The Right Distributive

Property says that (A � B)C � AC � BC, but AC � BC �CA � CB since the Commutative Property does not hold formatrix multiplication in most cases. 5. undefined

7. � 9. � 11. [45 55 65], � 280165120

350320180

2441

2032

�5�8

1524

810

79

246

135

s

p

O 8

8

16

24

32

40

16 24 32

0.30p � 0.15s � 6

528261

806765

260820912620

264538

403229

1257987

1380

192012

222326

785622710

71711

181210

566482530

1.001.50

1.001.50

2.251.75

1.501.00

245228319227117

149130108

1207572

184124182

232164160

2 4�23

�

1 5

6 �1

4�642

383218

�1�1�4

�24

�7

�5�12

� 3 9

10�23

� 1�23

� �2�12

�

39

1.54.5

�13�323

�2�16

4

8�10�12

�46

�14

456,873405,163340,480257,586133,235

16,43914,54512,679

79315450

549,499477,960455,305321,416

83,411

16,76314,62014,486

90415234

29�22

�2112

824

�223

105

1�7

444

444

Co

st (

$)

1

0

2

3

4

5

6

Hours1 2 3 4 5

y

xO

y � �x � 10

y � �5x � 16

y � x 13

7595

7089

6079

WeekdayWeekend

y

xO

y

xO

Sele

cted A

nsw

ers

13. 4 2 15. undefined 17. undefined

19. [6] 21. not possible 23. �

25. � 27. yesAC � BC � � � � � � � �

� � � � � �

(A � B)C � ��

29. noC(A � B) � � � ��

� � � � � �

AC � BC � � � � � � � � � � � � � �

31. � 33. � 35. any two matrices � and � where bg � cf, a � d,

and e � h 37. � 39. $431 41. $26,360

43. a � 1, b � 0, c � 0, d � 1; the original matrix 45. B

47. � 49. � 51. (5, �9) 53. $2.50; $1.50

55. 8; �16 57.

59.


1. (6, 3) 3. (1, 3, 5) 5. � 7. � 9. not possible

Pages 178–181 Lesson 4-41.

3. Sample answer: � 5. A′(4, 3), B′(5, �6),

C′(�3, �7) 7. � 9. A′(0, �4), B′(5, �4), C′(5, 0),

D′(0, 0) 11. B 13. D′(�3, 6), E′(�2, �3), F′(�10, �4)

15. � 17. 19. X′(�1, 1),

Y′(�4, 2), Z′(�1, 7)

21. �

23.

25. J(�5, 3), K(7, 2), L(4, �1) 27. y

xO

T'

Q'

R'

R

Q

T

S

S'

y

xOD'

E'

F

ED

G

F'

G'

11

41

54

24

y

xOC'

A'

A

B'

C

B

�2.50

1.5�1.5

02

00

50

54

04

�41

�41

�41

33

41

149103

120159

159200

232134

y

xO

y

xO

yxO

x � y � 8 12

�4

�8

�12

�16

�4 4 8

212

�20�28

�621

12�3

96.5099.50

118117

fh

eg

bd

ac

14,28513,270

4295

210190

0

165240

75

290175110

�4�16

�2052

�13�8

�2126

9�8

126

1�4

52

23

�54

1�4

52

�23

14

6�24

�12�40

06

�48

1�4

52

23

�54

�23

14

1�4

52

�4�16

�2052

1�4

52

06

�48

1�4

52

23

�54

�23

14

�4�16

�2052

�13�8

�2126

9�8

126

1�4

52

23

�54

1�4

52

�23

14

16�5

�11

24�32�48

2�30

�251

129


reflection same same yes

rotation same same yes

translation same same yes

dilation changes same no

Transformation Size Shape Isometry

Sele

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nsw

ers

29. � 31. � 33. (�1.5, �1.5), (�4.5, �1.5), (�6, �3.75), (�3, �3.75)

35. � 37. (�8, 7), (�7, �8), and (8, �7) 39. Multiply the

coordinates by � , then add the result to � .41. (17, �2), (23, 2)43. Transformations are used in computer graphics to createspecial effects. You can simulate the movement of an object,like in space, which you wouldn’t be able to recreateotherwise. Answers should include the following.• A figure with points (a, b), (c, d), (e, f ), (g, h), and (i, j)

could be written in a 2 5 matrix � and

multiplied on the left by the 2 2 rotation matrix. • The object would get smaller and appear to be moving

away from you.

45. A 47. undefined 49. � 51. 53.

D � {x x � 0}, R � {all real numbers}; no

D � {3, 4, 5}, R � {�4, 5, 6}; yes

55. x � 2.8 57. x � 1 � 1 59. 6 61. 28 63. �94

�


1. Sample answer: � 3. It is not a square matrix.

5. Cross out the column and row that contains 6. The minoris the remaining 2 2 matrix. 7. �38 9. �40 11. �4313. 45 15. 20 17. �22 19. �29 21. 63 23. 32 25. 3227. �58 29. 62 31. 172 33. �22 35. �5 37. �14139. �6 41. 14.5 units2 43. about 26 ft2

45. Sample answer: 47. If you know the coordinates of the vertices of a triangle,you can use a determinant to find the area. This isconvenient since you don’t need to know any additionalinformation such as the measure of the angles. Answersshould include the following.• You could place a coordinate grid over a map of the

Bermuda Triangle with one vertex at the origin. By usingthe scale of the map, you could determine coordinates torepresent the other two vertices and use a determinant toestimate the area.

• The determinant method is advantageous since you don’tneed to physically measure the lengths of each side or themeasure of the angles between the vertices.

49. C 51. �36.9 53. �493 55. �3252 57. A′(�5, 2.5),B′(2.5, 5), C′(5, �7.5) 59. [�4] 61. undefined

63. [14 �8] 65. 138,435 ft 67. y � ��43

�x 69. y � �12

�x � 5

71. (1, 9) 73. (�1, 1) 75. (4, 7)

Pages 192–194 Lesson 4-61. The determinant of the coefficient matrix cannot be zero.3. 3x � 5y � �6, 4x � 2y � 30 5. (0.75, 0.5) 7. no solution

9. �6, ��12

�, 2 11. savings account, $1500; certificate of

deposit, $2500 13. (�12, 4) 15. (6, 3) 17. (�0.75, 3)

19. (�8.5625, �19.0625) 21. (4, �8) 23. ��23

�, �56

� 25. (3, �4)

27. (2, �1, 3) 29. ��12491

�, ��12092

�, �22494

� 31. ��12585

�, �17403

�, �617430

�33. race car, 5 plays; snowboard, 3 plays 35. silk, $34.99;cotton, $24.99 37. peanuts, 2 lb; raisins, 1 lb; pretzels, 2 lb39. Cramer’s Rule is a formula for the variables x and ywhere (x, y) is a solution for a system of equations.Answers should include the following.• Cramer’s Rule uses determinants composed of the

coefficients and constants in a system of linear equationsto solve the system.

• Cramer’s Rule is convenient when coefficients are largeor involve fractions or decimals. Finding the value of thedeterminant is sometimes easier than trying to find agreatest common factor if you are solving by usingelimination or substituting complicated numbers.

41. 111°, 69° 43. 40 45. � 47. 49.

(4, 3)

51. c � 10h � 35 53. �


1. � 3.

5. �58 7. 26 9. (4, �5)

y

x

A' A

B

C

D

C'

D'B' O

�2�1

1�4

4�1

12

9�23

7266

y

xO

(4, 3)

x � y � 7

x �y � �1 12

y

x

A'

A

B

C

C'

B'

13

13

13

111

111

111

14

28

y

xO

x � y 2

y

xO

�78

21

24�13

�8

111833

ij

gh

ef

cd

ab

60

0�1

10

34

�4�4

�44

44

4�4

44

�44

�4�4

4�4


Sele

cted A

nsw

ers


1. � 3. Sample answer: � 5. yes

7. no inverse exists 11. yes 13. no 15. yes 17. true

19. false 21. no inverse exists 23. �17

�� 25. �

14

�� 27. ��112�� 29. �

312��

31. 10� 33a. yes

33b. Sample answer:

35. � 37. dilation by a scale factor of �12

�

39. MEET_IN_THE_LIBRARY 41. BRING_YOUR_BOOK

43. a � �1, d � �1, b � c � 0 45. A 47. �

49. � 51. � 53. (2, �4)

55. (�5, 4, 1) 57. �14 59. 1 61. �5 63. �52

�

65. 7.82 tons/in2 67. 5�12

� 69. 3 71. 300 73. �2 75. 477. �34

Pages 205–207 Lesson 4-81. 2r � 3s � 4, r � 4s � �2 3. Tommy; a 2 1 matrix cannot be multiplied by a 2 2 matrix.

5. � � � � � 7. (5, �2) 9. (�3, 5)

11. h � 1, c � 12 13. � � � � �

15. � � � � � 17. � � � � � 19. � � � � � 21. (3, 4) 23. (6, 1)

25. ��13

�, 4 27. (�2, �2) 29. (0, 9) 31. ��32

�, �13

� 33. 2010

35. The solution set is the empty set or infinite solutions.

37. D 39. (�6, 2, 5) 41. (0, �1, 3) 43. � 45. (4, �2) 47. (�6, �8) 49. {�4, 10} 51. {2, 7}

Pages 209–214 Chapter 4 Study Guide and Review1. identity matrix 3. Scalar multiplication 5. determinant

7. dimensions 9. equal matrices 11. (�5, �1) 13. (�1, 0)

15. � 17. � 19. [�18] 21. not possible

23. A′(1, 0), B′(8, �2), C′(3, �7) 25. A′(3, 5), B′(�4, 3),

C′(1, �2) 27. 109 29. 0 31. �52 33. ��23

�, 5 35. (�1, �3)

37. (1, 2, �1) 39. ��114�� 41. �

214��

43. ��110�� 45. (4, 2) 47. (�3, 1)

Chapter 5 Polynomials

Page 221 Chapter 5 Getting Started1. 2 � (�7) 3. x � (�y) 5. 2xy � (�6yz)

7. �8x3 � 2x � 6 9. �x � 3 11. ��32

�a � 1 13. 6.3; reals,

rationals 15. 17; reals, rationals, integers, whole numbers,natural numbers 17. 4; reals, rationals, integers, wholenumbers, natural numbers

Pages 226–228 Lesson 5-11. Sample answer: (2x2)3 � 8x6 since (2x2)3 � (2x2)3 � (2x2)3 �(2x2)3 � 2x2 � 2x2 � 2x2 � 2x � x � 2x � x � 2x � x � 8x6

3. Alejandra; when Kyle used the Power of a Productproperty in his first step, he forgot to put an exponent of

�2 on a. Also, in his second step, (�2)�2 should be �14

�, not 4.

5. 16b4 7. �6y2 9. 9p2q3 11. �c2

9d2� 13. 4.21 105

15. 3.762 103 17. about 1.28 s 19. b4 21. z10 23. �8c3

25. �y3z2 27. �21b5c3 29. �24r7s5 31. 90a4b4

33. �a32

bc4

2� 35. ��

m4

3n9� 37. �

8xy6

3� 39. �

v31w6� 41. �

25xz

3y7

2� 43. 7

45. 4.32 104 47. 6.81 10�3 49. 6.754 108

51. 6.02 10�5 53. 6.2 1010 55. 1.681 10�7

57. 2 10�7 m 59. about 330,000 times61. Definition of an exponent63. Economics often involves large amounts of money.Answers should include the following.• The national debt in 2000 was five trillion, six hundred

seventy-four billion, two hundred million or 5.6742 1012 dollars. The population was two hundred eighty-onemillion or 2.81 108.

• Divide the national debt by the population.�5.

26.78412

11008

12� � $2.0193 104 or about $20,193 per person.

65. B 67. (�3, 3) 69. � 71. 7 73. (2, 0, 4)

75. Sample answer using (0, 4.9) and (28, 8.3): y � 0.12x � 4.9 77. 7 79. 2x � 2y 81. 4x � 883. �5x � 10y

Pages 231–232 Lesson 5-21. Sample answer: x5 � x4 � x3

3.

5. yes, 3 7. 10a � 2b 9. 6xy � 18x 11. y2 � 3y � 7013. 4z2 � 1 15. 7.5x2 � 12.5x ft2 17. yes, 3 19. no21. yes, 7 23. �3y � 3y2 25. 10m2 � 5m � 15

x

x

x

x

x

x

x 2x

x x x

x 2 x 2

2

��12

� �32

�

1 �2

�20

�4�5

�42

63

�23

�2�4

�29

1�14

0�6

�3�2

�59

4�7

2115

�7

rst

616

�3

�5�12

8

311

�5

911

�1

xyz

23

�3

�5�7

0

314

�43�10

mn

�75

36

29

xy

�75

43

8�5

gh

3�7

2�4

�13

�

0

��13

�

1

�23

�

��83

�

�1

��13

�

�73

�

��15

�

��25

�

�35

�

�15

�

�9�11

�5�6

88

412

�44

00

y

xA'A''A

B

C

C'

B'

C''

B''

O

�34

� ��58

�

��15

� �130�

52

1�6

0�2

6�5

�7�3

�6�2

�13

14

33

33

0001

0010

0100

1000


Sele

cted A

nsw

ers

27. 7x2 � 8xy � 4y2 29. 12a3 � 4ab31. 6x2y4 � 8x2y2 � 4xy5 33. 2a4 � 3a3b � 4a4b4

35. �0.001x2 � 5x � 500 37. p2 � 2p � 24 39. b2 � 2541. 6x2 � 34x � 48 43. a6 � b2 45. x2 � 6xy � 9y2

47. d2 � 2 � �d14� 49. 27b3 � 27b2c � 9bc2 � c3

51. 9c2 � 12cd � 7d2 53. R2 � 2RW � W2

55. The expression for how much an amount of money willgrow to is a polynomial in terms of the interest rate.Answers should include the following.• If an amount A grows by r percent for n years, the

amount will be A(1 � r)n after n years. When thisexpression is expanded, a polynomial results.

• 8820(1 � r)3, 8820r3 � 26,460r2 � 26,460r � 8820• Evaluate one of the expressions when r � 0.04. For

example, 8820(1 � r)3 � 8820(1.04)3 or $9921.30 to thenearest cent. The value given in the table is $9921rounded to the nearest dollar.

57. B 59. 20r3t4 61. �4ba

2

2�

63. 65.

67. 2y3 69. 3a2

Pages 236–238 Lesson 5-31. Sample answer: (x2 � x � 5) (x � 1) 3. Jorge; Shelly is subtracting in the columns instead of adding.5. 5b � 4 � 7a 7. 3a3 � 9a2 � 7a � 6 9. x2 � xy � y2

11. b3 � b � 1 13. 3b � 5 15. 3ab � 6b2 17. 2c2 � 3d �4d2 19. 2y2 � 4yz � 8y3z4 21. b2 � 10b 23. n2 � 2n � 3

25. x3 � 5x2 � 11x � 22 � �x

3�9

2� 27. x2 29. y2 � y � 1

31. a3 � 6a2 � 7a � 7 � �a �

31

�

33. x4 � 3x3 � 2x2 � 6x � 19 � �x

5�6

3� 35. g � 5

37. t4 � 2t3 � 4t2 � 5t � 10 39. 3t2 � 2t � 3

41. 3d2 � 2d � 3 � �3d

2� 2� 43. x3 � x � �

2x6� 3� 45. x � 3

47. x � 2 49. x2 � x � 3 51. $0.03x � 4 � �10

x00�

53. 170 � �t2

1�70

1� 55. x3 � x2 � 6x � 24 ft

57. x2 � 3x � 12 ft/s 59. Division of polynomials can beused to solve for unknown quantities in geometricformulas that apply to manufacturing situations. Answersshould include the following.• 8x in. by 4x � s in.• The area of a rectangle is equal to the length times the

width. That is, A � �w.• Substitute 32x2 � x for A, 8x for �, and 4x � s for w.

Solving for s involves dividing 32x2 � x by 8x.

A � �w32x2 � x � 8x(4x � s)

�32x

82

x� x� � 4x � s

4x � �18

� � 4x � s

�18

� � s

The seam is �18

� inch.

61. D 63. y4z4 � y3z3 � 3y2z 65. a2 � 2ab � b2

67. y � �x � 2 69. 9 71. 4 73. 6

Page 238 Practice Quiz 11. 6.53 108 3. �108x8y3 5. �

xz6

2� 7. 3t2 � 2t � 8

9. m2 � 3 � �m

1�9

4�

Pages 242–244 Lesson 5-41. Sample answer: x2 � 2x � 1 3. sometimes5. a(a � 5 � b) 7. (y � 2)(y � 4) 9. 3(b � 4)(b � 4)

11. (h � 20)(h2 � 20h � 400) 13. �y

2�

y4

� 15. 2x(y3 � 5)

17. 2cd2(6d � 4c � 5c4d) 19. (2z � 3)(4y � 3)21. (x � 1)(x � 6) 23. (2a � 1)(a � 1) 25. (2c � 3)(3c � 2)27. 3(n � 8)(n � 1) 29. (x � 6)2 31. prime33. (y2 � z)(y2 � z) 35. (z � 5)(z2 � 5z � 25)37. (p2 � 1)(p � 1)(p � 1) 39. (7a � 2b)(c � d)(c � d)41. (a � b)(5ax � 4by � 3cz) 43. (3x � 2)(x � 1)

45. 30 ft by 40 ft 47. �xx

��

56

� 49. �x2 �x �

2x4� 4

� 51. x � 2

53. 16x � 16 ft/s 55. (8pn � 1)2 57. B 59. yes 61. no;(2x � 1)(x � 3) 63. t2 � 2t � 1 65. x2 � 267. 4x2 � 3xy � 3y2 69. [�2] 71. 15 in. by 28 in. 73. no75. Associative Property (�) 77. irrational 79. rational81. rational

Pages 247–249 Lesson 5-51. Sample answer: 64 3. Sometimes; it is true when x � 0.5. �2.668 7. 4 9. �3 11. x 13. 6 a b2 15. about 3.01 mi17. �12.124 19. 2.066 21. �7.830 23. 3.890 25. 4.647

27. 59.161 29. �13 31. 18 33. �2 35. �15

� 37. �0.4

39. � x 41. 8a4 43. �c2 45. 4z2 47. 6x2z2

49. 3p6 q3 51. �3c3d4 53. p � q 55. z � 4 57. not areal number 59. �5 61. about 1.35 m 63. x � 0 and y � 0, or y � 0 and x � 0 65. B 67. 7xy2(y � 2xy3 � 4x2)

69. (2x � 5)(x � 5) 71. 4x2 � x � 5 � �x �

82

�

73. � 75. (1, �3) 77. x2 � 11x � 24

79. a2 � 7a � 18 81. x2 � 9y2


1. Sometimes; ��n1

a��

na� only when a � 1. 3. The product

of two conjugates yields a difference of two squares. Eachsquare produces a rational number and the difference of two rational numbers is a rational number. 5. 2x y �

4x�

7. �24�35� 9. 2a2b2�3� 11. 22�3

2� 13. 2 � �5�15. 9�3� 17. 3�

32� 19. 5x2�2� 21. 3 x y�2y�

23. 6y2z�3

7� 25. �13

�c d �4

c� 27. ��3

26�� 29. �a

2�b2

b�� 31. 36�7�33. ��

26�� 35. 3�3� 37. 7�3� � 2�2�

39. 25 � 5�2� � 5�6� � 2�3� 41. 13 � 2�22�43. �28 �

137�3�� 45. ��1 �

2�3�� 47. 49. 6 � 16�2� yd,

24 � 6�2� yd2 51. 0 ft/s 53. about 18.18 m 55. x and yare nonnegative. 57. B 59. 12z4 61. y � 2

63. �xx

��

14

� 65. � 67. consistent and independent

69. �5 71. �2, 4 73. {x x � 6} 75. �14

� 77. �56

� 79. �1234�

81. �38

�

4�4

1�5

�x2 � 1��

23202504

8101418

y

xO

2x � y � 1

y

xO

y � � x � 2 13


Sele

cted A

nsw

ers

Page 256 Practice Quiz 21. x2y(3x � y � 1) 3. a(x � 3)2 5. 6 x y3 7. 2n � 39. �1 � �7�

Pages 260–262 Lesson 5-71. Sample answer: 64 3. In exponential form �

nbm� is equal

to (bm)�n1

�. By the Power of a Power Property, (bm)�n1

�� b�

mn�.

But, b�mn� is also equal to �b�n

1�m

by the Power of a Power Property. This last expression is equal to ��n

b�m. Thus,

�nbm� � ��n

b�m. 5. �3

x2� or ��3

x�2 7. 6�13�x�

53�y�

73� 9. �

13

� 11. 2

13. x�23� 15. a�

32�b�

23� 17. 19. �3� 21. �

56�

23. �5

c2� or ��5

c�2 25. 23�12� 27. 2z�

12� 29. 2 31. �

15

� 33. �19

�

35. 81 37. �23

� 39. �43

� 41. y4 43. b�15� 45. 47. t�

14�

49. 51. 53. �5� 55. 17�6

17� 57. �4

5x2y2�

59. �xy�

z

z�� 61. �2� 63. 2�6� � 5 65. 2�

32�

� 3�12�

67. 880 vibrations per second 69. about 33671. The equation that determines the size of the regionaround a planet where the planet’s gravity is stronger thanthe Sun’s can be written in terms of a fractional exponent.Answers should include the following.• The radical form of the equation is

r � D�5 ��M

Mp

s�2� or r � D�5 �

M

M

2p2s

��. Multiply the fraction

under the radical by �M

M

3

3s

s�.

r � D�5�M

M

2p2s

� ��M

M

3

3s

s��

� D�5�M

M

2pM

5s

�3s

�� D

� �D�5

M

M

s

2p M3

s��

The simplified radical form is r � �D�5

M

M

s

2p M3

s�� .

• If Mp and Ms are constant, then r increases as D increasesbecause r is a linear function of D with positive slope.

73. C 75. 36�2� 77. 8 79. �12

�x2 81. x � 2

83. x � 2�x� � 1

Pages 265–267 Lesson 5-81. Since x is not under the radical, the equation is a linearequation, not a radical equation. The solution is

x � ��3�

2� 1�. 3. Sample answer: �x� � �x � 3� � 3 5. �9

7. 15 9. 31 11. 0 � b � 4 13. 16 15. no solution 17. 919. �1 21. �20 23. no solution 25. x � 1 27. x � �1129. no solution 31. 3 33. 0 � x � 2 35. b � 5 37. 339. 1152 lb 41. 34 ft 43. Since �x � 2� � 0 and �2x � 3� � 0, the left side of the equation is nonnegative.Therefore, the left side of the equation cannot equal �1.

Thus, the equation has no solution. 45. D 47. 5�37�

49. (x2 � 1)�23� 51. �

�3

11000�

�

53. x � y � 7, 30x � 20y � 160; (2, 5)

55. 1 � y 57. �11 59. �3 � 10x � 8x2

Pages 273–275 Lesson 5-91a. true 1b. true 3. Sample answer: 1 � 3i and 1 � 3i

5. 5i xy �2� 7. �180�3� 9. 6 � 3i 11. �177� � �

1171�i

13. �2i�2� 15. 3, �3 17. 10 � 3j amps 19. 9i21. 10a2 b i 23. �12 25. �75i 27. 1 29. �i 31. 6

33. 4 � 5i 35. 6 � 7i 37. �8 � 4i 39. �1107� � �

167�i

41. �25

� � �15

�i 43. 20 � 15i 45. ��13

� � i

47. (5 � 2i)x2 � (�1 � i)x � 7 � i 49. �4i 51. �2i�3�53. �2i�10� 55. ��

�2

5��i 57. 4, �3 59. �

53

�, 4 61. �6171�, �

1191�

63. 13 � 18j volts65. Case 1: i � 0

Multiply each side by i to get i2 � 0 � i or �1 � 0. Thisis a contradiction.Case 2: i � 0Since you are assuming i is negative in this case, youmust change the inequality symbol when you multiplyeach side by i. The result is again i2 � 0 � i or �1 � 0, acontradiction.Since both possible cases result in contradictions, theorder relation “�” cannot be applied to the complexnumbers.

67. C 69. �1, �i, 1, i, �1, �i, 1, i, �1 71. 12 73. 4

75. y�13� 77. � 79. �

81. sofa: $1200, love seat: $600, coffee table: $25083. 85. 0

Pages 276–280 Chapter 5 Study Guide and Review1. scientific notation 3. FOIL method 5. extraneous

solution 7. square root 9. principal root 11. �f13� 13. 8xy4

15. 1.7 108 17. 9 102 19. 4x2 � 22x � 34

21. x3y � x2y4 23. 4a4 � 24a2 � 36 25. 2x3 � x � �x �

33

�

27. x � 4 29. 50(2x � 1)(2x � 1) 31. (5w2 � 3)(w � 4)33. (s � 8)(s2 � 8s � 64) 35. �16 37. 8 39. x4 � 341. 2m2 43. 2�6

2� 45. �5�3� 47. 20 � 8�6� 49. 9

y

xO

x � y � 1

x � 2y � 4

�2�1

12

2�3

�21

1�2

23

2�2��

y

xO

(2, 5)

x � y � 7

30x � 20y � 160

�5 M2pM3

s��

�5 M5s�

y2 � 2y�32�

�y � 4

a�152�

�6a

w�15�

�w

z(x � 2y)�12�

��x � 2y


Sele

cted A

nsw

ers

51. �2�10�7

� �5�� 53. 81 55. 57. 3x�

53�

� 4x�83� 59. 343

61. 4 63. 5 65. 8 67. 8m6i 69. 72 71. 23 � 14i 73. i

75. ��31�0

21i�

Chapter 6 Quadratic Functions and Inequalities


1. 3.

5. 7x2 � 16x � 48 7. 9x2 � 6x � 1 9. (x � 6)(x � 5)11. (x � 8)(x � 7) 13. prime 15. (x � 11)2 17. 1519. 6�5� 21. 5i 23. 3i�30�

Pages 290–293 Lesson 6-11. Sample answer: f(x) � 3x2 � 5x � 6; 3x2, 5x, �63a. up; min. 3b. down; max. 3c. down; max.3d. up; min. 5a. 0; x � �1; �15b. 5c.

7a. 3; x � �4; �47b. 7c.

9a. 0; x � ��53

�; ��53

�

9b. 9c.

11. min.; ��245� 13. $8.75 15a. 0; x � 0; 0

15b. 15c.

17a. �9; x � 0; 017b. 17c.

19a. 1; x � 0; 019b. 19c.

21a. 9; x � 4.5; 4.521b. 21c.

23a. 36; x � �6; �623b. 23c.

xO

f(x)

2

4

6

�8 �4�12�16(�6, 0)

f (x) � x 2 � 12x � 36

2

4 8 12

�4

�8

�12 (4 , �11 )12

14

xO

f(x)

f (x) � x 2 � 9x � 9

f (x)

xO

(0, 1)f (x) � 3x 2 � 1

xO�2�4 2 4

�4

4

(0, �9) f (x) � x 2 � 9

f(x)

(0, 0) xO

f (x) � �5x 2

f(x)

�4 �2 2

4

�4

�8

�12

xO

f(x)

f (x) � 3x 2 � 10x(� , � )53

253

xO

f(x)

�4�8�10

�8

�4

�12f(x) � x 2 � 8x � 3

(�4, �13)

f(x)

xO

(�1, �1)f(x) � x 2 � 2x

y

xO

y � x2 � 4

y

xO

y � 2x � 3

y�35�

�y


�3 3�2 0�1 �1

0 01 3

x f(x)

�6 �9�5 �12�4 �13�3 �12�2 �9

x f(x)

�3 �3�2 �8

��53

� ��235�

�1 �70 0

x f(x)

�2 �5�1 �8

0 �91 �82 �5

x f(x)

3 �94 �114.5 �11.255 �116 �9

x f(x)

�2 13�1 4

0 11 42 13

x f(x)

�2 �20�1 �5

0 01 �52 �20

x f(x)

�8 4�7 1�6 0�5 1�4 4

x f(x)

Sele

cted A

nsw

ers

25a. �3; x � 2, 225b. 25c.

27a. 0; x � ��54

�; ��54

�

27b. 27c.

29a. 0; x � �6; �629b. 29c.

31a. ��89

�; x � �13

�; �13

�

31b. 31c.

33. max.; �9 35. min.; �11 37. max.; 12

39. max.; ��78

� 41. min.; �11 43. min.; �10�13

� 45. 40 m

47. The y-intercept is the initial height of the object.49. 60 ft by 30 ft 51. $11.50 53. 5 in. by 4 in.55. If a quadratic function can be used to model ticket priceversus profit, then by finding the x-coordinate of the vertexof the parabola you can determine the price per ticket thatshould be charged to achieve maximum profit. Answersshould include the following. • If the price of a ticket is too low, then you won’t make

enough money to cover your costs, but if the ticket priceis too high fewer people will buy them.

• You can locate the vertex of the parabola on the graph ofthe function. It occurs when x � 40. Algebraically, this is

found by calculating x � ��2ba� which, for this case, is

x � �2�(�40

5000)

� or 40. Thus the ticket price should be set at

$40 each to achieve maximum profit.57. C 59. 3.20 61. 3.38 63. 1.56 65. �1 � 3i 67. 23

69. 4 71. [5 �13 8] 73. � 75. 5 77. �2

Pages 297–299 Lesson 6-21a. The solution is the value that satisfies an equation.1b. A root is a solution of an equation. 1c. A zero is the x value of a function that makes the function equal to 0.1d. An x-intercept is the point at which a graph crosses thex-axis. The solutions, or roots, of a quadratic equation arethe zeros of the related quadratic function. You can find thezeros of a quadratic function by finding the x-intercepts ofits graph. 3. The x-intercepts of the related function arethe solutions to the equation. You can estimate the solutionsby stating the consecutive integers between which the x-intercepts are located. 5. �2, 1 7. �7, 0 9. �7, 411. between �2 and �1, 3 13. �2, 7 15. 3 17. 019. no real solutions 21. 0, 4 23. between �1 and 0;

between 2 and 3 25. 3, 6 27. 6 29. ��12

�, 2�12

� 31. �2�12

�, 3

33. between 0 and 1; between 3 and 4 35. between �3 and �2; between 2 and 3 37. no real solutions39. Let x be the first number.

Then, 7 � x is the other number.x(7 � x) � 14

�x2 � 7x � 14 � 0Since the graph of the relatedfunction does not intersect thex-axis, this equation has no realsolutions. Therefore no suchnumbers exist. 41. �2, 1443. 3 s 45. about 35 mph47. �4 and �2; The value of thefunction changes from negativeto positive, therefore the valueof the function is zero between

these two numbers. 49. A 51. �1 53. 3, 5 55. �1.33

57. 4, x � 3; 3 59. 4; x � �6; �6

61. �1103� � �

123�i 63. 24 65. �60 67. x(x � 5)

69. (x � 7)(x � 4) 71. (3x � 2)(x � 2)

Pages 303–305 Lesson 6-31. Sample answer: If the product of two factors is zero, thenat least one of the factors must be zero. 3. Kristin; the ZeroProduct Property applies only when one side of theequation is 0. 5. {�8, 2} 7. {3} 9. {�3, 4}

f(x)

xO

�12 �8 �4

8

4

�4

14f (x) � x2 � 3x � 4

(�6, �5)(3, �5)

xO

f (x) � x 2 � 6x � 4

f(x)

y

xO

y � �x2 � 7x � 14

6 0 �24

14 ��23

� �8

xO

f(x)

( , �1)13

f (x) � x 2 � x �23

89

xO

f(x)

�4

8

4

�4�8

(�6, 9)

f (x) � �0.25x2 � 3x

xO

f(x)

(� , � )54

258

f (x) � 2x 2 � 5x

xO

f(x) (2, 5)

f (x) � �2x 2 � 8x � 3


0 �31 32 53 34 �3

x f(x)

�3 �3�2 �2

��54

� ��285�

�1 �30 0

x f(x)

�1 �79

�

0 ��89

�

�13

� �1

1 ��59

�

2 1�79

�

x f(x)

�8 8�7 8.75�6 9�5 8.75�4 8

x f(x)

Sele

cted A

nsw

ers

11. 6x2 � 11x � 4 � 0 13. D 15. {�4, 7} 17. {�9, 9}

19. {�3, 7} 21. �0, ��34

�� 23. {8} 25. ��14

�, 4� 27. ��23

�, ��32

��29. ��

34

�, �94

�� 31. {�3, 1} 33. 0, �3, 3 35. x2 � 5x � 14 � 0

37. x2 � 14x � 48 � 0 39. 3x2 � 16x � 5 � 041. 10x2 � 23x � 12 � 0 43. 14, 16 or �14, �1645. B � D2 � 8D � 1647. y � (x � p)(x � q)

y � x2 � px � qx � pqy � x2 � (p � q)x � pq

a � 1, b � �(p � q), c � �pq

axis of symmetry: x � ��2ba�

x � ��(

2p(1�

)q)

�

x � �p �

2q

�

The axis of symmetry is the average of the x-intercepts.Therefore the axis of symmetry is located halfway betweenthe x-intercepts. 49. �6 51. D 53. �5, 1 55. between�1 and 0; between 3 and 4 57. 3�2� � 2�3�59. 33 � 20�2� 61. (3, �5) 63. 2�2� 65. 3�3�67. 2i�3�


1. 4; x � 2; 2 3. 1�12

�, 4

5. 3x2 � 11x � 4 � 0

Pages 310–312 Lesson 6-41. Completing the square allows you to rewrite one side ofa quadratic equation in the form of a perfect square. Oncein this form, the equation is solved by using the SquareRoot Property. 3. Tia; before completing the square, youmust first check to see that the coefficient of the quadraticterm is 1. If it is not, you must first divide the equation by

that coefficient. 5. � � 7. �94

�; �x � �32

�2 9. {4 � �5�}

11. ��3 �4�33�� 13. Earth: 4.5 s, Jupiter: 2.9 s

15. {�2, 12} 17. {3 � 2�2�} 19. ��5 �3�11�� 21. {�1.6, 0.2}

23. about 8.56 s 25. 81; (x � 9)2 27. �449�; �x � �

72

�2

29. 1.44; (x � 1.2)2 31. �21

56�; �x � �

54

�2 33. {�12, 10}

35. {2 � �3�} 37. {–3 � 2i} 39. ��12

�, 1� 41. ��2 �3�10��

43. ��5 �6i�23�� 45. {0.7, 4} 47. ��

34

� � �2�� 49. �x1

�, �x �

11

�

51. Sample answers: The golden rectangle is found in muchof ancient Greek architecture, such as the Parthenon, aswell as in modern architecture, such as in the windows ofthe United Nations building. Many songs have their climaxat a point occurring 61.8% of the way through the piece,with 0.618 being about the reciprocal of the golden ratio.The reciprocal of the golden ratio is also used in the designof some violins. 53. 18 ft by 32 ft or 64 ft by 9 ft

55. D 57. x2 � 3x � 2 � 0 59. 3x2 � 19x � 6 � 061. between �4 and �3; between 0 and 1 63. �4, �1�

12

�

65. (2, �5) 67. x � (�257) � 2 69. 37 71. 121


1a. Sample answer: 1b. Sample answer:

1c. Sample answer:

3. b2 � 4ac must equal 0. 5a. 8 5b. 2 irrational

5c. �2 �2�2�� 7a. �3 7b. two complex 7c. ��3 �

2i�3��

9. �3, �2 11. ��5 �2

i�2�� 13. No; the discriminant of

�16t2 � 85t � 120 is �455, indicating that the equation hasno real solutions. 15a. 240 15b. 2 irrational

15c. 8 � 2�15� 17a. �23 17b. 2 complex 17c. �1 � i2�23��

19a. 49 19b. 2 rational 19c. �2, �13

� 21a. 24

21b. 2 irrational 21c. �1 � �6� 23a. 0 23b. one rational

23c. ��52

� 25a. �135 25b. 2 complex 25c. ��1 �4i�15��

27a. 1.48 27b. 2 irrational 27c. 29. �i

31. ��3 �2�15�� 33. �

92

� 35. �5 �3�46�� 37. 0, ��

130� 39. �2, 6

41. This means that the cables do not touch the floor of thebridge, since the graph does not intersect the x-axis and theroots are imaginary. 43. 1998 45a. k � �6 45b. k � �6or k � 6 45c. �6 � k � 6 47. D 49. �14, �4

51. �1 �22�2�� 53. �2, 7 55. a4b10 57. 4b2c2

59.

61. no 63. yes; (2x � 3)2 65. no

y

xO 2 4 6 8

8642

�6�4

�4�6x � y � 3

x � y � 9

y � x � 4

�21��

�1 � 2�0.37��

y

xO

y

xO

y

xO

4 � �2��

�4

�8

8 124

4

x

f(x)

(2, �8)

f (x) � 3x2 � 12x � 4

O


Sele

cted A

nsw

ers


1a. y � 2(x � 1)2 � 5 1b. y � 2(x � 1)2

1c. y � 2(x � 3)2 � 3 1d. y � 2(x � 2)2 � 3

1e. Sample answer: y � 4(x � 1)2 � 3 1f. Sample answer:y � (x � 1)2 � 3 1g. y � �2(x � 1)2 � 3 3. Sampleanswer: y � 2(x � 2)2 � 1 5. (�3, �1); x � �3; up

7. y � �3(x � 3)2 � 38; (�3, 38); x � � 3; down

9.

11. y � 4(x � 2)2 13. y � ��12

�(x � 2)2 � 3 15. (�3, 0);

x � �3; down 17. (0, �6); x � 0; up

19. y � �(x � 2)2 � 12; (�2, 12); x � �2; down

21. y � �3(x – 2)2 � 12; (2, 12); x � 2; down

23. y � 4(x � 1)2 � 7; (�1, �7); x � �1; up

25. y � 3�x � �12

�2� �

74

�; ��12

�, ��74

�; x � ��12

�; up

27. 29.

31.

33.

35.

37. Sample answer: the graph of y � 0.4(x � 3)2 � 1 isnarrower than the graph of y � 0.2(x � 3)2 � 1.39. y � 9(x � 6)2 � 1 41. y � ��

23

�(x � 3)2 43. y � �13

�x2 � 545. y � �2x2 47. 34,000 feet; 32.5 s after the aircraft beginsits parabolic flight 49. d(t) � �16t2 � 8t � 5051. Angle A; the graph of the equation for angle A is higherthan the other two since 3.27 is greater than 2.39 or 1.53.53. y � ax2 � bx � c

y � a�x2 � �ba

�x � c

y � a�x2 � �ba

�x � ��2ba�2 � c � a��

2ba�2

y � a�x � �2ba�2

� c � �4ba

2�

The axis of symmetry is x � h or ��2ba�. 55. D

57. 12; 2 irrational 59. �23; 2 complex 61. {3 � 3i}

63. 2t2 � 2t � �t �3

1� 65. n3 � 3n2 � 15n � 21

67a. Sample answer using (1994, 76,302) and (1997, 99,448):y � 7715x � 15,307,408 67b. 161,167 69. no 71. no


1. {�7 � 2�3�} 3. �11; 2 complex 5. ��9 �25�5��

7. y � �23

�(x � 2)2 � 5 9. y � �(x � 6)2; (6, 0), x � 6; down

Pages 332–335 Lesson 6-71. y � (x � 3)2 � 1 3a. x � �1, 5 3b. x � �1 or x � 53c. �1 � x � 55. 7.

9. {x �1 � x � 7} 11. � 13. about 6.1 s15. 17.

y

xO

y � x 2 � 4x84�4

4

8

12

y

xO

y � �x 2 � 7x � 8

O�2 2 4 6

12

8

4

y

x

y � �x 2 � 5x � 6

�4�8

�12

�20

�2�4

1284

2 4

y

xO

y � x 2 � 16

y

xO

12

272

y � � x 2 � 5x �

y

xO

y � �4x 2 � 16x � 11

y

xO

y � x 2 � 6x � 2

y

xO

y � (x � 2)2 � 414

y

xOy � 4(x � 3)2 � 1

y

xO

y � (x � 1)2 � 313


Sele

cted A

nsw

ers

19. 21.

23. 25.

27. �2 � x � 6 29. x � �7 or x � �3 31. {x �7 � x � 4}33. {x x � �6 or x � 4} 35. {x x � �7 or x � 1}37. all reals 39. {x x � 7} 41. � 43. 0 to 10 ft or 24 to 34 ft 45. The width should be greater than 12 cm andthe length should be greater than 18 cm 47. 6

49.

51. C 53. {x all reals, x � 2} 55. {x x � �9 or x � 3}57. {x �1.2 � x � �0.4} 59. y � (x � 1)2 � 8; (1, 8),

x � 1; up 61. y � �12

�(x � 6)2; (�6, 0), x � �6; up

63. ��5 �2

i�3�� 65. 4a2b2 � 2a2b � 4ab2 � 12a � 7b

67. xy3 � y � �1x

� 69. � 71. x � 0.08 � 0.002;

0.078 � x � 0.082

Pages 336–340 Chapter 6 Study Guide and Review1. f 3. a 5. i 7. c 9a. 20; x � �3; �39b. 9c.

11a. 7; x � 4; 411b. 11c.

13a. �3; x � �2; �213b. 13c.

15. min.; ��8196� 17. max.; 7 19. 2, �5 21. between �3 and

�2; between �38 and �37 23. 2, �8 25. {�1}

27. {�11, 2} 29. ��13

�, ��32

�� 31. x2 � 3x � 70 � 0

33. 289; (x � 17)2 35. �4196�; �x � �

74

�2 37. 3 � 2�5� 39a. �24

39b. 2 complex 39c. �1 � �6�i 41a. 73 41b. 2 irrational

41c. ��7 �6�73�� 43. y � 5�x � �

72

�2� �

143�; ��

72

�, ��143�;

x � ��72

�; up

45. 47.

49. y � �12

�(x � 2)2 � 3

51. 53.

55. all reals 57. �x x � ��12

� or x � 3�59. �x x � �

3 �32�6�� or x � �

3 �32�6��

y

xO

y � �x 2 � 7x � 11

7531

�10

25

15

5

y

xO

y � x 2 � 5x � 15

y

xO

y � �9x 2 � 18x � 6

y

xO

y � (x � 2)2 � 2

xO

f(x)f (x) � �x 2 � 4x � 3

(�2, 1)

xO

f(x)

124 8

�8

�4

4

(4, �9)

f (x) � x 2 � 8x � 7

xO

f(x)

4�8 8�4

24

16

8(�3, 11)

f (x) � x 2 � 6x � 20

4822

�21�13

y

xO

y � �x 2 � 4

y � x 2 � 4

y

xO

y � 2x 2 � x � 3

1062

�8

6

2

�4

y

xO

y � �x 2 � 13x � 36

�4 4�12 �8

20

12

4

�4

y

xO

y � �x2 � 7x � 10y

xO

y � x2 � 6x � 5


�4 �3�3 0�2 1�1 0

0 �3

x f(x)

�5 15�4 12�3 11�2 12�1 15

x f(x)

2 �53 �84 �95 �86 �5

x f(x)

Sele

cted A

nsw

ers

Chapter 7 Polynomial Functions

Page 345 Chapter 7 Getting Started1. between 0 and 1, between 4 and 5 3. between �5 and �4,

between 0 and 1 5. ��32

�, ��17

� 7. 3x � 4 9. �19

11. 18b2 � 3b � 6

Pages 350–352 Lesson 7-11. 4 � 4x0; x � x1 3. Sample answer given.

5. 6; 5 7. �21; 3 9. 2a9 � 6a3 � 12 11. 6a3 � 5a2 � 8a � 4513a. f(x) → �� as x → ��, f(x) → �� as x → �� 13b. even13c. 0 15. 109 lumens 17. 3; 1 19. 4; 6 21. No, this is not

a polynomial because the term �1c

� cannot be written in the

form xn, where n is a nonnegative integer. 23. 12; 1825. 1008; �36 27. 86; 56 29. 7; 4 31. 12a2 � 8a � 2033. 12a6 � 4a3 � 5 35. 3x4 � 16x2 � 26 37. �x6 � x3 �2x2 � 4x � 2 39a. f(x) → �� as x → ��, f(x) → �� as x → �� 39b. odd 39c. 3 41a. f(x) → �� as x → ��, f(x) →�� as x → �� 41b. even 41c. 0 43a. f(x) → �� as x →��, f(x) → �� as x → �� 43b. odd 43c. 1 45. 5.832 units47. f(x) → �� as x → ��; f(x) → �� as x → �� 49. �

12

�

51. f(x) �12

�x3 � �32

�x2 � 2x 53. 4 55. 8 points 57. C

59. {x 2 � x � 6} 61. �x �1 � x � �45

��63.

65. �4 � 3�2�� 67. 23,450(1 � p); 23,450(1 � p)3

69.

Pages 356–358 Lesson 7-21. There must be at least one real zero between two pointson a graph when one of the points lies below the x-axis andthe other point lies above the x-axis.

3.

5.

7. between �2 and �1, between �1 and 0,between 0 and 1, and between 1 and 2

9. Sample answer: rel. max. at x � 0, rel. min. at x � �2 and at x � 2

11. rel. max. between x � 15 and x � 16, and no rel. min.;f(x) → �� as x → ��, f(x) → �� as x → ��.

13a.

13b. at x � �4 and x � 0 13c. Sample answer: rel. max. at x � 0, rel. min. at x � �3

xO

�2

�4

�8

42

4

f (x)

f (x) � �x 3 � 4x 2

xO�4 �2

�4

4 2

4

8

f(x)

f(x) � x 4 � 8x 2 � 10

xO

f(x)

f (x) � x 4 � 4x 2 � 2

xO

f(x)

�4 �2

�4

�8

42

4

8

f (x) � x 4 � 7x 2 � x � 5

xO

f(x)

y

xO

y � �x2 � 6x � 5

y

xO�12 �8

�2

�4

2

y � (x � 5)2 � 113

xO

f(x)


�3 20

�2 �9

�1 �2

0 5

1 0

2 �5

3 26

x f(x)

�5 25

�4 0

�3 �9

�2 �8

�1 �3

0 0

1 �5

2 �24

x f(x)

Sele

cted A

nsw

ers

15a.

15b. at x � 1, between �1 and 0, and between 2 and 315c. Sample answer: rel. max. at x � 0, rel. min. at x � 217a.

17b. between 0 and 1, at x � 2, and at x � 417c. Sample answer: rel. max. at x � 3, rel. min. at x � 119a.

19b. between �2 and �1 and between 1 and 219c. Sample answer: no rel. max., rel. min. at x � 021a.

21b. between �3 and �2, between �1 and 0, between 0and 1, and between 1 and 2 21c. Sample answer: rel. max.at x � �2 and at x � 1.5, rel. min. at x � 023a.

23b. between 0 and 1, between 1 and 2, between 2 and 3,and between 4 and 5 23c. Sample answer: rel. max. at x � 2, rel. min. at x � 0.5 and at x � 425a.

25b. between �4 and �3, between �2 and �1, between �1and 0, between 0 and 1, and between 1 and 2 25c. Sampleanswer: rel. max. at x � �3 and at x � 0, rel. min. at x ��1 and at x � 1 27. highest: 1982; lowest: 2000 29. 5

31.

33. 0 and between 5 and 6 35. 3.4 s

37. 39.

41. D 43. �1.90; 1.23 45. 0; �1.22, 1.22 47. 24a3 � 4a2 � 249. 8a4 � 10a2 � 4 51. 2x4 � 11x2 � 16

53. 55.y

xO

y � x 2 � 2x

y

xO

y � x 2 � 4x � 6

O

y

xO

y

x

y

x0

2520

30

35

40

45

50

55606570

14 16 18128 1062 4

G(x )

B(x )

Ave

rag

e H

eig

ht

(in

.)

Age (yrs)

�4 �2 2 4

8

16

24

xO

f (x)

f(x) � x 5 � 4x 4 � x 3 � 9x 2 � 3

xO

f (x)

�2

�4

�8

2 4

4

f(x) � x 4 � 9x 3 � 25x 2 � 24x � 6

xO�4 �2

�4

�8

42

8

4

f (x)

f (x ) � �x 4 � 5x 2� 2x � 1

xO�4 �2

�4

�8

2

4

f (x)

f(x) � x 4 � 8

xO�4

�8

�4

�2 2 4

4

f(x) � �3x 3 � 20x 2 � 36x � 16

f (x)

xO

f (x)

f(x) � x 3 � 3x 2� 2


�2 �18

�1 �2

0 2

1 0

2 �2

3 2

4 18

x f(x)

�1 75

0 16

1 �3

2 0

3 7

4 0

5 �39

x f(x)

�3 73

�2 8

�1 �7

0 �8

1 �7

2 8

3 73

x f(x)

�4 �169

�3 �31

�2 7

�1 5

0 �1

1 1

2 �1

3 �43

x f(x)

�1 65

0 6

1 �1

2 2

3 �3

4 �10

5 11

x f(x)

�4 �77

�3 30

�2 7

�1 �2

0 3

1 �2

2 55

x f(x)

0 2 4 6 8 10 12 14 16 18 20

25 34 40 45 50 54 59 64 68 71 71

26 33 39 44 49 53 56 59 61 61 60

x

B(x)

G(x)

Sele

cted A

nsw

ers

57. (�3, �2) 59. (1, 3) 61. (x � 5)(x � 6)63. (3a � 1)(2a � 5) 65. (t � 3)(t2 � 3t � 9)

Pages 362–364 Lesson 7-31. Sample answer: 16x4 � 12x2 � 0; 4[4(x2)2 � 3x2] � 03. Factor out an x and write the equation in quadratic formso you have x[(x2)2 � 2(x2) � 1] � 0. Factor the trinomial andsolve for x using the Zero Product Property. The solutionsare �1, 0, and 1. 5. 84(n2)2 � 62(n2) 7. �4, �1, 4, 1 9. 6411. 2(x2)2 � 6(x2) � 10 13. 11(n3)2 � 44(n3) 15. not

possible 17. 0, �4, �3 19. ��3�, �3�, �i�3�, i�3�21. 2, �2, 2�2�, �2�2� 23. �9, �9 � 9

2i�3��, �9 � 9

2i�3��

25. 81, 625 27. 225, 16 29. 1, �1, 4 31. w � 4 cm, � �8 cm, h � 2 cm 33. 3 3 in. 35. h2 � 4, 3h � 2, h � 337. Write the equation in quadratic form, u2 � 9x � 8 � 0,

where u � a � 3 . Then factor and use the Zero ProductProperty to solve for a; 11, 4, 2, and �5. 39. D41.

43. 17; 27 45. �17

315�; 135 47. A′(�1, �2), B′(3, �3), C′(1, 3)

49. x2 � 5x � 4 51. x3 � 6x � 20 � �x

5�4

3�

Page 364 Practice Quiz 11. 2a3 � 6a2 � 5a � 13. Sample answer: maximum 5. �3, 3, �i�3�, i�3�at x � �2, minimum at x � 0.5

Pages 368–370 Lesson 7-41. Sample answer: f(x) � x2 � 2x � 3 3. dividend: x3 �6x � 32; divisor: x � 2; quotient: x2 � 2x � 10; remainder:12 5. 353, 1186 7. x � 1, x � 2 9. x � 2, x2 � 2x � 411. $2.894 billion 13. �9, 54 15. 14, �42 17. �19, �24319. 450, �1559 21. x � 1, x � 2 23. x � 4, x � 1

25. x � 3, x � �12

� or 2x � 1 27. x � 7, x � 4

29. x � 1, x2 � 2x � 3 31. x � 2, x � 2, x2 � 1

33. 3 35. 1, 4 37.

39. 7.5 ft/s, 8 ft/s, 7.5 ft/s 41. By the Remainder Theorem,the remainder when f(x) is divided by x � 1 is equivalent tof(1), or a � b � c � d � e. Since a � b � c � d � e � 0, theremainder when f(x) is divided by x � 1 is 0. Therefore, x � 1 is a factor of f(x). 43. $16.70 45. No, he will stillowe $4.40. 47. D 49. (x2)2 � 8(x2) � 4 51. not possible53. Sample answer: rel. max. and x � �1 and x � 1.5, rel. min. at x � 1

55. (4, �2) 57. A

59. S 61. �9 �6�57��

Pages 375–377 Lesson 7-51. Sample answer: p(x) � x3 � 6x2 � x � 1; p(x) has either 2 or 0 positive real zeros, 1 negative real zero, and 2 or 0imaginary zeros. 3. 6 5. �7, 0, and 3; 3 real 7. 2 or 0; 1;

2 or 4 9. 2, 1 � i, 1 � i 11. 2 � 3i, 2 � 3i, �1 13. ��83

�; 1

real 15. 0, 3i, �3i; 1 real, 2 imaginary 17. 2, �2, 2i, and�2i; 2 real, 2 imaginary 19. 2 or 0; 1; 2 or 0 21. 3 or 1; 0;2 or 0 23. 4, 2, or 0; 1; 4, 2, or 0 25. �2, �2 � 3i, �2 � 3i

27. 2i, �2i, �2i�, ��

2i� 29. ��

32

�, 1 � 4i, 1 � 4i 31. 4 � i,

4 � i, �3 33. 3 � 2i, 3 � 2i, �1, 1 35. f(x) � x3 � 2x2 �19x � 20 37. f(x) � x4 � 7x2 � 144 39. f(x) � x3 � 11x2 �23x � 4541a. 41b.

41c.

43. 1 ft 45. radius � 4 m, height � 21 m 47. �24.1, �4.0,0, and 3.1

[�30, 10] scl: 5 by [�20, 20] scl: 5

O

f (x )

x

O

f (x )

xO

f (x )

x

xO

f (x)

�2�4

�4

2 4

4

8

f (x) � �x 4 � 2x 3 � 3x 2 � 7x � 4

xO

f (x) � x 3 � 2x 2 � 4x � 6

f (x)

�2�4

�4

�8

2 4

4

8

xO

f(x) � x 3 � 4x 2 � x � 5

f (x)


�2 �21

�1 �1

0 5

1 3

2 �1

3 �1

4 9

5 35

x f(x)

1 �14 69 �140 100

1 �9 24 �20 0

5 �45 120 �1005

Sele

cted A

nsw

ers

49. Sample answer: f(x) � x3 � 6x2 � 5x � 12 and g(x) �2x3 � 12x2 � 10x � 24; each have zeros at x � 4, x � �2,and x � 3.51. If the equation models the level of a medication in apatient’s bloodstream, a doctor can use the roots of theequation to determine how often the patient should takethe medication to maintain the necessary concentration inthe body. Answers should include the following.• A graph of this equation reveals that only the first

positive real root of the equation, 5, has meaning for thissituation, since the next positive real root occurs after themedication level in the bloodstream has dropped below 0 mg. Thus according to this model, after 5 hours there isno significant amount of medicine left in the bloodstream.

• The patient should not go more than 5 hours beforetaking their next dose of medication.

53. C 55. �254, 915 57. min.; �13 59. min.; �7

61. (6p � 5)(2p � 9) 63. � 65. � 67. ��

12

�, �1, ��52

�, �5 69. ��19

�, ��13

�, �1, �3

Pages 380–382 Lesson 7-61. Sample answer: You limit the number of possible solutions.

3. Luis; Lauren found numbers in the form �pq

�, not �pq

� as Luis

did according to the Rational Zero Theorem. 5. �1, �2,

��12

�, ��13

�, ��16

�, ��23

� 7. �2, �4, 7 9. �2, 2, �72

� 11. 10 cm

11 cm 13 cm 13. �1, �2, �3, �6 15. �1, �2, �3, �6,

�9, �18 17. �1, ��13

�, ��19

�, �3, �9, �27 19. �1, �1, 2

21. 0, 9 23. 0, 2, �2 25. �2, �4 27. �12

�, ��13

�, �2

29. ��12

�, �13

�, �12

�, �34

� 31. �45

�, 0, 33. �1, �2, 5, i, �i

35. 2, �3 � i�3�; 2 37. V � 2h3 � 8h2 � 64h

39. V � �13

��3 � 3�2 41. � � 30 in., w � 30 in., h � 21 in.

43. The Rational Zero Theorem helps factor large numbersby eliminating some possible zeros because it is notpractical to test all of them using synthetic substitution.Answers should include the following.• The polynomial equation that represents the volume of

the compartment is V � w3 � 3w2 � 40w. • Reasonable measures of the width of the compartment

are, in inches, 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 22, 28, 33, 36, 42, 44, 63, 66, 77, and 84. The solution shows that w � 14 in., � � 22 in., and d � 9 in.

45. Sample answer x5 � x4 � 27x3 � 41x2 � 106x � 12047. �4, 2 � i, 2 � i 49. �7, 5 � 2i, 5 � 2i 51. x � 4,

3x2 � 2 53. �3xy�2x� 55. 6 cm, 8 cm, 10 cm57. 4x2 � 8x � 3 59. x5 � 7x4 � 8x3 � 106x2 � 85x � 25

61. x2 � x � 4 � �x �

51

�

Page 382 Practice Quiz 21. �930, �145 3. x4 � 4x3 � 7x2 � 22x � 24 � 0 5. ��

32

�

Pages 386–389 Lesson 7-71. Sometimes; sample answer: If f(x) � x � 2, g(x) � x � 8,then f ° g � x � 6 and g ° f � x � 6.3. Danette; [g ° f ](x) � g[ f(x)] means to evaluate the ffunction first and then the g function. Marquan evaluated the

functions in the wrong order. 5. x2 � x � 1; x2 � x � 7;

x3 � 4x2 � 3x � 12; �xx

2

��

43

�, x � 4 7. {(2, �7)}; {(1, 0), (2, 10)}

9. x2 � 11; x2 � 10x � 31 11. 11 13. p(x) � �34

�x; c(x) � x � 5

15. $33.74; price of CD when coupon is subtracted and then

25% discount is taken 17. 2x; 18; x2 � 81; �xx

��

99

�, x � 9

19. 2x2 � x � 8; 2x2 � x � 8; �2x3 � 16x2; �82�x2

x�, x � 8

21. �x3 �

x �x2

1� 1

�, x � �1; �x3 � x

x

2

��

12x � 1�, x � �1; x2 � x,

x � �1; �x3 � x2

x� x � 1�, x � 0 23. {(1, �3), (�3, 1), (2, 1)};

{(1, 0), (0, 1)} 25. {(0, 0), (8, 3), (3, 3)}; {(3, 6), (4, 4), (6, 6), (7, 8)} 27. {(5, 1), (8, 9)}; {(2, �4)} 29. 8x � 4; 8x � 131. x2 � 2; x2 � 4x � 4 33. 2x3 � 2x2 � 2x � 2; 8x3 � 4x2 �2x � 1 35. �12 37. 39 39. 25 41. 2 43. 79 45. 22647. P(x) � �50x � 1939 49. p(x) � 0.70x; s(x) � 1.0575x51. $110.30 53. 373 K; 273 K 55. $700, $661.20, $621.78,$581.73, $541.04 57. Answers should include the following.• Using the revenue and cost functions, a new function

that represents the profit is p(x) � r(c(x)).• The benefit of combining two functions into one function

is that there are fewer steps to compute and it is lessconfusing to the general population of people readingthe formulas.

59. C 61. �1, ��12

�, ��14

�, �2, �3, ��32

�, ��34

�, �6 63. x3 � 4x2 �

17x � 60 65. 6x3 � 13x2 � 9x � 2 67. x3 � 9x2 � 31x � 39

69. 10 � 2j 71. � 73. ��12

�� 75. ��

116�� 77. y � �

1��

54xx2

� 79. t � �pIr�

81. m � �GF

Mr2�

Pages 393–394 Lesson 7-81. no 3. Sample answer: f(x) � 2x, f�1(x) � 0.5x; f[f�1(x)]� f�1[f(x)] � x 5. {(4, 2), (1, �3), (8, 2)}7. f�1(x) � �x 9. y � 2x � 10

11. no 13. 15.24 m/s2 15. {(8, 3), (�2, 4), (�3, 5)}17. {(�2, �1), (�2, �3), (�4, �1), (6, 0)}19. {(8, 2), (5, �6), (2, 8), (�6, 5)}

21. g�1(x) � ��12

�x 23. g�1(x) � x � 4

xO 2 4

4

2

�4

�2

�2�4

g(x)

g�1(x) � x � 4

g(x ) � x � 4xO 2 4

4

2

�2

�2�4

g(x)

g(x ) � �2x

g�1(x ) � � x12

x

y

O 4 8 12

8

12

4

�4 y �1 � 2x �10

y � x � 512

xO 2 4

4

2

�4

�2

�2�4

f (x)

f �1(x ) � �xf (x) � �x

�22

�5�3

�2�4

�1�3

�21

3�1

5 � i�3��

�89

�16

298

16

2�4

9

�33

�2


Sele

cted A

nsw

ers


25. y � ��12

�x � �12

� 27. f�1(x) � �85

�x

29. f�1(x) � �54

�x � �345� 31. f�1(x) � �

87

�x � �47

�

33. no 35. yes 37. yes 39. y � �12

�x � �121� 41. I(m) � 320 �

0.04m; $4500 43. It can be used to convert Celsius toFahrenheit.

45. Inverses are used to convert between two units ofmeasurement. Answers should include the following.• Even if it is not necessary, it is helpful to know the

imperial units when given the metric units because mostmeasurements in the U.S. are given in imperial units so itis easier to understand the quantities using our system.

• To convert the speed of light from meters per second tomiles per hour,

f(x) � �3.0

1 s1e0c

8

omnd

eters��

36010

hse

ocuornds

� � �16010

mm

ieleters

�

� 675,000,000 mi/hr

47. B 49. g[h(x)] � 6x � 10; h[g(x)] � 6x 51. �7, �2, 3

53. 64 55. 3 57. 117 59. �7 61. �245�

Pages 397–399 Lesson 7-91. In order for it to be a square root function, only thenonnegative range can be considered. 3. Sample answer:

y � �2x � 4�5. 7.

D: x � 0; R: y � 0 D: x � 1; R: y � 3

9. 11.

13. Yes; sample answer: 15.The advertised pump will reach a maximum height of 87.9 ft.

D: x � 0, R: y � 0

17. 19.

D: x � 0, R: y � 0 D: x � 7, R: y � 0

21. 23.

D: x � 0.6, R: y � 0 D: x � �4, R: y � 5

25. 27.

D: x � 0.75, R: y � 3

y

Ox

8

6

4

2

2�4 �2

y � �x � 5

y

O x

8

6

4

2

�3 �2 �1

y � 2�3 � 4x � 3

y

Ox

8

6

4

2

2 4�2�4

y � 5 ��x � 4

y

O x

87654321

1 2 3 4 5 6 7 8

y � �5x � 3

y

O x

87654321

1 2 3 4 5 6 7 8

y � �x � 7

y

O x

87654321

1 2 3 4 5 6 7 8

y � �x 12

1 2 3 4 5 6 7 8

y � ��5x

�1�2�3�4�5�6�7�8

yO x

y

O x

4321

1 2

�2�3�4

3 4 5 6�2

y � �x � 2 � 1

2

4

6

8

2�2 4 6

y � �2x � 4

y

O x

y

O x

87654321

1 2 3 4 5 6 7 8

y � �x � 1 � 3

y

O x

87654321

1 2 3 4 5 6 7 8

y � �4x

xO 2 4

4

2

�4

�2

�2�4

f �1(x) � x � 87

47

f (x)

f (x) � 7x � 48

f (x)x

f (x) � x � 745

f (x)�1 � x � 54

354

�30 �20 �10

�10

�20

�30

�40

�40 O

xO 2 4

4

2

�4

�2

�2�4

f (x)

f �1(x) � x85

f (x) � x58

xO 2 4

4

�4

�2

�2�4

y

y �1 � � x � 12

12

y � �2x � 1

Sele

cted A

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29. 31.

33. 317.29 mi37. Square root functions are used in bridge design becausethe engineers must determine what diameter of steel cableneeds to be used to support a bridge based on its weight.Answers should include the following.• Sample answer: When the weight to be supported is less

than 8 tons.• 13,608 tons39. D 41. no 43. 2x � 2; 8; x2 � 2x � 15; �x

x��

53

�, x � 3

45. , x � ��32

�; ,

x��32

�; 2x � 3, x � ��32

�; 8x3 � 12x2 � 18x � 27, x � ��32

�

47. 2x2 � 4x � 16 49. a3 � 1

Pages 400–404 Chapter 7 Study Guide and Review1. f 3. a 5. e 7. �6; x � h � 2 9. �21; 6x � 6h � 311. 20; x2 � 2xh � h2 � x � h13a. 13b. at x � 3

13c. Sample answer: rel. max. at x � �1.4, rel. min. at x � 1.4

15a. 15b. between �2 and �315c. Sample answer: rel. max. at x � �1.6, rel. min. at x � 0.8

17a. 17b. between �2 and �1,between 0 and 1, andbetween 1 and 217c. Sample answer: rel. max. at x � �1, rel. min. at x � 0.9

19. �53

�, �3, 0 21. 4, �2 � 2i�3� 23. 2, �2 25. 4, �1

27. 20, �20 29. x2 � 2x � 3 31. 1; 0; 2 33. 3 or 1; 1; 0 or 2 35. 2 or 0; 2 or 0; 4, 2, or 0 37. �1, �1

39. 1, 2, 4, �3 41. �12

�, 2 43. x2 � 1; x2 � 6x � 11

45. �15x � 5; �15x � 25

47. x � 4 ; x � 4 49. f�1(x) � ��x

2� 3�

51. f�1(x) � �2x

��3

1� 53. y�1 � ��

12

��x� � �32

�

55. D: x � �35

�, R: y � 0 57.

Chapter 8 Conic SectionsPage 411 Chapter 8 Getting Started

1. {�4, �6} 3. ��32

�, �4� 5a. � 5b. � 5c. � 7. y

xO

x �y � 3

4�5

9�3

3�1

5�3

5�3

5�3

�1�2

40

�22

x21 3 4 5 6 7 8

56

4321

y

O

�2

y � �x � 2

x21 3 4 5 6 7 8

78

654

1

32

y

O

y � �5x � 3

xO 2 4

4

2

�2

�2�4

y

y � (2x � 3)2

y � � �x �12

32

xO 2 4

4

2

�2

�2�4

f (x)

f (x) � �3x � 12

f �1(x) � 2x � 1�3

xO 2 4

2

�4

�2

�2�4

f �1(x) � �x � 32

f (x) � �2x � 3

f (x)

xO

r (x)

r (x ) � 4x 3 � x 2� 11x � 3

xO 4 8

4

�8

�4

�4�8

p (x)

p (x ) � x 5 � x 4 � 2x 3 � 1

xO 4 8

�4

�8

�12

�4�8

h (x)

h (x) � x 3 � 6x � 9

8x3 � 12x2 � 18x � 28��

2x � 38x3 � 12x2 � 18x � 26��

2x � 3

y

O x

87654321

1 2 3 4 5 6 7 8

y � �6x � 2 � 1

y

O x

87654321

1 2 3 4 5 6 7 8

y � �5x � 8

Sele

cted A

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ers


Pages 414–416 Lesson 8-11. Since the sum of the x-coordinates of the given points isnegative, the x-coordinate of the midpoint is negative. Sincethe sum of the y-coordinates of the given points is positive,the y-coordinate of the midpoint is positive. Therefore, themidpoint is in Quadrant II. 3. Sample answer: (0, 0) and (5, 2) 5. (2.5, 2.25) 7. �122� units 9. D 11. (�4, �2)

13. ��127�, �

227� 15. (3.1, 2.7) 17. ��

214�, �

58

� 19. (7, 11)

21. Sample answer: Draw several line segments across theU.S. One should go from the northeast corner to thesouthwest corner; another should go from the southeastcorner to the northwest corner; another should go across themiddle of the U.S. from east to west; and so on. Find themidpoints of these segments. Locate a point to represent allof these midpoints.25. 25 units 27. 3�17� units 29. �70.25� units 31. 1 unit

33. ��

18213�

� units 35. 7�2� � �58� units, 10 units2

37. �130� units 39. about 0.9 h 41. The slope of the line

through (x1, y1) and (x2, y2) is �yx

2

2

�

�

yx

1

1� and the point-slope

form of the equation of the line is y � y1 � �yx

2

2

�

�

yx

1

1�(x � x1).

Substitute ��x1 �

2x2�, �

y1 �

2y2� into this equation. The

left side is �y1 �

2y2� � y1 or �

y2 �

2y1�. The right side is

�yx

2

2

�

�

yx

1

1��x1 �

2x2� � x1 � �

yx

2

2

�

�

yx

1

1��x2 �

2x1� or �

y2 �

2y1�. Therefore,

the point with coordinates ��x1 �

2x2�, �

y1 �

2y2� lies on the line

through (x1, y1) and (x2, y2).

The distance from ��x1 �

2x2�, �

y1 �

2y2� to (x1, y1) is

��x1 ��x1 �

2x�2�

2� ��y1 � �

y�1 �

2y2��2� or

��x1 �

2x�2�

2� ��

y1 �

2y�2�

2�. The distance from ��x1 �

2x2�, �

y1 �

2y2�

to (x2, y2) is

��x2 ��x1 �

2x�2�

2� ��y2 � �

y�1 �

2y2��2� � ��x2 �

2x�1�

2� ��

y2 �

2y�1��2

or ��x1 �

2x�2�

2� ��

y1 �

2y�2�

2�. Therefore the point with

coordinates ��x1 �

2x2�, �

y1 �

2y2� is equidistant from (x1, y1) and

(x2, y2). 43. C 45. on the line with equation y � x

47. D � {x x � 2}, 49. D � {x x � 0}, R � {y y � 0} R � {y y � 1}

51. �1 � 13i 53. 4 � 3i 55. y � (x � 2)2 � 357. y � 3(x � 1)2 � 2 59. y � �3(x � 3)2 � 17


1. (3, �7), �3, �6�1156�, x � 3, y � �7�

116� 3. When she added 9

to complete the square, she forgot to also subtract 9. Thestandard form is y � (x � 3)2 � 9 � 4 or y � (x � 3)2 � 5.

5. (3, �4), �3, �3�34

�, x � 3, 7. ��43

�, ��23

�, ��43

�, ��34

�, x � ��43

�,

y � �4�14

�, upward, 1 unit y � ��172�, downward, �

13

� unit

9. y � �18

�(x � 3)2 � 6 11. x � �214�y2 � 6

13. x � (y � 7)2 � 29

15. x � 3�y � �56

�2� 11�

112�

17. (0, 0), ��12

�, 0, y � 0, 19. (1, 4), �1, 3�12

�, x � 1,

x � ��12

�, right, 2 units y � 4�12

�, downward, 2 units

21. (4, 8), (3, 8), y � 8, x � 5, left, 4 units

161412108642

�4 �3�2�1 1 2 3 4

y

xO

(y � 8)2 � �4(x � 4)

y

x

O

�2(y � 4) � (x � 1)2

y

xO

y 2 � 2x

y

xO

y � (x � 3) 2 � 618

y

xO

y � �3x2 � 8x � 6y

xO

y � (x � 3) 2 � 4

y

xO

y � 2�x � 1

y

xO

y � �x � 2

Sele

cted A

nsw

ers


23. (�24, 7), ��23�34

�, 7, y � 7, 25. (4, 2), �4, 2�112�, x � 4,

x � �24�14

�, right, 1 unit y � 1�1121�, upward, �

13

� unit

27. ��147�, �

34

�, ��6176�, �

34

�, y � �34

�, 29. (123, �18), �122�14

�, �18, y �

x � �61

96�, left, �

14

� unit �18, x � 123�34

�, left, 3 units

31. 1 33. y � ��23

� 35. 0.75 cm

37. x � ��214�(y � 6)2 � 8 39. y � �

116�(x � 1)2 � 7

41. x � �14

�(y � 3)2 � 4

43. about y � �0.00046x2 � 325 45. y � ��26,

1200�x2 � 6550

47. A parabolic reflector can be used to make a carheadlight more effective. Answers should include the following.• Reflected rays are focused at that point.• The light from an unreflected bulb would shine in all

directions. With a parabolic reflector, most of the light canbe directed forward toward the road.

49. A 51. 10 units

53.

55. 4 57. 9 59. 2�3� 61. 4�3�

Pages 428–431 Lesson 8-31. Sample answer: (x � 6)2 � (y � 2)2 � 16 3. Lucy; 36 isthe square of the radius, so the radius is 6 units.5. (x � 1)2 � (y � 5)2 � 4 7. (x � 3)2 � (y � 7)2 � 9

9. (0, 14), �34� units

11. ��23

�, �12

�, �2�3

2�� unit 13. (�2, 0), 2�3� units

15.

17. (x � 2)2 � (y � 1)2 � 4 19. (x � 8)2 � (y � 7)2 � �14

�

21. (x � 1)2 � �y � �12

�2� �

19445�

23. �x � �13�2 � (y � 42)2 � 177725. (x � 4)2 � (y � 2)2 � 4 27. (x � 5)2 � (y � 4)2 � 2529. (x � 2.5)2 � (y � 2.8)2 � 1600

y

x

Earth

Satellite

6400km 42,200

km

35,800km

y

xO

(x � 2)2 � y 2 � 12

y

xO

(x � )2 � (y � )2

�23

12

89

O�16 �8 8 16

�8

24

16

8

y

x

x 2 � (y � 14)2 � 34

y

xO y ��x � 1

y

xO

x � (y � 3)2� 414

�2�4�6�8

�4 �3�2�1 1 2 3 4 5 6

8642

y

xO

y � (x � 1)2� 7116

141210

1 2 3 4 5 6 7 8�2

8642

y

xO

x � � (y � 6)2� 8124

20

�20

�40

�60

�120 �60 60 120

y

xO

x � � y 2 � 12y � 1513

y

xO

x � �4y 2 � 6y � 2

y

xO

y � 3x 2 � 24x � 50

24

16

8

�8

�24 �16 �8 8

y

xO

x � y 2 � 14y � 25

Sele

cted A

nsw

ers


31. (0, 0), 12 units

33. (�3, �7), 9 units

35. (3, �7), 5�2� units

37. ��2, �3�, �29� units

39. (0, 3), 5 units

41. (9, 9), �109� units

43. ��32

�, �4, �3�2

17�� units 45. (�1, �2), �14� units

47. �0, ��92

�, �19� units

49. (x � 1)2 � (y � 2)2 � 5 51. A 53. y � ��16 � (�x � 3)�2�55.

57. (1, 0), ��1121�, 0, y � 0, 59. (�2, �4), ��2, �3�

34

�, x � 1�

112�, left, �

13

� unit x � �2, y � �4�14

�, upward, 1 unit

61. (�1, �2) 63. �4, �2, 1 65. 28 in. by 15 in. 67. 669. 25 71. 2�2�

y

xO

y � x 2 � 4x

y

xO

x � �3y 2 � 1

[�10, 10] scl:1 by [�10, 10] scl:1

yxO

4x 2� 4y 2 � 36y � 5 � 0

y

xO

x 2� y 2 � 2x � 4y � 9

42

�2�4�6�8

�10�12

�6�4�2 2 4 6 8 10

y

xO

x 2� y 2 � 3x � 8y � 20

x 2� y 2� 18x � 18y � 53 � 018

161412108642

�2�2 2 4 6 8 10 12 14 16 18

y

xO

y

xO

x 2� y 2 � 6y � 16 � 0

y

xO

x 2� (y � ��3)2 � 4x � 25

2

�6�4�2 2 4 6 8 10�2�4�6�8

�10�12�14

y

xO

(x � 3)2 � (y � 7)2 � 50

42

�12�10�8�6�4�2 2 4 6 8�2�4�6�8

�10�12�14�16

O

y

x

(x � 3)2 � (y � 7)2 � 81

8 16�16 �8

16

8

�8

�16

y

xO

x2 � y 2 � 144

Sele

cted A

nsw

ers



1. 13 units

3. (0, 0), �1�12

�, 0, y � 0, 5. (0, 4), 7 units

x � �1�12

�, right, 6 units

Pages 437–440 Lesson 8-41. x � �1, y � 2 3. Sample answer: �(x �

42)2

� � �(y �

15)2

� � 1

5. �(y �

364)2

� � �(x �

42)2

� � 1

7. (0, 0): (0, �3); 6�2�; 6

9. (0, 0); (�2, 0); 4�2�; 4

11. about �1.32x

2

1015� � �1.27

y

2

1015� � 1 13. �1x6

2� � �

y7

2� � 1

15. �1y6

2� � �

(x �4

2)2� � 1 17. �

(y �

644)2

� � �(x �

42)2

� � 1

19. �(x �64

5)2� � � 1 21. �

1x62

9� � �

2y5

2� � 1

23. about �2.02x

2

1016� � �2.00

y

2

1016� � 1 25. �2y0

2� � �

x42� � 1

27. (0, 0); �0, ��5�; 2�10�; 2�5�

29. (�8, 2); ��8 � 3�7�, 2; 24; 18

31. (0, 0); ��6�, 0; 6; 2�3�

33. (0, 0); �0, ��7�; 8; 6

35. (�3, 1); (�3, 5), (�3, �3); 4�6�; 4�2�

37. (2, 2); (2, 4), (2, 0); 2�7�; 2�3�

39. �1x22� � �

y9

2� � 1 41. C 43. about �1.35

x

2

1019�� 1.26

y

2

1019�� 1

45. (x � 4)2 � (y � 1)2 � 101 47. (x � 4)2 � (y � 1)2 � 16

y

xO

y

xO

y

xO

16x 2 � 9y 2 � 144

y

xO

3x 2 � 9y 2 � 27

8�8�16�24

16

8

�8

�16

y

xO

(x � 8)2 144

(y � 2)2 81

� � 1

y

xO

y 2

10 x 2

5 � � 1

(y � 4)2�

�841�

y

xO

4x 2 � 8y 2 � 32

y

xO

y 2

18 x 2

9 � � 1

12108642

�2�4

�8�6�4�2 2 4 6 8

y

xO

x 2 � (y � 4)2 � 49

y

xO

y 2 � 6x

Sele

cted A

nsw

ers


49.

51. Sample answer: 128,600,00053. 55.

57.

Pages 445–448 Lesson 8-51. sometimes 3. Sample answer: �

x42� � �

y9

2� � 1

5. �x12� � �

1y5

2� � 1

7. �1, �6 � 2�5�; 9. �4 � 2�5�, �2;

�1, �6 � 3�5�; �4 � 3�5�, �2;

y � 6 � ��2�

55�

�(x � 1) y � 2 � ��2

5��(x � 4)

11. �x42� � �

1y2

2� � 1 13. � �

x62� � 1

15. �2x52� � �

3y6

2� � 1 17. �(x �

492)2

� � �(y �

43)2

� � 1

19. �1x62� � �

y9

2� � 1

21. (�9, 0); ��130�, 0; y � ��79

�x

23. (0, �4); �0, ��41�; 25. ��2�, 0; ��3�, 0; y � ��

45

�x y � ��2

2��x

27. (0, �6); �0, �3�5�; 29. (�2, 0), (�2, 8); (�2, �1),

y � �2x (�2, 9); y � 4 � ��43

�(x � 2)

31. (�3, �3), (1, �3); 33. �1, �3 � 2�6�; ��1 ��13�, �3; �1, �3 � 4�2�;

y � 3 � ��32

�(x � 1) y � 3 � ��3�(x � 1)

AA08 10 C A2BLAC

x

y

�4�2

�6�8

�10

2 4 6 8

642

�2�4�6�8 O

y2 � 3x2 � 6y � 6x � 18 � 0

x

y

O

� 1 (x � 1)2

4 �(y

� 3)2

9 � 1 �

4

8

4

12

�4

�4

�8O

x

y

� 1 (y � 4)2

16(x

� 2)2

9 � 1 ��4

�12

168

8

16

�8�16 O x

y

y 2 � 36 � 4x 2

x

y

O

x 2 � 2y 2 � 2

x

y

�4�2

�6�8

2 6 84

42

68

�2�4�6�8 O

� 1 y 2

16 �x 2

25

x

y

�8�4

�12�16

4 12 168

84

1216

�4�8�12�16 O

� 1 x 2

81 �y 2

49

�y � �121�2

��245�

x

y

�8�4

�12�16

4 12 16 208

84

1216

�12�8�4 O

yxO

� 1 (y � 6)2

20 �(x � 1)2

25

y

xOy � 2 � �2(x � 1)

y

xO

y � x12

y

xOy � �2x

Peo

ple

(m

illio

ns)

00 2 4 6 8 10 12 14 16 18 20

120

118

116

114

112

110

108

106

104

0

Years Since 1980

Married Americans

Sele

cted A

nsw

ers


35. �1.1

x02

25� � �

7.8y9

2

75� � 1 37. 120 cm, 100 cm

39. about 47.32 ft 41. C

43. 45.

47. �(x �16

5)2� � �

(y �

12)2

� � 1 49. �(x �25

1)2� � �

(y �

94)2

� � 1

51. �4, �2 53. � 55. about 5,330,000 subscribers

per year 57. 2x � 17y 59. 1, �2, 9 61. 5, 0, �263. 0, 1, 0


1. �(y �

811)2

� � �(x �

323)2

� � 1

3. (�1, 1); ��1, 1 � �11�; 8; 2�5�

5. �(x �16

2)2� � �

(y �

52)2

� � 1

Pages 450–452 Lesson 8-61. Sample answer: 2x2 � 2y2 � 1 � 0 3. The standard formof the equation is (x � 2)2 � (y � 1)2 � 0. This is anequation of a circle centered at (2, �1) with radius 0. Inother words, (2, �1) is the only point that satisfies theequation.

5. �1y6

2� � �

x82� � 1, hyperbola 7. �(x �

41)2

� � �(y �

13)2

� � 1, ellipse

9. ellipse

11. 13. �y4

2� � �

x22� � 1, ellipse

15. �x42� � �

y1

2� � 1, hyperbola 17. y � (x � 2)2 � 4, parabola

19. (x � 2)2 � (y � 3)2 � 9, 21. �(x �32

4)2� � �

3y2

2� � 1,

circle hyperbola

23. x2 � (y � 4)2 � 5, circle 25. �x42� � �

(y �

31)2

� � 1, ellipse

27. y � �(x � 4)2 � 7, 29. �(x �25

3)2� � �

(y �

91)2

� � 1,

parabola ellipse

31. hyperbola 33. circle 35. parabola 37. ellipse

y

xO

4�8�12 �4

�4

�8

�12

�16

yxO

y

xO

y

xO

y

xO 2 4�8�6�12�10 �4�2

8642

�2�4�6�8

y

xO

y

xO

y

xO

y

xO

y

xO2 4 6 8 10 12 14

108642

�2�4�6

y

xO

x

y

O2 4 6 8�8�6�4�2

8642

�2�4�6�8

x

y

O

020

�75

y

xO

xy � �2

x

y

O

xy � 2

Sele

cted A

nsw

ers

39. parabola 41. b 43. c 45. The plane should bevertical and contain the axis of the double cone.

47. D 49. 0 � e � 1, e � 1 51. �(x �9

3)2� � �

(y �

46)2

� � 1

53. x12 55. �xy

7

4� 57. (2, 6) 59. (0, 2)

Pages 458–460 Lesson 8-71a. (�3, �4), (3, 4)

1b. (�1, 4)

3. Sample answer: x2 � y2 � 40, y � x2 � x5. (�4, �3), (3, 4) 7. (1, �5), (�1, �5)9.

11. (2, 4), (�1, 1) 13. ��1 � �17�, 1 � �17�, ��1 � �17�, 1 � �17� 15. ��5�, �5�, ��5�, ��5�17. (5, 0), (�4, �6) 19. (�8, 0) 21. no solution

23. (�5, 5), (�5, 1), (3, 3) 25. ��53

�, ��73

�, (1, 3) 27. 0.5 s

29. ��40 �524�5��, �45 �

512�5�� 31. No; the comet and Pluto

may not be at either point of intersection at the same time.

33.

35.

37. 39. none 41. none

43. Systems of equations can be used to represent thelocations and/or paths of objects on the screen. Answersshould include the following.• y � 3x, x2 � y2 � 2500• The y-intercept of the graph of the equation y � 3x is 0, so

the path of the spaceship contains the origin.

• ��5�10�, �15�10� or about (�15.81, �47.43)

45. B 47. Sample answer: x2 � y2 � 36, �(x �16

2)2� � �

y4

2� � 1

49. Sample answer: x2 � y2 � 81, �x42� � �

1y0

2

0� � 1

51. impossible

53. �(y �

93)2

� � �x42� � 1, ellipse

55. �7, 0 57. �7, 3 59. ��43

� 61a. 40

61b. two real, irrational 61c. � ��510�� 63. 2 � 9i

65. �85

� � �15

�i 67. 6 69. �51 71. y � 3x � 2

Pages 461–466 Chapter 8 Study Guide and Review1. true 3. true 5. true 7. true 9. False; the midpoint

formula is given by ��x1 �

2x2�, �

y1 �

2y2�. 11. ��

52

�, 413. ��

1470�, ��

4430� 15. �290� units

y

xO

y

xO

y

O x

y

O x

y

O x

y

O x

y � 5 � x 2

y � 2x 2 � 2

y

Ox

4x � 3y � 0

x 2 � y 2 � 25


Sele

cted A

nsw

ers

17. (1, 1); (1, 4); x � 1; y � �2; upward; 12 units

19. (4, �2); (4, �4); x � 4; y � 0; downward; 8 units

21. y � ��18

�x2 � 1

23. (x � 4)2 � y2 � �196� 25. (x � 1)2 � (y � 2)2 � 4

27. (�5, 11); 7 units

29. (�3, 1); 5 units

31. (0, 0); (0, �3); 10; 8

33. (1, �2); �1 � �3�, �2; 4; 2

35. (0, �2); �0, ��13�; y � ��23

�x

37. (0, �4); (0, �5); y � ��43

�x

39. y � (x � 2)2 � 4; parabola

x

y

O

x 2 � 4x � y � 0

x

y

9y 2 � 16x 2 � 1448

1216

8 12 16�16

�16

�8

�12�8

x

y y 2

4 x 2

9� � 1

x

y

O

x 2 � 4y 2 � 2x � 16y � 13 � 0

y

xO

x 2

16 y 2

25�

�8 8

4

8

�4

�8

� 1

y

xO�4 6

4

8

�4

�8x 2 � y 2 � 6x � 2y � 15 � 0

(�3, 1)

(�5, 11)

�12 �6�18 63

3

9

15

21

x

y

O

(x � 5)2 � (y � 11)2 � 49

y

xO

y � � x 2 � 118

y

xO

x2 � 8x � 8y � 32 � 0

y

xO

(x � 1)2 � 12(y � 1)


Sele

cted A

nsw

ers


41. �y4

2� � �

(x �1

1)2� � 1; hyperbola

43. ellipse 45. circle 47. (6, �8), (12, �16)49.

Chapter 9 Rational Expressions and Equations


1. �16

� 3. �58

� 5. 16 7. 2�12

� 9. 1�214�

11. 13. 12 15. 15 17. 15

19. 6 21. 7�12

�


1. Sample answer: �46

�, �46((xx

��

22

))

� 3. Never; solving the

equation using cross products leads to 15 � 10, which is

never true. 5. �a �

1b

� 7. �230cb

� 9. �65

� 11. cd2x 13. D

15. ��7nm

2� 17. �

3s

� 19. �12

� 21. �2aa��

11

� 23. ��247bca

� 25. �2p2

27. �xb2y

3

2� 29. �43

� 31. 1 33. �ww

��

34

� 35. �(a �2(a

2)�(a

5�)

2)�

37. �2p 39. �22

xx

�

�

yy

� 41. �43

� 43. a � �b or b

45. �1638,21729

��

ma

� 47. (2x2 � x � 15) m2

49. A rational expression can be used to express the fractionof a nut mixture that is peanuts. Answers should includethe following.• The rational expression �1

83

��

xx

� is in simplest form because

the numerator and the denominator have no commonfactors.

• Sample answer: �138�

�x

x� y

� could be used to represent the

fraction that is peanuts if x pounds of peanuts and y pounds of cashews were added to the original mixture.

51. A 53. ��17�, �2�2�

55. �(x �9

7)2� � �

(y �

12)2

� � 1; hyperbola

57. odd; 3 59. {�1, 4} 61. {0, 5}

63. � 65. �1�19

� 67. 1�145�

69. ��1181�

Pages 481–484 Lesson 9-21. Catalina: you need a common denominator, not acommon numerator, to subtract two rational expressions.3a. Always; since a, b, and c are factors of abc, abc is always a common denominator of �

1a

� � �1b

� � �1c

�. 3b. Sometimes; if a,

b, and c have no common factors, then abc is the LCD of

�1a

� � �1b

� � �1c

�. 3c. Sometimes; if a and b have no common

factors and c is a factor of ab, then ab is the LCD of �1a

� � �1b

� � �1c

�.

3d. Sometimes; if a and c are factors of b, then b is the

LCD of �1a

� � �1b

� � �1c

�. 3e. Always; since �1a

� � �1b

� � �1c

� � �abbcc

� �

�aabcc

� � �aabbc

�, the sum is always �bc �aabcc� ab�. 5. 80a2b3c

7. �2

x�

2yx3

� 9. �4327m� 11. �(a �

3a5�)(a

1�0

4)� 13.�2

1x3(xx2

��

1)4(xx

��

91)

� units

15. 180x2yz 17. 36p3q4 19. x2(x � y)(x � y)

21. (n � 4)(n � 3)(n � 2) 23. �1321v

� 25. �2x �

3y15y�

27. �255ba�2b2

7a3� 29. �110w

90�w

423� 31. �

aa

��

34

� 33. �(y �

y(y3)

�

(y9�

)3)

�

35. 37.�(x �x22

)�2(x

6� 3)

�

39. �(y2y

�

2 �

1)(yy

�

�

42)

� 41. �1 43. �aa

��

72

� 45. 12 ohms

47. �x

2�4

4� h 49.�(d � L

2)m2(d

d� L)2� or �(d2

2�md

L2)2�

51. Subtraction of rational expressions can be used todetermine the distance between the lens and the film if thefocal length of the lens and the distance between the lensand the object are known. Answers should include thefollowing.• To subtract rational expressions, first find a common

denominator. Then, write each fraction as an equivalentfraction with the common denominator. Subtract thenumerators and place the difference over the commondenominator. If possible, reduce the answer.

• �1q

� � �110� � �

610� could be used to determine the distance

between the lens and the film if the focal length of thelens is 10 cm and the distance between the lens and theobject is 60 cm.

53. C 55. �a(aa��

12)

�

�8d � 20��(d � 4)(d � 4)(d � 2)

4 8 12

8

4

x

y

O

� 1 � (y � 2)2

1(x � 7)2

9�4

�8

y

O x

� � 1 y 2

4 (x � 4)2

1

y

xO

x

y

O

(x � 1)2

1 y 2

4� � 1

Sele

cted A

nsw

ers


57. 59.

61.


1. �tt

��

23

� 3. ��3y2

2� 5. (w � 4)(3w � 4) 7. �

4aa��

b1

�

9. �(n �n

6�)(n

29� 1)

�


1. Sample answer: f(x) � �(x � 5)1(x � 2)� 3. x � 2 and y � 0

are asymptotes of the graph. The y-intercept is 0.5 and thereis no x-intercept because y � 0 is an asymptote.5. asymptote: x � �5; hole: x � 17.

9.

11. 13.

15. y � 0 and 0 � C � 1 17. asymptotes: x � �4, x � 219. asymptotes: x � �1, hole: x � 5 21. hole: x � 1

23. 25.

27. 29.

31. 33.

35. 37.

39.

xO

f (x)

f (x) � 1(x � 2)2

xO

f (x)

f (x) � x � 1x2 � 4

xO

f (x)

f (x) � �1(x � 2)(x � 3)

xO

f (x)

f (x) � x2 � 1x � 1

xO

f (x)

f (x) � x � 1x � 3

xO

f (x)

f(x) � 1(x � 3)2

xO

f (x)

f (x) � 5xx � 1

4

8

4 8�4

�4

�8

xO

f (x)

f (x) � � 5x � 1

2

6

4 8�4

�4

�8

�8xO

f (x)

f (x) � 3x

yO

10

6

2

�8�16

�4

8 16

C � y

y � 12

C

xO

f (x)

f(x) � x � 2x2 � x � 6

xO

f (x)

f (x) � x � 5x � 1

2

4

4 8�4

�2

�4

�8

xO

f (x)

2

4

4 8�4

�2

�4

�8

f (x) � 6(x � 2)(x � 3)

y

xO

2

2468

�8

�4

4 6�2�6

10

� � 116 25

(x � 2)2 (y � 5)2

x

y

8

2

6

�6

O

�2�8

� � 1 x 2 y 2

2016

x

y

O

x 2 � y � 4

(y � 3)2 � x � 2

Sele

cted A

nsw

ers


41. The graph is bell-shaped with a horizontal asymptote atf(x) � 0.43. 45. about �0.83

m/s

47. 49. It represents heroriginal free-throwpercentage of 60%.

51. A rational function can be used to determine how mucheach person owes if the cost of the gift is known and thenumber of people sharing the cost is s. Answers shouldinclude the following.

• • Only the portion in thefirst quadrant is significantin the real world becausethere cannot be a negativenumber of people nor anegative amount of moneyowed for the gift.

53. B 55. �(x �3x

3�)(x

1�6

2)�

57. (6, 2); 5 59. $65,892

61. {�12, 10} 63. 4.5 65. 20

Pages 495–498 Lesson 9-41a. inverse 1b. direct 3. Sample answers: wages andhours worked, total cost and number of pounds of apples;distances traveled and amount of gas remaining in thetank, distance of an object and the size it appears 5. direct;�0.5 7. 24 9. �8 11. 25.8 psi13.

15. joint; 5 17. direct; 3 19. direct; �7 21. inverse; 2.523. V � kt 25. 118.5 km 27. 20 29. 64 31. 4 33. 9.6

35. 0.83 37. �16

� 39. 100.8 cm3 41. m � 20sd 43. 1860 lb

45. joint 47. I � �dk2� 49. The sound will be heard �

14

� as

intensely. 51. about 127,572 calls 53. no; d � 055. A direct variation can be used to determine the totalcost when the cost per unit is known. Answers shouldinclude the following.• Since the total cost T is the cost per unit u times the

number of units n or T � un, the relationship is a directvariation. In this equation u is the constant of variation.

• Sample answer: The school store sells pencils for 20¢ each. John wants to buy 5 pencils. What is the totalcost of the pencils? ($1.00)

57. C 59. asymptotes: x � �4, x � 3 61. �y �

xx

�

63. �mm(m

��

51)

� 65. 0.4; 1.2 67. ��35

�; 3 69. A

71. P 73. C


1. 3. 49 5. 112


1. Sample answer: This graph is a rational function.It has an asymptote at x � �1.

3. The equation is a greatest integer function. The graphlooks like a series of steps. 5. inverse variation or rational7. c9. identity or direct 11. absolute valuevariation

13. absolute value 15. rational 17. quadratic 19. b 21. g

y

xO

y � x � 2

y

xO

y � x

P

dO

xO

f (x )

f(x) � x � 1x � 4

P

dO

P � 0.43d

xO

y

(x � 6)2 � (y � 2)2� 25

sO

c � 150s50

50 100

100

�50

�50

�100

�100

c � 0

s � 0

c

O

P(x)

x

4

8

4�12 �4

�4

�8

�8

P(x ) � 6 � x10 � x

O

Vf

m1

Vf � 5m1 � 7

m1 � 7

4

8�16

12

20

�8 �4

0 0

1 0.43

2 0.86

3 1.29

4 1.72

Depth (ft) Pressure (psi)

45. about �0.83 m/s

Sele

cted A

nsw

ers


23. constant 25. square root

27. rational 29. absolute value

31. C � 4.5m 33. a line slanting to the right and passingthrough the origin

35.

37a. absolute value 37b. quadratic 37c. greatest integer37d. square root 39. C 41. 22

43.

45. (8, �1); �8, ��78

�; x � 8; y � �1�18

�; up; �12

� unit

47. (5, �4); �5�34

�, �4; y � �4; x � 4�14

�; right; 3 units

49. impossible 51. ��13

�, 2 53. 1 55. ��167� 57. 45x3y3

59. 3(x � y)(x � y) 61. (t � 5)(t � 6)(2t � 1)


1. Sample answer: �15

� � �a �

22

� � 1 3. Jeff; when Dustin

multiplied by 3a, he forgot to multiply the 2 by 3a. 5. 2, 6

7. �6, �2 9. v � 0 or v � 1�16� 11. 2 13. �6, 1

15. �1 � a � 0 17. 11 19. t � 0 or t � 3 21. 0 � y � 2

23. 14 25. � 27. 7 29. ��3 �23�2�� 31. 32 33. band,

80 members; chorale, 50 members 35. 24 cm 37. 5 mL39. 6.1541. If something has a general fee and cost per unit, rationalequations can be used to determine how many units aperson must buy in order for the actual unit price to be agiven number. Answers should include the following.

• To solve �500x� 5x� � 6, multiply each side of the equation

by x to eliminate the rational expression. Then subtract 5xfrom each side. Therefore, 500 � x. A person would needto make 500 minutes of long distance minutes to makethe actual unit price 6¢.

• Since the cost is 5¢ per minute plus $5.00 per month, theactual cost per minute could never be 5¢ or less.

43. C 45. square root

47. 36 49. 2�130� 51. �137� 53. {x 0 � x � 4}

Pages 513–516 Chapter 9 Study Guide and Review1. false; point discontinuity 3. false; rational 5. true

7. ��343bac

� 9. (y � 3)(y � 6) 11. �n �

23

� 13. �7(xx��

54)

� 15. �139y�

17. ��230b�

y

O x

y � 2 x�

y

xO

3x � y 2 � 8y � 31

y

xO

2 4 6 10 12

8101214

642

�2�2

(y � 1) � (x � 8)212

f (x )

xO

f (x) � 8(x � 1)(x � 3)

y

x

Co

st (

cen

ts)

40

0

80

120

160

Ounces2 4 6 8 10

y

xO

y � 2x

y

xO

y � x2 � 1

x � 1

y

xO

y � �9x

y

xO

y � �1.5

Sele

cted A

nsw

ers


19. 21.

23.

25. �1�23

� 27. 8 29. 80 31. absolute value 33. 1�19

� 35. 3

37. 1�12

�

Chapter 10 Exponential andLogarithmic Relations


1. x12 3. ��172yx5z

3� 5. a � �14 7. y � �2

9. f �1(x) � ��12

�x 11. f �1(x) � �x � 1

13. g[h(x)] � 3x � 2; h[g(x)] � 3x � 215. g[h(x)] � x2 � 8x � 16; h[g(x)] � x2 � 4

Pages 527–530 Lesson 10-11. Sample answer: 0.8 3. c 5. b7. D � {x x is all real numbers.}, R � {y y � 0}

9. decay 11. y � 3��12

�x13. 22�7� or 4�7�

15. 33�2� or 27�2� 17. x � 0 19. y � 65,000(6.20)x

21. D � {x x is all real 23. D � {x x is all real numbers.}, R � {y y � 0} numbers.}, R � {y y � 0}

25. D � {x x is all real numbers.}, R � {y y � 0}

27. growth 29. decay 31. decay 33. y � �2��14

�x

35. y � 7(3)x 37. y � 0.2(4)x 39. 54 or 625 41. 74�2�

43. n2 � 45. n � 5 47. 1 49. ��83

� 51. n � 3 53. �3

55. 10 57. y � 100(6.32)x 59. y � 3.93(1.35)x

61. 2144.97 million; 281.42 million; No, the growth rate hasslowed considerably. The population in 2000 was muchsmaller than the equation predicts it would be.63. A(t) � 1000(1.01)4t 65. s � 4x 67. Sometimes; truewhen b � 1, but false when b � 1. 69. A

71.

The graphs have the same shape. The graph of y � 2x � 3 isthe graph of y � 2x translated three units up. Theasymptote for the graph of y � 2x is the line y � 0 and for y � 2x � 3 is the line y � 3. The graphs have the samedomain, all real numbers, but the range of y � 2x is y � 0and the range of y � 2x � 3 is y � 3. The y-intercept of thegraph of y � 2x is 1 and for the graph of y � 2x � 3 is 4.

73.

The graphs have the same shape. The graph of y � ��15

�x �2

is the graph of y � ��15

�x translated two units to the right. The

asymptote for the graph of y � ��15

�xand for y � ��

15

�x �2is

[�5, 5] scl: 1 by [�1, 9] scl: 1

[�5, 5] scl: 1 by [�1, 9] scl: 1

yx

O

y � �( )x15

y

xO

y � 0.5(4)x

y

xO

y � 2(3)x

y

xO

y � 2( )x13

xO

f �1(x) � �x � 1

f (x) � �x � 1

f (x)

xO

f �1(x) � � x12

f (x) � �2x

f (x)

O x

f (x) � 5(x � 1)(x � 3)

f (x)

O xf (x) � 2

x

f (x)

O x

f (x ) � 4x � 2

f (x)

Sele

cted A

nsw

ers


the line y � 0. The graphs have the same domain, all realnumbers, and range, y � 0. The y-intercept of the graph of

y � ��15

�xis 1 and for the graph of y � ��

15

�x �2is 25. 75. For

h � 0, the graph of y � 2x is translated h units to theright. For h � 0, the graph of y � 2x is translated h unitsto the left. For k � 0, the graph of y � 2x is translated k units up. For k � 0, the graph of y � 2x is translated k units down. 77. 1, 6 79. 0 � x � 3 or x � 681. greatest integer

83. � 85. �511�� 87. g[h(x)] � 2x � 6;

h[g(x)] � 2x � 11 89. g[h(x)] � �2x � 2; h[g(x)] � �2x � 11

Pages 535–538 Lesson 10-21. Sample answer: x � 5y and y � log5 x 3. Scott; the valueof a logarithmic equation, 9, is the exponent of theequivalent exponential equation, and the base of thelogarithmic expression, 3, is the base of the exponential

equation. Thus, x � 39 or 19,683. 5. log7 �419� � �2

7. 36�12�

� 6 9. �3 11. �1 13. 1000 15. �12

�, 1 17. 3

19. 107.5 21. log8 512 � 3 23. log5 �1125� � �3

25. log100 10 � �12

� 27. 53 � 125 29. 4�1 � �14

� 31. 8�23�

� 4

33. 4 35. �12

� 37. �5 39. 7 41. n � 5 43. �3 45. 1018.8

47. 81 49. 0 � y � 8 51. 7 53. x � 24 55. 4 57. 259. 5 61. a � 363. log5 25 � 2 log5 5 Original equation

log5 52 � 2 log5 51 25 � 52 and 5 � 51

2 � 2(1) Inverse Property of Exponents and Logarithms

2 � 2 � Simplify.65. log7 [log3 (log2 8)] � 0 Original equation

log7 [log3 (log2 23)] � 0 8 � 23

log7 (log3 3) � 0 Inverse Property ofExponents and Logarithms

log7 (log3 31) � 0 3 � 31

log7 1 � 0 Inverse Property ofExponents and Logarithms

log7 70 � 0 1 � 70

0 � 0 � Inverse Property ofExponents and Logarithms

67a.

67b. The graph of y � log2 x � 3 is the graph of y � log2 xtranslated 3 units up. The graph of y � log2 x � 4 is thegraph of y � log2 x translated 4 units down. The graph oflog2 (x � 1) is the graph of y � log2 x translated 1 unit tothe right. The graph of log2 (x � 2) is the graph of y � log2 xtranslated 2 units to the left. 69. 101.4 or about 25 times asgreat 71. 2 and 3; Sample answer: 5 is between 22 and 23.73. A logarithmic scale illustrates that values next to eachother vary by a factor of 10. Answers should include thefollowing.• Pin drop: 1 100; Whisper: 1 102; Normal conversation:1 106; Kitchen noise: 1 1010; Jet engine: 1 1012

•

• On the scale shown above, the sound of a pin drop andthe sound of normal conversation appear not to differ bymuch at all, when in fact they do differ in terms of theloudness we perceive. The first scale shows this differencemore clearly.75. D 77. b12 79. �3, �

154� 81.

83. 85. x10 87. 8a6b3 89. �yx2z

3

3�

Page 538 Practice Quiz 11. growth 3. log4 4096 � 6 5. �

43

� 7. �35

� 9. x � 26

Page 544–546 Lesson 10-31. properties of exponents 3. Umeko; Clemente incorrectlyapplied the product and quotient properties of logarithms. log7 6 � log7 3 � log7 (6 � 3) or log7 18

Product Property of Logarithmslog7 18 � log7 2 � log7 (18 2) or log7 9

Quotient Property of Logarithms

5. 2.6310 7. 6 9. 3 11. pH � 6.1 � log10 �CB

� 13. 1.3652

15. �0.2519 17. 2.4307 19. �0.4307 21. 2 23. 4 25. 14

27. 2 29. � 31. 10 33. �x43� 35. False; log2 (22 � 23) �

log2 12, log2 22 � log2 23 � 2 � 3 or 5, and log2 12 � 5, since25 � 12. 37. 2 39. about 0.4214 kilocalories per gram41. 3 43. About 95 decibels; L � 10 log10R, where L is theloudness of the sound in decibels and R is the relativeintensity of the sound. Since the crowd increased by afactor of 3, we assume that the intensity also increases by afactor of 3. Thus, we need to find the loudness of 3R.L � 10 log10 3RL � 10 (log103 � log10R)L � 10 log103 � 10 log10RL � 10(0.4771) � 90L � 4.771 � 90 or about 9545. 7.547. Let bx � m and by � n. Then logb m � x and logb n � y.

�bb

xy� � �

mn

�

bx � y � �mn

� Quotient Property

logb bx � y � logb �

mn

�

x � y � logb �mn

�

logb m � logb n � logb �mn

�

49. A 51. 4 53. 2x 55. �8

Replace x with logb m andy with logb n.

Inverse Property of Exponentsand Logarithms

Property of Equality forLogarithmic Equations

6x � 58��(x � 3)(x � 3)(x � 7)

5��73��

2 � 1011 4 � 1011 6 � 1011 8 � 1011 1 � 10120

Pindrop

Whisper(4 feet)

Normalconversation

Jetengine

Kitchennoise

y

xO

y � log2x � 3

y � log2(x � 2)

y � log2(x � 1)

y � log2x � 4

�6�5

311

01

10

y

xO

y � �2x �

Sele

cted A

nsw

ers


57. odd; 3 59. �3ab� 61. �

35x� 63. 1 65. x � �

53

�

Pages 549–551 Lesson 10-41. 10; common logarithms 3. A calculator is notprogrammed to find base 2 logarithms. 5. 1.3617

7. 1.7325 9. 4.9824 11. 11.5665 13. �lloo

gg

57

�; 0.8271

15. �lloo

gg

92

�; 3.1699 17. 0.6990 19. 0.8573 21. �0.0969

23. 11 25. 2.1 27. {x x � 2.0860} 29. {a a � 1.1590}31. 0.4341 33. 4.7820 35. �1.1909 37. {n n � �1.0178}

39. 3.7162 41. 0.5873 43. �7.6377 45. �lloogg

123

� � 3.7004

47. �lloo

gg

37

� � 0.5646 49. �2

lloo

gg

41.6

� � 0.6781 51. between

0.000000001 and 0.000001 mole per liter 53. Sirius55. Vega 57. about 3.75 yr or 3 yr 9 mo59. Comparisons between substances of different aciditiesare more easily distinguished on a logarithmic scale.Answers should include the following.Sample Answer:• Tomatoes: 6.3 10�5 mole per liter

Milk: 3.98 10�7 mole per literEggs: 1.58 10�8 mole per liter

• Those measurements correspond to pH measurements of5 and 4, indicating a weak acid and a stronger acid. Onthe logarithmic scale we can see the difference in theseacids, whereas on a normal scale, these hydrogen ionconcentrations would appear nearly the same. Forsomeone who has to watch the acidity of the foods theyeat, this could be the difference between an enjoyablemeal and heartburn.

61. C 63. 1.6938 65. 64 67. 62 69. (d � 2)(3d � 4)71. prime 73. 32 � x 75. log5 45 � x 77. logb x � y

Pages 557–559 Lesson 10-51. the number e 3. Elsu; Colby tried to write each side as apower of 10. Since the base of the natural logarithmicfunction is e, he should have written each side as a powerof e; 10ln 4x � 4x. 5. 0.0334 7. �2.3026 9. e0 � 1 11. 5x13. 1.0986 15. 0 � x � 403.4288 17. �90.017119. about 15,066 ft 21. 148.4132 23. 1.6487 25. 2.302627. �3.5066 29. about 49.5 cm 31. 2 � ln 6x 33. ex � 5.235. y 37. 45 39. �0.6931 41. x � 0.4700 43. 0.597345. x � �0.9730 47. 49.4711 49. 14.3891 51. 45.0086

53. 1 55. t � �100

rln 2� 57. t � �

11r0

� 59. about 55 yr

61. about 21 min63. The number e is used in the formula for continuouslycompounded interest, A � Pert. Although no banks actuallypay interest compounded continually, the equation is soaccurate in computing the amount of money for quarterlycompounding, or daily compounding, that it is often usedfor this purpose. Answers should include the following.• If you know the annual interest rate r and the principal P,

the value of the account after t years is calculated bymultiplying P times e raised to the r times t power. Use acalculator to find the value of ert.

• If you know the value A you wish the account to achieve,the principal P, and the annual interest rate r, the time tneeded to achieve this value is found by first taking thenatural logarithm of A minus the natural logarithm of P.Then, divide this quantity by r.

65. 1946, 1981, 2015; It takes between 34 and 35 years forthe population to double.

67. �loglo

0g.0647

� � �1.7065 69. 5 71. inverse; 4 73. direct; �7

75. 3.32 77. 1.43 79. 13.43


1. �lloo

gg

54

�; 1.1610 3. 3 5. 1.3863

Pages 563–565 Lesson 10-61. y � a(1 � r)t, where r � 0 represents exponential growthand r � 0 represents exponential decay 3. Sample answer:money in a bank 5. about 33.5 watts 7. y � 212,000e0.025t

9. C 11. at most $108,484.93 13. No; the bone is onlyabout 21,000 years old, and dinosaurs died out 63,000,000years ago. 15. about 0.0347 17. $12,565 billion19. after the year 2182 21. Never; theoretically, the amountleft will always be half of the previous amount.23. about 19.5 yr 25. ln y � 3 27. 4x2 � e8 29. p � 3.3219

31. �0.5(0

6.08p)� � �

0.5(04.08p)� 33. �

1p50� 35. ellipse 37. circle

39. 8 107

Pages 566–570 Chapter 10 Study Guide and Review1. true 3. false; common logarithm 5. true7. false; logarithmic function 9. false; exponential function

11. growth 13. y � 7��15

�x

15. �1 17. x � ��6� or

x � �6� 19. log5 �215� � �2 21. 43 � 64 23. 6�2 � �

316�

25. �5 27. 2 29. �32

� 31. �13

� � y � 3 33. �4, 3 35. 1.7712

37. 3 39. 6 41. 15 43. 5.7279 45. x � 7.3059

47. x � 5.8983 49. �lloogg

141

�; 1.7297 51. �lologg102000

�; 2.3059

53. ex � 7.4 55. 7x 57. x � 1.1632 59. 0 � x � 49.471161. 74.2066 63. 5.05 days 65. about 3.6%

Chapter 11 Sequences and Series


1. 6 3. �5 5. �12

�

7. 9.

11. 17 13. �312� 15. �

35

�

Pages 580–582 Lesson 11-11. The differences between the terms are not constant.3. Sample answer: 1, �4, �9, �14, … 5. �3, �5, �7, �97. 14, 12, 10, 8, 6 9. �112 11. 15 13. 56, 68, 80

15. 30, 37, 44, 51 17. 6, 10, 14, 18 19. �73

�, 3, �131�, �

133�

21. 5.5, 5.1, 4.7, 4.3 23. 2, 15, 28, 41, 54

25. 6, 2, �2, �6, �10 27. �43

�, 1, �23

�, �13

�, 0 29. 28 31. 94

Ox

y

42�8

8162432404856

6x

y

O

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33. 335 35. �236� 37. 27 39. 61 41. 37.5 in. 43. 30th

45. 82nd 47. an � �7n � 25

49. 13, 17, 21

51. Yes; it corresponds to n � 100. 53. 4, �255. 7, 11, 15, 19, 2357. Arithmetic sequences can be used to model the numbersof shingles in the rows on a section of roof. Answers shouldinclude the following.• One additional shingle is needed in each successive row.• One method is to successively add 1 to the terms of the

sequence: a8 � 9 � 1 or 10, a9 � 10 � 1 or 11, a10 � 11 �1 or 12, a11 � 12 � 1 or 13, a12 � 13 � 1 or 14, a13 � 14 �1 or 15, a14 � 15 � 1 or 16, a15 � 16 � 1 or 17. Anothermethod is to use the formula for the nth term: a15 � 3 �(15 � 1)1 or 17.

59. B 61. �0.4055 63. 146.4132 65. 2, 5, 8, 1167. 11, 15, 19, 23, 27

Pages 586–587 Lesson 11-21. In a series, the terms are added. In a sequence, they are

not. 3. Sample answer:4

�n�1

(3n � 4) 5. 230 7. 552

9. 260 11. 95 13. �6, 0, 6 15. 344 17. 1501 19. �921. 104 23. �714 25. 14 27. 10 rows 29. 721 31. 16233. 108 35. �195 37. 315,150 39. 1,001,000 41. 17, 26,35 43. �12, �9, �6 45. 265 ft 47. False; for example, 7 �10 � 13 � 16 � 46, but 7 � 10 � 13 � 16 � 19 � 22 � 25 �28 � 140. 49. C 51. 5555 53. 6683 55. �135

57. ��92

� 59. 61. 26�21� 63. 16 65. �227�

Pages 590–592 Lesson 11-31a. Geometric; the terms have a common ratio of �2.1b. Arithmetic; the terms have a common difference of �3.3. Marika; Lori divided in the wrong order when finding r.

5. 2, �4 7. �16

54� 9. �4 11. 3, 9 13. 15, 5 15. 54, 81

17. �22

07�, �

48

01� 19. �2.16, 2.592 21. 2, �6, 18, �54, 162

23. 243, 81, 27, 9, 3 25. �136� 27. 729 29. 243 31. 1

33. 78,125 35. �8748 37. 655.36 lb 39. an � 36��13

�n � 1

41. an � �2(�5)n � 1 43. �18, 36, �72 45. 16, 8, 4, 247. 8 days 49. False; the sequence 1, 4, 9, 16, …, forexample, is neither arithmetic nor geometric.51. The heights of the bounces of a ball and the heightsfrom which a bouncing ball falls each form geometricsequences. Answers should include the following.• 3, 1.8, 1.08, 0.648, 0.3888• The common ratios are the same, but the first terms are

different. The sequence of heights from which the ballfalls is the sequence of heights of the bounces with theterm 3 inserted at the beginning.

53. C 55. 203 57. �12, �16, �20 59. 127 61. �6811�

Page 592 Practice Quiz 11. 46 3. 187 5. 1


1. Sample answer: 4 � 2 � 1 � �12

�

3. Sample answer: The first term is a1 � 2. Divide thesecond term by the first to find that the common ratio is r � 6. Therefore, the nth term of the series is given by 2 � 6n � 1.There are five terms, so the series can be written as

5

�n�1

2 � 6n � 1. 5. 39,063 7. 165 9. 129 11. �10

993�

13. 3 15. 728 17. 1111 19. 244 21. 2101 23. �7238

�

25. 1040.984 27. 6564 29. 1,747,625 31. 3641 33. �541661

�

35. 2555 37. �3847

� 39. 3,145,725 41. 243 43. 2 45. 80

47. about 7.13 in. 49. If the number of people that eachperson sends the joke to is constant, then the total numberof people who have seen the joke is the sum of a geometricseries. Answers should include the following.• The common ratio would change from 3 to 4.• Increase the number of days that the joke circulates so

that it is inconvenient to find and add all the terms of theseries.

51. C 53. 3.99987793 55. ��14

�, �32

�, �9 57. 232

59.

61. Sample answer: 294 63. 2 65. �23

� 67. 0.6


1. Sample answer:�

�n�1

��12

�n

3. Beth; the common ratio for

the infinite geometric series is ��43

�. Since ��43

� � 1, the

series does not have a sum and the formula S � �1

a�

1r

� does

not apply. 5. does not exist 7. �34

� 9. 100 11. �7939�

13. 96 cm 15. does not exist 17. 45 19. �16 21. �554�

23. does not exist 25. 1 27. �23

� 29. �32

� 31. 2

33. 40 � 20�2� � 20 � … 35. 900 ft 37. 75, 30, 12

39. �8, �3�15

�, �1�275�, ��

16245

� 41. �19

� 43. �8929� 45. �

492979

� 47. �292990

�

49. The total distance that a ball bounces, both up anddown, can be found by adding the sums of two infinitegeometric series. Answers should include the following.

• an � a1 � rn � 1, Sn � �a1(

11

�

�

rrn)

�, or S � �1

a�

1r

�

• The total distance the ball falls is given by the infinitegeometric series 3 � 3(0.6) � 3(0.6)2 � … . The sum of

this series is �1 �3

0.6� or 7.5. The total distance the ball

bounces up is given by the infinite geometric series 1.8(0.6) � 1.8(0.6)2 � 1.8(0.6)3 � … . The sum of this

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900

800

700

600

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Years Since 19950 1 2 3 4 5 6

�3 �

2�89��

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series is �11.8�(0

0.6.6)

� or 2.7. Thus, the total distance the ball

travels is 7.5 � 2.7 or 10.2 feet.

51. C 53. �87

841

4� 55. 3 57. x � 5 59. �(x �

�3x)�(x

7� 1)

�

61. (x � 2)2 � (y � 4)2 � 36 63. ��12

�, �32

�, �72

� 65. x2 � 36 � 0

67. x2 � 10x � 24 � 0 69. The number of visitors was

decreasing. 71. 3 73. �12

� 75. �4

Pages 608–610 Lesson 11-61. an � an � 1 � d; an � r � an � 1 3. Sometimes; if f(x) � x2

and x1 � 2, then x2 � 22 or 4, so x2 � x1. But, if x1 � 1, thenx2 � 1, so x2 � x1. 5. �3, �2, 0, 3, 7 7. 1, 2, 5, 14, 419. 1, 3, �1 11. bn � 1.05bn � 1 � 10 13. �6, �3, 0, 3, 615. 2, 1, �1, �4, �8 17. 9, 14, 24, 44, 84 19. �1, 5, 4, 9, 13

21. �72

�, �74

�, �76

�, �78

�, �170� 23. 67 25. 1, 1, 2, 3, 5, …

27. $99,921.21, $99,841.95, $99,762.21, $99,681.99, $99,601.29,$99,520.11, $99,438.44, $99,356.28 29. tn � tn � 1 � n31. 16, 142, 1276 33. �7, �16, �43 35. �3, 13, 333

37. �52

�, �327�, �14

245� 39. $75.78

41. Under certain conditions, the Fibonacci sequence can beused to model the number of shoots on a plant. Answersshould include the following.• The 13th term of the sequence is 233, so there are 233

shoots on the plant during the 13th month.• The Fibonacci sequence is not arithmetic because the

differences �0, 1, 1, 2, … of the terms are not constant.The Fibonacci sequence is not geometric because the

ratios �1, 2, �32

�, … of the terms are not constant.

43. C 45. �16

� 47. �5208 49. 3x � 7 units 51. 5040

53. 20 55. 210

Pages 615–617 Lesson 11-71. 1, 8, 28, 56, 70, 56, 28, 8, 1 3. Sample answer: (5x � y)4

5. 17,160 7. p5 � 5p4q � 10p3q2 � 10p2q3 � 5pq4 � q5

9. x4 � 12x3y � 54x2y2 � 108xy3 � 81y4 11. 1,088,640a6b4

13. 362,880 15. 72 17. 495 19. a3 � 3a2b � 3ab2 � b3

21. r8 � 8r7s � 28r6s2 � 56r5s3 � 70r4s4 � 56r3s5 � 28r2s6 �8rs7 � s8 23. x5 � 15x4 � 90x3 � 270x2 � 405x � 24325. 16b4 � 32b3x � 24b2x2 � 8bx3 � x4 27. 243x5 � 810x4y �

1080x3y2 � 720x2y3 � 240xy4 � 32y5 29. �3a25� � �

58a4� � 5a3 �

20a2 � 40a � 32 31. 27x3 � 54x2 � 36x � 8 cm3 33. 45

35. 924x6y6 37. 5670a4 39. 145,152x6y3 41. ��683�x5

43. The coefficients in a binomial expansion give thenumbers of sequences of births resulting in given numbersof boys and girls. Answers should include the following.• (b � g)5 � b5 � 5b4g � 10b3g2 � 10b2g3 � 5bg4 � g5;

There is one sequence of births with all five boys, fivesequences with four boys and one girl, ten sequenceswith three boys and two girls, ten sequences with twoboys and three girls, five sequences with one boy andfour girls, and one sequence with all five girls.

• The number of sequences of births that have exactly kgirls in a family of n children is the coefficient of bn � kgk

in the expansion of (b � g)n. According to the BinomialTheorem, this coefficient is �(n �

n!k)!k!�.

45. C 47. 3, 5, 9, 17, 33 49. �lloo

gg

52

�; 2.3219 51. �lloo

gg

85

�;

1.2920 53. asymptotes: x � �4, x � 1 55. hyperbola

57. yes 59. True; �1(12� 1)� � �

1(22)� or 1. 61. True; �1

2(14� 1)2� �

�1(

44)� or 1.

Page 617 Practice Quiz 21. 1,328,600 3. 24 5. 1, 5, 13, 29, 61 7. 5, �13, 419. a6 � 12a5 � 60a4 � 160a3 � 240a2 � 192a � 64

Pages 619–621 Lesson 11-81. Sample answers: formulas for the sums of powers of thefirst n positive integers and statements that expressionsinvolving exponents of n are divisible by certain numbers3. Sample answer: 3n � 15. Step 1: When n � 1, the left side of the given equation is

�12

�. The right side is 1 � �12

� or �12

�, so the equation is true for

n � 1.

Step 2: Assume �12

� � �212� � �

213� � … � �

21k� � 1 � �

21k� for some

positive integer k.

Step 3: �12

� � �212� � �

213� � … � �

21k� � �

2k1� 1� � 1 � �

21k� � �

2k1� 1�

� 1 � �2k

2� 1� � �

2k1� 1�

� 1 � �2k

1� 1�

The last expression is the right side of the equationto be proved, where n � k � 1. Thus, the equation istrue for n � k � 1.

Therefore, �12

� � �212� � �

213� � … � �

21n� � 1 � �

21n� for all positive

integers n.7. Step 1: 51 � 3 � 8, which is divisible by 4. The statementis true for n � 1.Step 2: Assume that 5k � 3 is divisible by 4 for somepositive integer k. This means that 5k � 3 � 4r for somepositive integer r.Step 3: 5k � 3 � 4r

5k � 4r � 35k � 1 � 20r � 15

5k � 1 � 3 � 20r � 125k � 1 � 3 � 4(5r � 3)Since r is a positive integer, 5r � 3 is a positiveinteger. Thus, 5k � 1 � 3 is divisible by 4, so thestatement is true for n � k � 1.

Therefore, 5n � 3 is divisible by 4 for all positive integers n.9. Sample answer: n � 311. Step 1: When n � 1, the left side of the given equation is1. The right side is 1[2(1) � 1] or 1, so the equation is truefor n � 1.Step 2: Assume 1 � 5 � 9 � … � (4k � 3) � k(2k � 1) forsome positive integer k.Step 3: 1 � 5 � 9 � … � (4k � 3) � [4(k � 1) � 3]

� k(2k � 1) � [4(k � 1) � 3]� 2k2 � k � 4k � 4 � 3� 2k2 � 3k � 1� (k � 1)(2k � 1)� (k � 1)[2(k � 1) � 1]

The last expression is the right side of the equation to be proved, where n � k � 1. Thus, the equation is true for n � k � 1.

Therefore, 1 � 5 � 9 � … � (4n � 3) � n(2n � 1) for allpositive integers n.13. Step 1: When n � 1, the left side of the given equation is

13 or 1. The right side is �12(1

4� 1)2� or 1, so the equation is

true for n � 1.Step 2: Assume 13 � 23 � 33 � … � k3 � �

k2(k4� 1)2� for some

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positive integer k.Step 3: 13 � 23 � 33 � … � k3 � (k � 1)3

� �k2(k

4� 1)2� � (k � 1)3

�

�

�

�

�


Therefore, 13 � 23 � 33 � … � n3 � �n2(n

4� 1)2� for all

positive integers n.15. Step 1: When n � 1, the left side of the given equation is �

13

�.

The right side is �12

��1 � �13

� or �13

�, so the equation is true for n � 1.

Step 2: Assume �13

� � �312� � �

313� � … � �

31k� � �

12

��1 � �31k� for some

positive integer k.

Step 3: �13

� � �312� � �

313� � … � �

31k� � �

3k1� 1� � �

12

��1 � �31k� � �

3k1� 1�

� �12

� � �2 �

13k� � �

3k1� 1�

� �3k �

21

��3k

3�1

� 2�

� �32k

�

�

31

k�� 1

1�

� �12

��3k

3�

k

1

��1

1�

� �12

��1 � �3k

1� 1�

The last expression is the right side of the equation to be proved, where n � k � 1. Thus, the equation istrue for n � k � 1.

Therefore, �13

� � �312� � �

313� � … � �

31n� � �

12

��1 � �31n� for all positive

integers n.

17. Step 1: 81 � 1 � 7, which is divisible by 7. The statementis true for n � 1.Step 2: Assume that 8k � 1 is divisible by 7 for somepositive integer k. This means that 8k � 1 � 7r for somewhole number r.Step 3: 8k � 1 � 7r

8k � 7r � 18k � 1 � 56r � 8

8k � 1 � 1 � 56r � 78k � 1 � 1 � 7(8r � 1)Since r is a whole number, 8r � 1 is a whole number. Thus, 8k � 1 � 1 is divisible by 7, so the statement istrue for n � k � 1.

Therefore, 8n � 1 is divisible by 7 for all positive integers n.

19. Step 1: 121 � 10 � 22, which is divisible by 11. Thestatement is true for n � 1.Step 2: Assume that 12k � 10 is divisible by 11 for somepositive integer k. This means that 12k � 10 � 11r for somepositive integer r.Step 3: 12k � 10 � 11r

12k � 11r � 1012k � 1 � 132r � 120

12k � 1 � 10 � 132r � 11012k � 1 � 10 � 11(12r � 10)Since r is a positive integer, 12r � 10 is a positiveinteger. Thus, 12k � 1 � 10 is divisible by 11, so thestatement is true for n � k � 1.

Therefore, 12n � 10 is divisible by 11 for all positiveintegers n.21. Step 1: There are 6 bricks in the top row, and 12 � 5(1) �6, so the formula is true for n � 1.Step 2: Assume that there are k2 � 5k bricks in the top krows for some positive integer k.Step 3: Since each row has 2 more bricks than the oneabove, the numbers of bricks in the rows form an arithmeticsequence. The number of bricks in the (k � 1)st row is 6 �[(k � 1) � 1](2) or 2k � 6. Then the number of bricks in thetop k � 1 rows is k2 � 5k � (2k � 6) or k2 � 7k � 6.k2 � 7k � 6 � (k � 1)2 � 5(k � 1), which is the formula tobe proved, where n � k � 1. Thus, the formula is true for n � k � 1.Therefore, the number of bricks in the top n rows is n2 � 5nfor all positive integers n.23. Step 1: When n � 1, the left side of the given equation

is a1. The right side is �12

�[2a1 � (1 � 1)d] or a1, so the

equation is true for n � 1.Step 2: Assume a1 � (a1 � d) � (a1 � 2d) � … �

[a1 � (k � 1)d] � �2k

�[2a1 � (k � 1)d] for some positive integer k.

Step 3: a1 � (a1 � d) � (a1 � 2d) � … � [a1 � (k � 1)d] �[a1 � (k � 1 � 1)d]

� �2k

�[2a1 � (k � 1)d] � [a1 � (k � 1 � 1)d]

� �2k

�[2a1 � (k � 1)d] � a1 � kd

�

�

�

�

� �k �

21

�(2a1 � kd)

� �k �

21

�[2a1 � (k � 1 � 1)d]

The last expression is the right side of the formula tobe proved, where n � k � 1. Thus, the formula istrue for n � k � 1.

Therefore, a1 � (a1 � d) � (a1 � 2d) � … � [a1 � (n � 1)d] ��n2

�[2a1 � (n � 1)d] for all positive integers n.25. Sample answer: n � 3 27. Sample answer: n � 229. Sample answer: n � 11 31. Write 7n as (6 � 1)n. Thenuse the Binomial Theorem.7n � 1 � (6 � 1)n � 1

� 6n � n � 6n � 1 � �n(n

2� 1)�6n � 2 � … � n � 6 � 1 � 1

� 6n � n � 6n � 1 � �n(n

2� 1)�6n � 2 � … � n � 6

Since each term in the last expression is divisible by 6, thewhole expression is divisible by 6. Thus, 7n � 1 is divisibleby 6. 33. C 35. x6 � 6x5y � 15x4y2 � 20x3y3 � 15x2y4 �6xy5 � y6 37. 256x8 � 1024x7y � 1792x6y2 � 1792x5y3 �1120x4y4 � 448x3y5 � 112x2y6 � 16xy7 � y8 39. 2, 14, 78241. 0, 1

(k � 1)2a1 � k(k � 1)d��

2

(k � 1)2a1 � (k2 � k � 2k)d��

2

k � 2a1 � (k2 � k)d � 2a1 � 2kd��

2

k[2a1 � (k � 1)d] � 2(a1 � kd)��

2

(k � 1)2�(k � 1) � 1 2��

4

(k � 1)2(k � 2)2��

4

(k � 1)2(k2 � 4k � 4)��

4

(k � 1)2�k2 � 4(k � 1) ��

4

k2(k � 1)2 � 4(k � 1)3��

4

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Pages 622–626 Chapter 11 Study Guide and Review1. partial sum 3. sigma notation 5. Binomial Theorem7. arithmetic series 9. 38 11. �11 13. �3, 1, 5 15. 6, 3,0, �3 17. 2322 19. �220 21. 32 23. 3

25. 6, 12 27. 4, 2, 1, �12

� 29. 1452 31. �14

1,1697

� 33. 72

35. ��11

63� 37. 3, 2, �2, �18, �82 39. 1, 3, 4, 7, 11 41. 10,

66, 458 43. �1, 4, �31 45. x4 � 8x3 � 24x2 � 32x � 1647. 160x3y3

49. Step 1: When n � 1, the left side of the given equation is 1. The right side is 21 � 1 or 1, so the equation is true forn � 1.Step 2: Assume 1 � 2 � 4 � … � 2k � 1 � 2k � 1 for somepositive integer k.Step 3: 1 � 2 � 4 � … � 2k � 1 � 2(k � 1) � 1 � 2k � 1 � 2k

� 2 � 2k � 1� 2k � 1 � 1


Therefore, 1 � 2 � 4 � … � 2n � 1 � 2n � 1 for all positiveintegers n.

Chapter 12 Probability and Statistics


1. �16

� 3. �12

� 5. �23

�

7.

9.

11. 3 13. �13� 15. a3 � 3a2b � 3ab2 � b3

17. m5 � 5m4n � 10m3n2 � 10m2n3 � 5mn4 � n5

Pages 634–637 Lesson 12-11. HHH, HHT, HTH, HTT, THH, THT, TTH, TTT 3. Theavailable colors for the car could be different from those forthe truck. 5. dependent 7. 256 9. D 11. independent13. dependent 15. 16 17. 30 19. 1024 21. 10,08023. 362,880 25. 27,216 27. 80029. The maximum number of license plates is a productwith factors of 26s and 10s, depending on how many lettersare used and how many digits are used. Answers shouldinclude the following.• There are 26 choices for the first letter, 26 for the second,

and 26 for the third. There are 10 choices for the firstnumber, 10 for the second, and 10 for the third. By theFundamental Counting Principle, there are 263 � 103 or17,576,000 possible license plates.

• Replace positions containing numbers with letters.

31. C 33. 20 mi 35. 28x6y2 37. 7 39. �12

� 41. ��x �

x5y

�

43. �1, �2 45. y � (x � 3)2 � 2 47. y � ��12

�x2 � 8 49. 3

51. �17

� � 53. no inverse exists 55. y � �23

�x � �13

�

57. 30 59. 720 61. 15 63. 1

Pages 641–643 Lesson 12-21. Sample answer: There are six people in a contest. How

many ways can the first, second, and third prizes beawarded? 3. Sometimes; the statement is only true when r � 1. 5. 120 7. 6 9. permutation; 5040 11. 8413. 9 15. 665,280 17. 70 19. 210 21. 126023. combination; 28 25. permutation; 12027. permutation; 3360 29. combination; 455 31. 6033. 111,540 35. 80,089,12837. C(n � 1, r) � C(n � 1, r � 1)

� �(n �

(n1�

�1)

r!)!r!

� �

� �(n �

(nr�

�1)

1!)!r!

� � �(n �

(nr)�!(r

1�)!

1)!�

� �(n �

(nr�

�1)

1!)!r!

� � �nn

��

rr

� � �(n �

(nr)�!(r

1�)!

1)!� � �

rr

�

� �(n �

(n1�)!(

rn)!r

�!

r)� � �

((nn

��

r1))!!rr!

�

�

� �((nn

��

1r))!!rn!

�

� �(n �

n!r)!r!�

� C(n, r)39. D 41. 24 43. 120 45. 80 47. Sample answer: n � 2

49. x � 0.8047 51. 20 days 53. �(y �

94)2

� � �(x �

44)2

� � 1

55. –4; 128 57. {�2, 5} 59. 8�2� 61. 4�5� 63. (0, 2)

65. ��67

� 67. {�7, 15} 69. �35

� 71. �15

�

Pages 647–650 Lesson 12-31. Sample answer: The event July comes before June has aprobability of 0. The event June comes before July has aprobability of 1. 3. There are 6 � 6 or 36 possible outcomesfor the two dice. Only 1 outcome, 1 and 1, results in a sum

of 2, so P(2) � �316�. There are 2 outcomes, 1 and 2 as well as

2 and 1, that result in a sum of 3, so P(3) � �326� or �

118�. 5. �

27

�

7. 8:1 9. 2:7 11. �1101� 13. �

18

� 15. �110� 17. �

225� 19. �

565�

21. �2585� 23. �

11115

� 25. �1615� 27. �

12145

� 29. 0 31. 0.007

33. 0.109 35. 3:5 37. 5:3 39. 1:4 41. 3:1 43. �130�

45. �49

� 47. �19

� 49. �35

� 51. 2:23 53. 1:4 55. �210� 57. �

290�

59. �290� 61. �

1120� 63. Probability and odds are good tools for

assessing risk. Answers should include the following.

• P(struck by lightning) � �s �s

f� � �750

1,000�, so Odds �

1:(750,000 � 1) or 1:749,999. P(surviving a lightning

strike) � �s �s

f� � �

34

�, so Odds � 3:(4 � 3) or 3:1.

• In this case, success is being struck by lightning orsurviving the lightning strike. Failure is not being struckby lightning or not surviving the lightning strike.

65. D 67. experimental; about 0.307 69. theoretical; �117�

71. permutation; 1260 73. 16 75. direct variation

77. (4, 4) 79. �365� 81. �

14

� 83. �290�

Page 650 Practice Quiz 11. 24 3. 18,720 5. 56 7. combination; 20,358,520 9. �

11032

�

Pages 654–657 Lesson 12-41. Sample answer: putting on your socks, and then yourshoes 3. Mario; the probabilities of rolling a 4 and rolling

(n � 1)!(n � r � r)��

(n � r)!r!

(n � 1)!��[n � 1 � (r � 1)]!(r � 1)!

�13

14

20 30 40 9080706050

20 25 30 35 40

Sele

cted A

nsw

ers


a 2 are both �16

�. 5. �14

� 7. �6

463� 9. �

14

� 11. dependent; �22210

�

13. �112� 15. �

2356� 17. �

16

� 19. �56

� 21. �419� 23. �

1201� 25. 0 27. �

125�

29. �125� 31. independent; �

28

51� 33. dependent; �

211�

35. dependent; �284101�

37. First SpinBlue Yellow Red

�13

� �13

� �13

�

Blue BB BY BR

�13

� �19

� �19

� �19

�

Second Yellow YB YY YRSpin �

13

� �19

� �19

� �19

�

Red RB RY RR

�13

� �19

� �19

� �19

�

39. �13

� 41. �1,16109,054� 43. �

2603,82275

� 45. about 4.87% 47. no

49. Sample answer: As the number of trials increases, theresults become more reliable. However, you cannot beabsolutely certain that there are no black marbles in the bagwithout looking at all of the marbles. 51. Probability canbe used to analyze the chances of a player making 0, 1, or 2 free throws when he or she goes to the foul line to shoot 2 free throws. Answers should include the following.• One of the decimals in the table could be used as the

value of p, the probability that a player makes a givenfree throw. The probability that a player misses both freethrows is (1 � p)(1 � p) or (1 � p)2. The probability that aplayer makes both free throws is p � p or p2. Since the sumof the probabilities of all the possible outcomes is 1, theprobability that a player makes exactly 1 of the 2 freethrows is 1 � (1 � p)2 � p2 or 2p(1 � p).

• The result of the first free throw could affect the player’sconfidence on the second free throw. For example, if theplayer makes the first free throw, the probability of he orshe making the second free throw might increase. Or, ifthe player misses the first free throw, the probability thathe or she makes the second free throw might decrease.

53. C 55. �3

340� 57. 1440 ways 59. 36 61. x, x � 4

63. 65. 153 67. b 69. (1, 2)

71. (�2, 4) 73. �56

� 75. �1121�

77. 1�152�

Pages 660–663 Lesson 12-51. Sample answer: mutually exclusive events: tossing a coinand rolling a die; inclusive events: drawing a 7 and adiamond from a standard deck of cards 3. The events arenot mutually exclusive, so the chance of rain is less

than 100%. 5. �13

� 7. �12

� 9. �23

� 11. inclusive; �143� 13. �

56�

15. �24

52� 17. �

13453

� 19. �1

343� 21. �

13483

� 23. mutually

exclusive; �79

� 25. inclusive; �23

14� 27. �

143� 29. �

25251

� 31. �168683

�

33. �18

� 35. �14

� 37. �7180� 39. �

1930� 41. �

71810

� 43. �35� 45. �

1277�

47. Subtracting P(A and B) from each side and adding P(A or B) to each side results in the equation P(A or B) �P(A) � P(B) � P(A and B). This is the equation for theprobability of inclusive events. If A and B are mutuallyexclusive, then P(A and B) � 0, so the equation simplifies toP(A or B) � P(A) � P(B), which is the equation for theprobability of mutually exclusive events. Therefore, the

equation is correct in either case. 49. C 51. �2

116� 53. �

2116�

55. 4:1 57. 2:5 59. 254 61. (�8, �10) 63. (x � 1)2(x � 1)(x2 � 1) 65. min: (�0.42, 0.62); max: (�1.58, 1.38)67. (1, 3), (1, �1), (3, 3), (3, 5);

max: f(3, 5) � 23; min: f(1, �1) � �369. direct variation71. 35.4, 34, no mode, 7273. 63.75, 65, 50 and 65, 3075. 12.98, 12.9, no mode, 4.7

Pages 666–670 Lesson 12-61. Sample answer: {10, 10, 10, 10, 10, 10}

3. � � ��n1

��n

i=1(x�i � x�)2� 5. 8.3, 2.9 7. $7300.50, $5335.25

9. 2500, 50 11. 3.1, 1.7 13. 37,691.2, 194.1 15. 82.9, 9.117. 77.7; 32; 19 19. Mean; it is highest. 21. $1047.88,$1049.50, $695 23. Mean or median; they are nearly equaland are more representative of the prices than the mode.25. Mode; it is lowest. 27. 19.3 29. 19.5 31. 59.8, 7.733. 100% 35. Sample answer: The first graph might beused by a sales manager to show a salesperson that he orshe does not deserve a big raise. It appears that sales aresteady but not increasing fast enough to warrant a big raise.37. A: 2.5, 2.5, 0.7, 0.8; B: 2.5, 2.5, 1.1, 1.039. The statistic(s) that best represent a set of test scoresdepends on the distribution of the particular set of scores.Answers should include the following.• mean, 73.9; median, 76.5; mode, 94• The mode is not representative at all because it is the

highest score. The median is more representative than the mean because it is influenced less than the mean bythe two very low scores of 34 and 19.

• Each measure is increased by 5.

41. D 43. 1.9 45. inclusive; �143� 47. �

1169� 49. �

21034

�

51. (0, �9); �0, ��106�; ��95

� 53. 17 55. 12 cm3 57. (1, 5)

59. 136 61. 380 63. 396


1. �230� 3. �

29

� 5. �16

� 7. �34

� 9. 23.6, 4.9

Pages 673–675 Lesson 12-71. Sample answer:

The use of cassettes since CDs were introduced.

y

xO

y

xO

y � x2 � 4

Sele

cted A

nsw

ers


3. Since 99% of the data is within 3 standard deviations of the mean, 1% of the data is more than 3 standarddeviations from the mean. By symmetry, half of this, or0.5%, is more than 3 standard deviations above the mean.5. 68% 7. 95% 9. 250 11. 81.5% 13. normallydistributed 15. 68% 17. 0.5% 19. 50% 21. 95%23. 815 25. 16% 27. The mean would increase by 25; thestandard deviation would not change; and the graph wouldbe translated 25 units to the right. 29. A 31. 17.5, 4.2

33. �123� 35. �

143� 37. �3, 2, 4 39. �

14

�, 1 41. 0.76 h 43. 56c5d3

Pages 678–680 Lesson 12-81. Sample answer: In a 5-card hand, what is the probabilitythat at least 2 cards are hearts? 3a. Each trial has morethan two possible outcomes. 3b. The number of trials is

not fixed. 3c. The trials are not independent. 5. �18

�

7. �28,

1561� 9. �

22

78

,,65

46

81

� 11. about 0.37 13. �116� 15. �

14

� 17. �1161�

19. �3182858

� 21. �62438

� 23. �10

124� 25. �

15

31

52

� 27. �55132

� 29. �150152

�

31. �35

11

92

� 33. about 0.44 35. about 0.32 37. �372�

39. Getting a right answer and a wrong answer are theoutcomes of a binomial experiment. The probability is fargreater that guessing will result in a low grade than in ahigh grade. Answers should include the following.• Use (r � w)5 � r5 � 5r4w � 10r3w2 � 10r2w3 � 5rw4 � w5

and the chart on page 48 to determine the probabilities ofeach combination of right and wrong.

• P(5 right): r5 � ��14

�5� �

10124� or about 0.098%;

P(4 right, 1 wrong): �110

524� or about 1.5%;

P(3 right, 2 wrong): 10r3w2 � 10��14

�3��34

�2� �

54152

� or about

8.8%; P(3 wrong, 2 right): 10r2w3 � 10��14

�2��34

�3� �

153152

� or

about 26.4%; P(4 wrong, 1 right): 5rw4 � 5��14

��34

�4� �

1400254

�

or about 39.6%; P(5 wrong): w5 � ��34

�5� �

1204234

� or

about 23.7%.41. B 43. normal distribution 45. 1047. Mean; it is highest.49.

51. 0.1 53. 0.039 55. 0.041

Pages 684–685 Lesson 12-91. Sample answer: If a sample is not random, the results ofa survey may not be valid. 3. The margin of samplingerror decreases when the size of the sample n increases. As

n increases, �p(1

n� p)� decreases. 5. No; these students

probably study more than average. 7. about 4% 9. Theprobability is 0.95 that the percent of Americans ages 12and older who listen to the radio every day is between 72%and 82% 11. No; you would tend to point toward themiddle of the page. 13. Yes; a wide variety of people

would be called since almost everyone has a phone.15. about 8% 17. about 4% 19. about 3% 21. about 4%23. about 3% 25. about 2% 27. about 983 29. A politicalcandidate can use the statistics from an opinion poll toanalyze his or her standing and to help plan the rest of thecampaign. Answers should include the following.• The candidate could decide to skip areas where he or she

is way ahead or way behind, and concentrate on areaswhere the polls indicate the race is close.

• about 3.5%• The margin of error indicates that with a probability of

0.95 the percent of the Florida population that favoredBush was between 43.5% and 50.5%. The margin of errorfor Gore was also about 3.5%, so with probability 0.95 thepercent that favored Gore was between 40.5% and 47.5%.Therefore, it was possible that the percent of the Floridapopulation that favored Bush was less than the percentthat favored Gore.

31. C 33. �352� 35. 95% 37. 97.5%

Pages 687–692 Chapter 12 Study Guide and Review1. c 3. a 5. d 7. f 9. 5040 codes 11. 4 13. 1:3 15. 7:5

17. 2:3 19. independent; �316� 21. dependent; �

17

�

23. mutually exclusive; �23

� 25. inclusive; �173� 27. 341.0, 18.5

29. 3400 31. 800 33. �312� 35. �2,176,7

182,336� 37. �2,

1147,64,3778,25,03036

�

39. 460 mothers

Chapter 13 Trigonometric FunctionsPage 699 Chapter 13 Getting Started

1. 10 3. 16.7 5. x � 7, y � 7�2� 7. x � 4�3�, y � 8

9. f�1(x) � x � 3 11. f�1(x) � ��x � 4�

Pages 706–708 Lesson 13-11. Trigonometry is the study of the relationships betweenthe angles and sides of a right triangle. 3. Given only themeasures of the angles of a right triangle, you cannot find

the measures of its sides. 5. sin � � ; cos � � �161�;

tan � � ; csc � � ; sec � � �161�; cot � �

7. cos 23° � �3x2�; x � 34.8 9. B � 45°, a � 6, c � 8.5

11. a � 16.6, A � 67°, B � 23° 13. 1660 ft 15. sin � � �141�;

cos � � ; tan � � ; csc � � �141�; sec � � ;

cot � � 17. sin � � ; cos � � �34

�; tan � � ;

csc � � ; sec � � �43

�; cot � � 19. sin � � ; �5��

3�7��

4�7��

�7��7�

��105��

11�105��

4�105��105�

�

6�85��

11�85��85�

�

�85��

xO

f (x)

f (x) � x 2 � 4

f �1(x) � �x � 4 ��

xO

f (x)

f �1(x) � x � 3

f (x) � x � 3

y

xO

x � y � 4

Sele

cted A

nsw

ers


cos � � ; tan � � �12

�; csc � � �5�; sec � � ;

cot � � 2 21. tan 30° � �1x0�, x � 5.8 23. sin 54° � �

17x.8�,

x � 22.0 25. cos x° � �13

56�, x � 65

27a. sin 30° � �oh

pyp

p� sine ratio

sin 30° � �2xx� Replace opp with x and hyp with 2x.

sin 30° � �12

� Simplify.

27b. cos 30° � �haydpj

� cosine ratio

cos 30° �

cos 30° � Simplify.

27c. sin 60° � �oh

pyp

p� sine ratio

sin 60° �

sin 60° � Simplify.

29. B � 74°, a � 3.9, b � 13.5 31. B � 56°, b � 14.8, c � 17.933. A � 60°, a � 19.1, c � 22 35. A � 72°, b � 1.3, c � 4.137. A � 63°, B � 27°, a � 11.5 39. A � 49°, B � 41º, a � 8, c � 10.6 41. about 300 ft 43. about 6° 45. 93.53 units2

47. The sine and cosine ratios of acute angles of righttriangles each have the longest measure of the triangle, thehypotenuse, as their denominator. A fraction whosedenominator is greater than its numerator is less than 1.The tangent ratio of an acute angle of a right triangle does

not involve the measure of the hypotenuse, �oapdpj

�. If the

measure of the opposite side is greater than the measure ofthe adjacent side, the tangent ratio is greater than 1. If themeasure of the opposite side is less than the measure of theadjacent side, the tangent ratio is less than 1. 49. C 51.No; band members may be more likely to like the samekinds of music. 53. �3

8� 55. �1

156� 57. {�2, �1, 0, 1, 2} 59.

20 qt 61. 12 m2

Pages 712–715 Lesson 13-21. reals3. 5.

7. 9. ��18� 11. 135° 13. 1140°

15. 785°, �295° 17. 21 h

19. 21.

23. 25.

27. �23� 29. ��

12� 31. �

113� 33. �

7990� 35. 150°

37. �45° 39. 1305° 41. �16

20� � 515.7°

43. Sample answer: 585°, �135° 45. Sample answer: 345°,�375° 47. Sample answer: 8°, �352° 49. Sample answer:

�11

4�, ��

54� 51. Sample answer: �

34�, ��

134�

53. Sample answer: �132�, ��

32� 55. 2689° per second;

47 radians per second 57. about 188.5 m2 59. about640.88 in2 61. Student answers should include the following.• An angle with a measure of more than 180° gives anindication of motion in a circular path that ended at a pointmore than halfway around the circle from where it started. • Negative angles convey the same meaning as positiveangles, but in an opposite direction. The standardconvention is that negative angles represent rotations in aclockwise direction.• Rates over 360° per minute indicate that an object isrotating or revolving more than one revolution per minute.63. D 65. A � 22°, a � 5.9, c � 15.9 67. c � 0.8, A � 30°, B � 60° 69. about 7.07% 71. combination, 3573. [g° h](x) � 4x2 � 6x � 23, [h° g](x) � 8x2 � 34x � 4475. 1418.2 or about 1418; the number of sports radio

stations in 2008 77. 79. 81.

Page 715 Practice Quiz 11. B � 42°, a � 13.3, c � 17.93.

5. �1198� 7. 210° 9. 305°; �415°


1. False; sec 0° � �rr

� or 1 and tan 0° � �0r

� or 0. 3. To find the value of a trigonometric function of �, where� is greater than 90°, find the value of the trigonometricfunction for ��, then use the quadrant in which the terminal

O

y

x�60˚

�10��10�

�3�5��

O

y

x

�

O

y

x

�150˚

O

y

x790˚

O

y

x

235˚

O

y

x�45˚

O

y

x

300˚

O

y

x

�70˚

290˚

�3��

Replace opp with �3�x andhyp with 2x.

�3�x�

�3��

Replace adj with �3�x andhyp with 2x.

�3x��

�5��

2�5��

Sele

cted A

nsw

ers


side of � lies to determine the sign of the trigonometricfunction value of �. 5. sin � � 0, cos � � �1, tan � � 0, csc � � undefined, sec � � �1, cot � � undefined7. 55° 9. 60º

11. �1 13. � 15. sin � � � , cos � � ,

tan � � ��2�, csc � � � , sec � � �3� 17. sin � � �2245�,

cos � � �275�, tan � � �

274�, csc � � �

22

54�, sec � � �

275�, cot � � �

274�

19. sin � � � , cos � � , tan � � ��85

�,

csc � � � , sec � � , cot � � ��58

� 21. sin � � �1,

cos � � 0, tan � � undefined, csc � � �1,

sec � � undefined, cot � � 0 23. sin � � � ,

cos � � , tan � � �1, csc � � ��2�, sec � � �2�,

cot � � �1

25. 45° 27. 30°

29. �4

� 31. �7

�

33. � 35. ��3� 37. undefined 39. �3�

41. undefined 43. 45. 0.2, 0, �0.2, 0, 0.2, 0, and

�0.2; or about 11.5°, 0°, �11.5°, 0°, 11.5°, 0°, and �11.5°

47. sin � � ��45

�, tan � � ��43

�, csc � � ��54

�, sec � � �53

�,

cot � � ��34

� 49. cos � � � , tan � � � , csc � � 3,

sec � � � , cot � � �2�2� 51. sin � � �

cos � � � , tan � � 3, csc � � � , cot � � �31

�

53. about 173.2 ft 55. 9 meters 57. II59. Answers should include the following.• The cosine of any angle is defined as �

xr�, where x is the

x-coordinate of any point on the terminal ray of the angleand r is the distance from the origin to that point. Thismeans that for angles with terminal sides to the left of they-axis, the cosine is negative, and those with terminal sidesto the right of the y-axis, the cosine is positive. Thereforethe cosine function can be used to model real-world datathat oscillate between being positive and negative.• If we knew the length of the cable we could find thevertical distance from the top of the tower to the rider.Then if we knew the height of the tower we could subtractfrom it the vertical distance calculated previously. This willleave the height of the rider from the ground.

61. ��52

�, � 63. 300° 65. sin 28° � �1x2�, 5.6

67. sin x° � �153�, 23 69. (7, 2) 71. (5, �4) 73. 15.1

75. 32.9° 77. 39.6°

Pages 729–732 Lesson 13-41. Sometimes; only when A is acute, a � b sin A or a � band when A is obtuse, a � b.3. Gabe said there is not enough information to do this problem. That is not correct. By using the Law of Sines, he can find ∠ B. Therefore, hecan find ∠ C. ∠ C � 180° � (64° � m∠ B). Once ∠ C is found, A � �

12� ba sin C will yield the area of the triangle.

5. 6.4 cm2 7. B � 80°, a � 32.0, b � 32.6 9. no solution 11. one; B � 24°, C � 101°, c � 12.0 13. 5.5 m15. 19.5 yd2 17. 62.4 cm2 19. 14.6 mi2 21. C � 73°, a � 55.6, b � 48.2 23. B � 46°, C � 69°, c � 5.125. A � 40°, B � 65°, b � 2.8 27. A � 20°, a � 22.1, c � 39.8 29. one; B � 36°, C � 45°, c � 1.8 31. no 33. one; B � 18°, C � 101°, c � 25.8 35. two; B � 85°, C � 15°, c � 2.4; B � 95°, C � 5°, c � 0.8 37. two; B � 65°, C � 68°, c � 84.9; B � 115°, C � 18°, c � 28.339. 7.5 mi from Ranger B, 10.9 mi from Ranger A41. 107 mph 43. Answers should include the following.• If the height of the triangle is not given, but the measureof two sides and their included angle are given, then theformula for the area of a triangle using the sine functionshould be used.• You might use this formula to find the area of atriangular piece of land, since it might be easier to measuretwo sides and use surveying equipment to measure theincluded angle than to measure the perpendicular distancefrom one vertex to its opposite side.

• The area of �ABC is �12

�ah.

sin B � �hc

� or h � c sin BA

C

B

a

h

b

c

5�3��

�10��

3�10��

10

3�10��

103�2��

�2��

2�2��

�3��

�3��

O

y

x

13�7

�'O

y

x

5�4

�'

O

y

x

�210˚

�'

O

y

x

315˚

�'

�2��

�2��

�89��89�

�

5�89��

8�89��

�6��

�3��6�

�2�3��

O

y

x

�240˚

�'

O

y

x

235˚

�'

A B

C

8 m 15 m

64°

Sele

cted A

nsw

ers


• Area � �12

�ah or Area � �12

�a(c sin B)

45. B � 78°, a � 50.1, c � 56.1 47. 49. 660°, �60°

51. �17

6�, ��

76� 53. �

25251

� 55. 5.6 57. 39.4°

Pages 735–738 Lesson 13-51. Mateo; the angle given is not between the two sides;therefore the Law of Sines should be used.3. Sample answer:

5. sines; B � 70°, a � 9.6, b � 14 7. cosines; A � 23°, B � 67°, C � 90° 9. 94.3° 11. cosines; A � 48°, B � 63°, C � 70° 13. sines; B � 102°, C � 44°, b � 21.015. A � 80°, a � 10.9, c � 5.4 17. cosines; A � 30°, B � 110°, C � 40° 19. sines; C � 77°, b � 31.7, c � 31.621. no 23. cosines; A � 52°, C � 109°, b � 21.0 25. cosines; A � 24°, B � 125°, C � 31° 27. sines; B � 49°, C � 91°, c � 9.3 29. about 100.1° 31. 4.4 cm,9.0 cm 33. 91.6°35. Answers should include the following.• The Law of Cosines can be used when you know allthree sides of a triangle or when you know two sides andthe included angle. It can even be used with two sides andthe nonincluded angle. This set of conditions leaves aquadratic equation to be solved. It may have one, two, orno solution just like the SSA case with the Law of Sines.• Given the latitude of a point on the surface of Earth, youcan use the radius of the Earth and the orbiting height of asatellite in geosynchronous orbit to create a triangle. Thistriangle will have two known sides and the measure of theincluded angle. Find the third side using the Law ofCosines and then use the Law of Sines to determine theangles of the triangle. Subtract 90 degrees from the anglewith its vertex on Earth’s surface to find the angle at whichto aim the receiver dish.37. A 39. Sample answer: 100.2° 41. one; B � 46°,

C � 79°, c � 9.6 43. sin � � �11

23�, cos � � �

153�, tan � � �

152�,

csc � � �11

32�, sec � � �

153�, cot � � �

152� 45. sin � � ,

cos � � , tan � � , csc � � , sec � � ,

cot � � 47. {x x � �0.6931} 49. 405°, �315°

51. 540°, �180° 53. �19

6�, ��

56�


1. sin � � ; cos � � � ; tan � � ��32

�;

csc � � ; sec � � � ; cot � � ��23

� 3. 27.7 m2

5. cosines; c � 15.9, C � 59°, B � 43°

Pages 742–745 Lesson 13-61. The terminal side of the angle � in standard positionmust intersect the unit circle at P(x, y). 3. Sample answer:The graphs have the same shape, but cross the x-axis at

different points. 5. sin � � ; cos � � 7. ��12

�

9. 2 s 11. sin � � �45

�; cos � � ��35

� 13. sin � � �1157�; cos � � �

187�

15. sin � � ; cos � � ��12

� 17. ��12

� 19. �1 21. 1

23. �14

� 25. 27. �3�3� 29. 6 31. 2 33. �4140� s

35. ��12

�, , ��12

�, , (�1, 0), ��12

�, � , ��12

�, � 37. �

yx

� 39. ��xy

� 41. �3� 43. sine: D � {all reals}, R � {�1 �

y � 1}; cosine: D � {all reals}, R � {�1 � y � 1} 45. A47. cosines; c � 12.4, B � 59°, A � 76° 49. 27.0 in2

51. 6800 53. 5000 55. 250 57. does not exist 59. 861. 2x � 9 63. 2y � 7 � �

y �5

3� 65. 110° 67. 80° 69. 89°

Pages 749–751 Lesson 13-71. Restricted domains are denoted with a capital letter.3. They are inverses of each other. 5. � � Arccos 0.5 7. 0°9. � 3.14 11. 0.75 13. 0.58 15. � � Arcsin �17. y � Arccos x 19. Arccos y � 45° 21. 60° 23. 45°25. 45° 27. 2.09 29. 0.52 31. 0.5 33. 0.60 35. 0.8

37. 0.5 39. �0.5 41. 0.71 43. 0.96 45. 60° south of west

47. No; with this point on the terminal side of the throwingangle �, the measure of � is found by solving the equation

tan � � �1178�. Thus � � tan�1 �

1178� or about 43.4°, which is

greater than the 40° requirement. 49. 31° 51. SupposeP(x1, y1) and Q(x2, y2) lie on the line y � mx � b. Then

m � �yx

2

2

�

�

yx

1

1�. The tangent of the angle � the line makes with

the positive x-axis is equal to the ratio �oapdpj

� or �yx

2

2

�

�

yx

1

1�. Thus

tan � � m.

53. 37°55.

57. From a right triangle perspective, if an acute angle � has

a given sine, say x, then the complementary angle �

2� � �

has that same value as its cosine. This can be verified bylooking at a right triangle. Therefore, the sum of the anglewhose sine is x and the angle whose cosine is x should be �

2�.

59. �1 61. sines; B � 69°, C � 81°, c � 6.1 or B � 111°, C � 39°, c � 3.9 63. 46, 39 65. 11, 109

y

xO

P (x1, y1)

Q (x2, y2)

�

x2 � x1

y2 � y1

y � mx � b

�3��3�

��3��3�

�

1 � �3��

�3��

�2��2�

�

�13��13�

�

2�13��

3�13��

�15��

2�10��

2�6��15�

��10��

�6��

15

13

9

�3��

0 �12

� 1 ��12

� � � �1

�2

� �2

� �2

� �2

� �2

� �2

� �2

� �2

� �2

�

�3��

2�2��

2�3��

2�2��

2x

y

Sele

cted A

nsw

ers


Pages 752–756 Chapter 13 Study Guide and Review1. false, coterminal 3. true 5. true 7. false, an angle thathas its terminal side on an axis where x or y is equal to zero9. false, terminal 11. B � 65°, a � 2.5, b � 5.413. A � 7°, a � 0.7, c � 5.6 15. A � 76°, B � 14°, b � 1.0,

c � 4.1 17. ��76� 19. �720° 21. 320°, �400°

23. �4

�; ��15

4� 25. sin � � ��

187�, cos � � �

11

57�, tan � � ��1

85�,

csc � � ��187�, sec � � �

11

75�, cot � � ��

185� 27. ��3�

29. 31. two; B � 53°, C � 87°, c � 12.4; B � 127°,

C � 13°, c � 3.0 33. no 35. one; A � 51°, a � 70.2, c � 89.7 37. sines; C � 105°, a � 28.3, c � 38.639. cosines; A � 34°, B � 81°, c � 6.4 41. cosines; B � 26°,

C � 125°, a � 8.3 43. �12

� 45. � 47. ��3� 49. 1.05

51. 0

Chapter 14 Trigonometric Graphs andIdentitiesPage 761 Chapter 14 Getting Started

1. ��2

2�� 3. 0 5. ��

�2

2�� 7. ��

12

� 9. ��2

3�� 11. 1 13. not

defined 15. �12

� 17. �5x(3x � 1) 19. prime

21. (2x � 1)(x � 2) 23. 8, �3 25. �8, 5 27. �4, ��32

�

Pages 766–768 Lesson 14-11. Sample answer: Amplitude is half the difference betweenthe maximum and minimum values of a graph; y � tan �has no maximum or minimum value. 3. Jamile; Theamplitude is 3 and the period is 3. 5. amplitude: 2;period: 360° or 2

7. amplitude : does not exist; period: 180° or

9. amplitude: 4; period: 180° or

11. amplitude: does not exist; period: 120° or �23�

13. 12 months; Sample answer: The pattern in thepopulation will repeat itself every 12 months.

15. amplitude: 3; period: 360° or 2

17. amplitude: does not exist; period: 360° or 2

90˚�180˚�270˚ �90˚ 180˚ 270˚

2

45

1

3

�2�3�4�5

y

�O

y � 2 csc �

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � 3 sin �

30˚�60˚ �30˚ 60˚ 90˚ 120˚ 150˚

1

2

0.5

1.5

�1�1.5

�2

y

�

O

y � sec 3�12

90˚ 180˚ 270˚ 360˚

2

4

1

3

5

�2�1

�3�4�5

y

�O

y � 4 sin 2�

90˚�90˚ 180˚�180˚ 270˚�270˚

1

2

0.5

1.5

�1�1.5

�2

y

�O

y � tan �14

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � 2 sin �

�2��

2�3��

Sele

cted A

nsw

ers


19. amplitude: �15

�; period: 360° or 2

21. amplitude: 1; period 90° or �2

�

23. amplitude: does not exist; period: 120° or �23�


27. amplitude: 6; period: 540° or 3


31. amplitude: does not exist; period: 180° or

33.

y � �35

� sin 4� 35. �1107� 37. Sample answer: The amplitudes

are the same. As the frequency increases, the perioddecreases.

39. y � 2 sin �5

�t

45̊�45˚ 90˚�90̊ 135˚�135˚

2

4

1

3

5

�2�3�4�5

y

�O

y � sin 4�35

�90˚ 180˚90˚�180˚�270˚ 270˚

4

810

2

6

�4�6�8

�10

y

�O

2y � tan �

180˚�360˚�540˚ �180˚ 360˚ 540˚

4

810

2

6

�4�6�8

�10

y

�O

y � 3 csc �12

90˚�90˚ 180˚�180˚ 270˚�270˚

4

8

2

6

10

�4�6�8

�10

y

�O

y � 6 sin �23

270˚�270˚ 540˚�540˚ 810˚�810˚

4

810

2

6

�4�6�8

�10

y

�O

y � 4 tan �13

30˚�60˚ �30˚ 60˚

2

45

1

3

�2�3�4�5

y

�O

y � sec 3�

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � sin 4�

90˚�90˚ 180˚�180˚ 270˚�270˚

0.4

0.8

0.2

0.6

1

�0.4�0.6�0.8

�1

y

�O

y � sin �15

Sele

cted A

nsw

ers

41. about 1.9 ft 43. A 45. 90° 47. 45° 49. ��2

2�� 51. �

1136�

53.

55.

Pages 774–776 Lesson 14-21. vertical shift: 15; amplitude: 3; period: 180°; phase shift:45° 3. Sample answer: y � sin (� � 45°) 5. no amplitude;180°; �60°

7. no amplitude; 2; ��3

�

9. �5; y � �5; no amplitude; 360°

11. 0.25; y � 0.25; 1; 360°

13. �6; no amplitude; 60°; �45°

15. �2; �23

� ; 4; ��6

�

2��2� ��

1

�1

�2

�3

3��3�

y

�O

y � cos [ (� � )] � 212

23

�6

45˚�45˚

1

�2�3�4�5�6�7�8�9

�10�11

y

�O

y � 2 cot (3� � 135 )̊ � 6

90˚

1

0.5

1.5

�0.5

�1

�1.5

180˚ 270˚ 360˚

y

�O

y � sin � � 0.25

90˚�180˚�270˚ �90˚ 180˚ 270˚

4

810

2

6

�4�6�8

�10

y

�O

y � sec � � 5

��

234

1

�1�2�3�4

y

O

y � sec (� � )�3

�2�

�2

3�2�

3�2

�

90˚�90˚ 180˚�180˚ 270˚�270˚

2

45

1

3

�2�3�4�5

y

�O

y � tan (� � 60 )̊

�4�8 4 8

3

79

1

5

1315

11

�3�5

y

O x

y � 2x 2

y � 2(x � 1)2

�4�8 4 8

3

79

1

5

1315

11

�3�5

y

O x

y � x 2

y � 3x 2


Sele

cted A

nsw

ers

17. h � 4 � cos �2

�t or h � 4 � cos 90°t

19. 1; 360°; �90°

21. 1; 2; �4

�

23. no amplitude; 180°; �22.5°

25. �1; y � �1; 1; 360°

27. �5; y � �5; 1; 360°

29. �12

�; y � �12

�; �12

� ; 360°

31.

translation �4

� units left and 5 units up

33. 1; 2; 120°; 45°

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�

O

y � 2 sin [3(� � 45 )̊] � 1

��4

�2

�4�

�2

3�4

3�4

12

1618

2468

10

14

y

�O

y � 5 � tan (� � )�4

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � sin � �12

12

90˚�90˚ 180˚�180˚ 270˚�270˚

21

�2�3�4�5�6�7�8

y

�O

y � cos � � 5

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�

O

y � sin � � 1

�45̊ 90˚45˚�90˚�135˚ 135˚

2

45

1

3

�2�3�4�5

y

�O

y � tan (� � 22.5˚)14

��

y � sin (� � )�4

�2

�2

3�2

3�2

��

2

4

1

3

5

�2�3�4�5

y

�

O

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � cos (� � 90 )̊


Sele

cted A

nsw

ers

35. �3.5; does not exist; 720°; �60°

37. 1; �14

� ; 180°; 75°

39. 3; 2; ; ��4

�

41.

The graphs are identical. 43. c 45. 300; 14.5 yr

47. h � 9 � 6 sin ��9

�(t � 1.5)

49. Sample answer: You can use changes in amplitude andperiod along with vertical and horizontal shifts to show ananimal population’s starting point and display changes tothat population over a period of time. Answers shouldinclude the following information.• The equation shows a rabbit population that begins at

1200, increases to a maximum of 1450 then decreases to aminimum of 950 over a period of 4 years.

• Relative to y � a cos bx, y � a cos bx � k would have avertical shift of k units, while y � a cos [b(x � h)] has ahorizontal shift of h units.

51. D 53. amplitude: 1; period: 720° or 4

55. 0.75 57. 0.83 59. 35 61. 0.66 63. �(a �5a

2�)(a

1�3

3)�

65. �23(yy

2

�

�

51)0(yy

�

�

35)

� 67. �1 69. �12

� 71. ��33�

� 73. 1

Pages 779–781 Lesson 14-31. Sample answer: The sine function is negative in the third and fourth quadrants. Therefore, the terminal side of the angle must lie in one of those two quadrants.

3. Sample answer: Simplifying a trigonometric expressionmeans writing the expression as a numerical value or interms of a single trigonometric function, if possible.

5. ��54

� 7. �2� 9. tan2 � 11. csc � 13. �12

� 15. ��5�

17. �54

� 19. 21. �34

� 23. � 25. cot � 27. cos �

29. 2 31. cot2 � 33. 1 35. csc2 � 37. about 11.5°

39. about 9.4° 41. No; R2 � �I tan �

Ecos �� simplifies to

E � �I s

Rin

2�

�. 43. P � I2R � �1 � ta

I2

nR2 2 ft�.

45. Sample answer: You can use equations to find theheight and the horizontal distance of a baseball after it hasbeen hit. The equations involve using the initial angle theball makes with the ground with the sine function. Answersshould include the following information.• Both equations are quadratic in nature with a leading

negative coefficient. Thus, both are inverted parabolaswhich model the path of a baseball.

• model rockets, hitting a golf ball, kicking a rock

4�7��3�

�

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � sin �2

2

4

1

3

5

�2�3�4�5

y

�O

y � 3 � cos �12

y � 3 � cos (� � �)12

��2

�2

�� 3�2

3�2

5

7

4321

6

8

�2

y

�O��

�2

�2

�� 3�2

3�2

y � 3 � 2 sin [2(� � )]�4

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � cos (2� � 150 )̊ � 114

90˚�180˚�270˚ �90˚ 180˚ 270˚

2

68

4

�4�6�8

�10�12

y

�O

y � 3 csc [ (� � 60 )̊] � 3.512


Sele

cted A

nsw

ers

47. A 49. 12; y � 12; no amplitude; 180°

51. amplitude: 1; period: 120° or �23�

53. 93 55. Symmetric (�) 57. Multiplication (�)


1. �34

� , 720° or 4

3. ��35

� 5. ��2

5��

Pages 784–785 Lesson 14-41. sin � tan � � sec � � cos �

sin � tan � � �co

1s �� cos � sec � � �

co1s ��

sin � tan � � �co

1s ��

ccooss2

��

� Multiply by the LCD, cos �.

sin � tan � � �1 �

cocso�s2 �

� Subtract.

sin � tan � � �scions2

��

� 1 � cos2 � � sin2 �

sin � tan � � sin � � �csoin

s��

� Factor.

sin � tan � � sin � tan � �csoins

��

�� tan �

3. Sample answer: sin2 � � 1 � cos2 �; it is not an identitybecause sin2 � � 1 � cos2 �.

5. tan2 � cos2 � � 1 � cos2 �

�csoin

s2

2��

� cos2 � � sin2 �

sin2 � � sin2 �

7. �1 �

cstcan

�

2 �� tan2 �

�scescc

2

2

��

� � tan2 �

� tan2 �

�cos

12 �� sin2 � � tan2 �

tan2 � � tan2 �

9. �setca�n

��

1� � �

setca�n

��

1�

�se

tca�n

��

1� � �

setca�n

��

1� � �

sseecc

��

��

11

�

�se

tca�n

��

1� ��

tans�e�c2(s

�ec

��

1� 1)

�

�se

tca�n

��

1� ��

tan � �ta(sne2c�� 1)

�

�se

tca�n

��

1� � �

setca�n

��

1�

11. cos2 � � tan2 � cos2 � � 1

cos2 � � �csoin

s2

2��

� � cos2 � � 1

cos2 � � sin2 � � 11 � 1

13. 1 � sec2 � sin2 � � sec2 �

1 � �cos

12 �� sin2 � � sec2 �

1 � tan2 � � sec2 �sec2 � � sec2 �

15. �11��

ccoo

ss

��

� � (csc � � cot �)2

�11

��

ccoo

ss

��

� � csc2 � � 2 cot � csc � � cot2 �

�11

��

ccoo

ss

��

� � �sin

12 �� 2 � �

csoin

s��

� � �sin

1�

� � �csoin

s2

2

��

�

�11

��

ccoo

ss

��

� � �sin

12 ��

2si

cno2s��

� � �csoin

s2

2

��

�

�11

��

ccoo

ss

��

� �

�11

��

ccoo

ss

��

� �

�11

��

ccoo

ss

��

� �

�11

��

ccoo

ss

��

� � �11

��

ccoo

ss

��

�

17. cot � csc � � �sciont

��

��

tcasnc �

��

cot � csc � �

cot � csc � �

�co

ssi�n

��

1�

cot � csc � �

cot � csc � � �co

ssi�n

��

1� ��sin �(

ccooss

�� 1)�

cot � csc � � �csoin

s��

� � �sin

1�

�

cot � csc � � cot � csc �

�co

ssi�n

��

1�

��sin �(

ccooss

�� 1)�

sin � cos � � sin ��

cos �

�csoin

s��

� � �sin

1�

�

��sin � � �

csoin

s��

�

(1 � cos �)(1 � cos �)��(1 � cos �)(1 � cos �)

(1 � cos �)(1 � cos �)��

1 � cos2 �

1� 2 cos � � cos2 ��

sin2 �

�cos

12 ��

��sin

12 ��

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � sin �34

12

45˚�45˚ 90˚�90˚ 135˚�135˚

2

4

1

3

5

�2�3�4�5

y

�O

y � cos 3�

90˚�90˚ 180˚�180˚ 270˚�270˚

10

15

20

5

�5

y

�O

y � tan � � 12


Sele

cted A

nsw

ers

19. �sseinc

��

� � �csoin

s��

� � cot �

� �csoin

s��

� � cot �

�sin �

1cos ��

sinsi

�n2

co�s �

� � cot �

�s1in�

�scino

2

s��

� � cot �

�sin

co�sc2

o�s �

� � cot �

�csoin

s��

� � cot �

cot � � cot �

21. �1 �sin

si�n �

� � �cs

cco�t2

��

1�

�1 �

sinsi

�n �

� � �cs

cco�t2

��

1� � �

ccsscc

��

��

11

�

�1 �

sinsi

�n �

� � �cot2

cs�c(2cs

�c��

11)

�

�1 �

sinsi

�n �

� � �cot2 �

c(ocst2c

�� 1)�

�1 �

sinsi

�n �

� � csc � � 1

�1 �

sinsi

�n �

� � �sin

1�

� � �ssiinn

��

�

�1 �

sinsi

�n �

� � �1 �

sinsi

�n �

�

23. �sec

12 ��

csc12 �� 1

cos2 � � sin2 � � 1

1 � 1

25. 1 � tan4 � � 2 sec2 � � sec4 �(1 � tan2 �)(1 � tan2 �) � sec2 � (2 � sec2 �)

[1 � (sec2 � � 1)](sec2 �) � (2 � sec2 �)(sec2 �)(2 � sec2 �)(sec2 �) � (2 � sec2 �)(sec2 �)

27. �1 �

sinco

�s �

� � �1 �

sinco

�s �

�

�1 �

sinco

�s �

� � �11

��

ccoo

ss

��

� � �1 �

sinco

�s �

�

�sin

1��(1

c�os

c2

o�s �)

�� 1 �

sinco

�s �

�

�sin �(

s1in

�

2 �cos �)

�� 1 �

sinco

�s �

�

�1 �

sinco

�s �

� � �1 �

sinco

�s �

�

29. tan � sin � cos � csc2 � � 1

�csoin

s��

� � sin � � cos � � �sin

12 �� 1

1 � 1

31. �

� �2vg2� � �

csoin

s2

2��

� � �cos

12 ��

� �v2 s

2ign2 ��

33. Sample answer: Consider a right triangle ABC withright angle at C. If an angle, say A, has a sine of x, thenangle B must have a cosine of x. Since A and B are both in aright triangle and neither is the right angle, their sum mustbe �

2

�. 35. D

37.

is not

39.

may be41.

may be

43. 45. 47. 1: 360°; 30°

49. 3; 2; ��2

�

51. 53.

Pages 788–790 Lesson 14-51. sin (� � �) � sin � � sin �sin � cos � � cos � sin � ≠ sin � � sin �

�6� � 2�2��6�

�

��2

�2

�� 3�2

3�2

2

4

1

3

5

�2�3�4�5

y

�O

y � 3 cos (� � )�2

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � cos (� � 30 )̊

�193��5�

�

[�360, 360] scl: 90 by [�5, 5] scl: 1

[�360, 360] scl: 90 by [�5, 5] scl: 1

[�360, 360] scl: 90 by [�5, 5] scl: 1

v2 �csoin

s2

2��

�

��2g �cos

12 ��

v2 tan2 ��2 sec2 �

�co

1s ��

�sin �


Sele

cted A

nsw

ers

3. Sometimes; sample answer: The cosine

function can equal 1. 5. 7.

9. ��12

�

11. sin �� 2

� � cos �

sin � cos �2

� � cos � sin �2

� � cos �

sin � · 0 � cos � · 1 � cos �cos � � cos �

13. �15�

�

5��

3�3�

� 15. 17.

19. 21. ��2

2�� 23. ��2

2�� 25.

27.

29. cos (90° � �) � cos 90° cos � � sin 90° sin �� 0 � 1 sin �� sin �

31. sin (90° � �) � cos �sin 90° cos � � cos 90° sin � � cos �

1 · cos � � 0 · sin � � cos �cos � � 0 � cos �

cos � � cos �33. cos ( � �) � �cos �

cos cos � � sin sin � � �cos ��1 · cos � � 0 · sin � � �cos �

�cos � � �cos �35. sin ( � �) � sin �

sin cos � � [cos sin �] � sin �0 · cos � � [�1 · sin �] � sin �

0 � [�sin �] � sin �sin � � sin �

37. sin �� 3

� � cos �� 6

�� sin � cos �

3

� � cos � sin �3

� � cos � cos �6

� � sin � sin �6

�

� �12

� sin � � ��2

3�� cos � � �

�2

3�� cos � � �

12

� sin �

� �12

� sin � ��12

� sin �

� sin �

39. cos (� � �) � �1 �

setca�n

s�ecta

�n �

�

cos (� � �) �

cos (� � �) � � �ccoo

ss

��

ccoo

ss

��

�

cos (� � �) �

cos (� � �) � cos (� � �)41. Destructive; the resulting graph has a smaller amplitudethan the two initial graphs. 43. 0.4179 E 45. 0.5564 E47. Sample answer: To determine communicationinterference, you need to determine the sine or cosine of thesum or difference of two angles. Answers should includethe following information.• Interference occurs when waves pass through the same

space at the same time. When the combined waves havea greater amplitude, constructive interference results andwhen the combined waves have a smaller amplitude,destructive interference results.

49. C51. sin2 � � tan2 � � (1 � cos2 �) � �

scescc

2

2

��

�

sin2 � � tan2 � � sin2 � � �scescc

2

2

��

�

sin2 � � tan2 � � sin2 � � �cos

12 ��

sin12 ��

sin2 � � tan2 � � sin2 � � �csoin

s2

2��

�

sin2 � � tan2 � � sin2 � � tan2 �

53. �tsaenc �

�� csc �

�co

1s ��

csoin

s��

� � csc �

�co

1s ��

csoin

s��

� � csc �

�sin

1�

� � csc �

csc � � csc �55. 4 57. 2 sec � 59. sin � � ��

45

� , cos � � ��35

� , tan � � �43

�,

csc � � ��54

� , sec � � ��53

� , cot � � �34

� 61. 360 63. 56

65. about 228 mi 67. ��25�

� 69. ��5

5�� 71. ��

�2

6��

73.

Pages 794–797 Lesson 14-61. Sample answer: If x is in the third quadrant, then �

x2

� is

between 90° and 135°. Use the half-angle formula for cosineknowing that the value is negative. 3. Sample answer: Theidentity used for cos 2� depends on whether you know thevalue of sin �, cos � or both values.

5. , ��19

�, ��630��, ��

�6

6�� 7. ��

3�8

7��, ��

18

�,

��8 �

42��7��, ��8 �

42��7�� 9. ��2 �

2��3��

�

11. cos2 2x � 4 sin2 x cos2 x � 1cos2 2x � sin2 2x � 1

1 � 1

13. ��112609

�, �11619

9�, �5�

2626�

�, ��2626�� 15. �4�

92�

�, ��79

�, ��36�

�, ��3

3��

17. ��3�

3255�

�, �2332�, ��8 �

4��55��, ��8 �

4��55��

19. ��1835��, ��

1178�, ��6

15��, ��6

21�� 21. ��

4�9

2��, �

79

�, ��18�6� 12�2��,

��18�

6� 12�2�� 23. �4�

95�

�, ��19

�, ��66�

�, ��360�

� 25. ��2 �

2��3��

�

27. ��2 �

2��2��

� 29. ��2 �2

��2��

31. sin 2x � 2 cot x sin2 x

2 sin x cos x � 2 �csoin

sxx

� � sin2 x

2 sin x cos x � 2 sin x cos x33. sin4 x � cos4 x � 2 sin2 x � 1

(sin2 x � cos2 x)( sin2 x � cos2 x) � 2 sin2 x � 1(sin2 x � cos2 x) � 1 � 2 sin2 x � 1

[sin2 x � (1 � sin2 x)] � 1 � 2 sin2 x � 1sin2 x � 1 � sin2 x � 2 sin2 x � 1

2 sin2 x � 1 � 2 sin2 x � 1

35. tan2 �x2

� � �11

��

ccoo

ss

xx

�

� �11

��

ccoo

ss

xx

�

� �11

��

ccoo

ss

xx

�

�11

��

ccoo

ss

xx

� � �11

��

ccoo

ss

xx

�

��1 �

2co�s x��2

��

sin2 �x2

�

�cos2 �

x2

�

4�5��

��6� �� 2��

2

cos � cos � � sin � sin ��

1

1 � �csoin

s��

� � �csoin

s��

�

��co

1s ��

co1s ��

1 � �csoin

s��

� � �csoin

s��

�

��co

1s ��

co1s ��

��6� � �2��

�2�� 6��

��6� � �2��

��6� � �2��2�� 6�

��

�3��

�6� � �2��


Sele

cted A

nsw

ers

37. 46.3° 39. 2 � �3� 41. �14

� tan � 43. The maxima occur

at x � ��2

� and ��32�. The minima occur at x � 0, � and

�2. 45. The graph of f(x) crosses the x-axis at the pointsspecified in Exercise 43. 47. Sample answer: The soundwaves associated with music can be modeled usingtrigonometric functions. Answers should include thefollowing information.• In moving from one harmonic to the next, the number of

vibrations that appear as sine waves increase by 1.• The period of the function as you move from the nth

harmonic to the (n � 1)th harmonic decreases from

�2n� to �

n2�

1�.

49. B 51. ��6� �4

�2�� 53. ��

�2

3�� 55. �

12

�

57. cos �(cos � � cot �) � cot � cos �(sin � � 1)

cos �(cos � � cot �) � �csoin

s��

� cos � sin � � cot � cos �

cos �(cos � � cot �) � cos2 � � cot � cos �cos �(cos � � cot �) � cos �(cos � � cot �)

59. 102.5 or about 316 times greater 61. 1, �1 63. �52

� , �265. 0, ��

12

�

Page 797 Practice Quiz 21. sin � sec � � tan �

sin � � �co

1s �� tan �

�csoin

s��

� � tan �

tan � � tan �

3. sin � � tan � � �sin �(

ccooss

�� 1)�

sin � � tan � �

sin � � tan � � �sin

c�os

co�s �

� � �csoin

s��

�

sin � � tan � � sin � � tan �

5. cos ��32� � � � �sin �

cos �32� cos � � sin �

32� sin � � �sin �

0 � (�1 � sin �) � �sin �

�sin � � �sin �

7. ��23�

� 9. ��2 �2

��3��


1. Sample answer: If sec � � 0 then �co1s �� 0. Since no

value of � makes �co1s �� 0, there are no solutions.

3. Sample answer: sin � � 2 5. 135°, 225° 7. �6

�

9. 0 � k 11. 60° � k � 360°, 300° � k � 360°

13. �6

� � 2k, �56� � 2k , �

2

� � 2k or 30° � k � 360°,

150° � k � 360°, 90° � k � 360° 15. 60°, 300° 17. 210°, 330°

19. �6

�, �56�, �

32� 21. �

76�, �11

6� 23. �

3

� � 2k, �53� � 2k

25. �23�� 2k, �

43� � 2k 27. �

3

� � 2k, �53� � 2k

29. 45° � k � 180° 31. 270° � k � 360° 33. 0° � k � 180°,

60° � k � 180° 35. 0 � 2k, �2

� � 2k, �32� � 2k or 0° �

k � 360°, 90° � k � 360°, 270° � k � 360° 37. 0 � k or 0° �

k � 180° 39. 0 � 2k, �3

� � 2k, �53� � 2k, or 0° � k � 360°,

60° � k � 360°, 300° � k � 360° 41. S � �ta3n52

�� or S � 352 cot �

43. y � �32

� � �32

�sin (t)

45. (4.964, �0.598) 47. D 49. �2245�, �

275�, ��10

10��, �3�

1010�

�

51. , �178�, ��6

3��, ��6

33�� 53. ��

�2

3�� 55. b � 11.0, c � 12.2,

m � C � 78

Pages 805–808 Chapter 14 Study Guide and Review

1. h 3. d 5. e 7. g 9. amplitude: �12

�; period: 360° or 2π

11. amplitude: 1; period: 720° or 4π

13. amplitude: does not exist; period: 540° or 3π

90˚�90˚

2345

1

�1�2�3�4�5

180˚�180˚ 270˚�270˚ 360˚�360˚

y

O

y � csc �12

23

�

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � sin �12

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�

O

y � � cos �12

5�11��

1 2 3 4 5 6 7 8 9�1

2.5

3.5

21.5

10.5

3

4

�1

y

tO

y � � sin (� t )32

32

sin � cos � � sin ��

cos �


Sele

cted A

nsw

ers

15. �1, �12

�, 180°, 60°

17. 1, does not exist, 4π, ��π4

�

19. ��43

� 21. sin2 � 23. sec �

25. �1 �

sinco

�s �

� � csc � � cot �

�1 �

sinco

�s �

� � �sin

1�

� � �csoin

s��

�

�1 �

sinco

�s �

� � �1 �

sincs

�c �

�

�1 �

sinco

�s �

� � �1 �

sinco

�s �

� � �11

��

ccoo

ss

��

�

�1 �

sinco

�s �

� � �sin

1�

�(1

c�os

c2

o�s �)

�

�1 �

sinco

�s �

� � �sin �

s(1in

�

2 �cos �)

�

�1 �

sinco

�s �

� � �1 �

sinco

�s �

�

27. sec �(sec � � cos �) � tan2 �

�co

1s ��co

1s �� cos � � tan2 �

�cos

12 �� 1 � tan2 �

sec2 � � 1 � tan2 �

tan2 � � tan2 �

29. 31. 33.

35. sin (30 � �) � cos (60 � �)sin 30° cos � � cos 30° sin � � cos 60° cos � � sin 60° sin �

�12

� cos � � sin � � �12

� cos � � sin �

37. �cos � � cos (π � �)�cos � � cos π cos � � sin π sin ��cos � � �1 � cos � � 0 � sin ��cos � � �cos �

39. �112609

�, �116199

�, , � 41. ��112609

�, �116199

�, , �

43. 0° 45. �π6

� � 2kπ, �56π� � 2kπ

5�26��26�

��26��

5�26��

�3��3�

�

��6� � �2��2� � �6�

��6� � �2��

4

810

2

6

�4�6�8

�10

y

�O

y � 3 sec [ (� � )] � 112

��2��3� � 2� 3�

�4

90˚�90˚ 180˚�180˚ 270˚�270˚

2

4

1

3

5

�2�3�4�5

y

�O

y � sin [2(� � 60 )̊] � 112


Photo CreditsP

hoto

Cre

dit

s

R86 Photo Credits

Cover Vanni Archive/CORBIS; x D & K Tapparel/GettyImages; xi Telegraph Colour Library/Getty Images; xii DEXImages Inc./CORBIS Stock Market; xiii AFP/CORBIS; xivCORBIS; xix Jenny Hager/ImageState; xv BrownieHarris/CORBIS Stock Market; xvi Ray F. Hillstrom Jr.; xviiKunio Owaki/CORBIS Stock Market; xviii Jane Burton/Bruce Coleman; xx Food & Drug Administration/SPL/PhotoResearchers; xxi R. Ian Lloyd/Masterfile; xxii Getty Images;2 David De Lossy/Getty Images; 2–3 Bryan Peterson/GettyImages; 4 Johnny Stockshooter/International Stock;4–5 Orion/International Stock; 6 Mark Harmel/Getty Images;14 Amy C. Etra/PhotoEdit; 16 Archivo Iconografico,S.A./CORBIS; 19 Aaron Haupt; 20 SuperStock; 23 MichaelNewman/PhotoEdit; 26 Pictor; 28 Robert Yager/GettyImages; 31 E.L. Shay; 38 Lawrence Migdale; 40 IndexStock/Ewing Galloway; 43 PhotoDisc; 44 Rudi VonBriel/PhotoEdit; 54–55 Jack Dykinga/Getty Images;56 William J. Weber; 61 Bettmann/CORBIS; 64 D & KTapparel/Getty Images; 67 Lynn M. Stone; 72 (l)SuperStock,(r)Richard T. Nowitz/CORBIS; 80 VCG/Getty Images;82 John Evans; 85 Matt Meadows; 94 David Ball/CORBISStock Market; 99 Getty Images; 108 PhotoDisc;108–109 NASA/TSADO/Tom Stack & Assotes; 111 DaveStarrett/Masterfile; 114 Telegraph Colour Library/GettyImages; 121 Will Hart/PhotoEdit; 124 NASA; 126 DougMartin; 129 AFP/CORBIS; 131 Caroline Penn/CORBIS;138 S. Carmona/CORBIS; 140 M. Angelo/CORBIS;143 Andy Lyons/Allsport; 152 CORBIS; 152–153 WilliamSallaz/DUOMO; 157 Bettman/CORBIS; 161 Tui DeRoy/Bruce Coleman, Inc.; 165 PhotoDisc; 169 Andy LyonsSTF/Allsport; 172 Mark Tomalty/Masterfile; 175 MarkRichards/PhotoEdit; 180 Michael Denora/Getty Images;187 Jonathan Blair/CORBIS; 190 FDR Library; 193 DEXImages Inc./CORBIS Stock Market; 195 Jose Luis PelaezInc./CORBIS Stock Market; 197 Volker Steger/SPL/PhotoResearchers; 203 Ken Eward/Science Source/PhotoResearchers; 218 www.comstock.com; 218–219 RafaelMarcia/Photo Researchers; 220–221 Carl Purcell/Words andPictures/PictureQuest; 225 AFP/CORBIS; 227 K.G.Murti/Visuals Unlimited; 229 David Umberger/PurdueUniversity Photo; 243 Steve Rayer/CORBIS; 249 RogerRessmeyer/CORBIS; 255 Roy Ooms/Masterfile;259 AFP/CORBIS; 262 Victoria & Albert Museum,London/Art Resource, NY; 267 Lori Adamski Peek/GettyImages; 274 Kaluzny/Thatcher/Getty Images; 284 AllsportConcepts/Getty Images; 284–285 Ed Pritchard/GettyImages; 291 Aidan O’Rourke; 292 Getty Images;298 SuperStock; 304 Matthew McVay/Stock Boston;306 DUOMO/CORBIS; 311 CORBIS; 313 Jeff Kaufman/Getty Images; 318 Bruce Hands/Getty Images; 327 NASA;329 Nick Wilson/Allsport; 331 Todd Rosenberg/Allsport;334 Aaron Haupt; 344–345 Guy Grenier/Masterfile;346 Brownie Harris/CORBIS Stock Market; 351 MarthaSwope/Timepix; 355 VCG/Getty Images; 357 MichaelNewman/PhotoEdit; 363 Gregg Mancuso/Stock Boston;365 Boden/Ledingham/Masterfile; 372 National Library ofMedicine/Mark Marten/Photo Researchers; 376 VCG/GettyImages; 381 Bob Krist/CORBIS; 383 Ed Bock/CORBIS StockMarket; 388 SuperStock; 392 Matt Meadows; 394 SuperStock;395 Raymond Gehman/CORBIS; 396 Frank Rossotto/

Stocktreck/CORBIS Stock Market; 398 Getty Images;408 Michael S. Yamashita/CORBIS; 408–409 Jose FusteRaga/eStock Photo; 410 Photographers Library LTD/eStockPhoto; 410–411 James Hackett/eStock Photo; 424 JamesRooney; 426 SuperStock; 432 Matt Meadows; 435 Ray F.Hillstrom Jr.; 439 James P. Blair/CORBIS; 443 CORBIS;446 Bob Krist/Getty Images; 459 (l)Space Telescope ScienceInstitute/NASA/SPL/Photo Researchers, (r)MichaelNewman/PhotoEdit; 470–471 David Fleetham/GettyImages; 477 AFP/CORBIS; 483 Pascal Rondeau/Allsport; 487 Aaron Haupt; 489 Bettmann/CORBIS;494 JPL/TSADO/Tom Stack & Associates; 496 Geoff Butler;497 Lynn M. Stone/Bruce Coleman, Inc.; 499 PicturePress/CORBIS; 503 Kunio Owaki/CORBIS Stock Market;505 (b)Phil Schermeister/CORBIS, (t)Bruce Ayres/GettyImages; 507 Reuters NewMedia Inc./CORBIS; 511 KeithWood/Getty Images; 520–521 Michael S. Yamashita/CORBIS; 522 Aaron Haupt; 525 Ariel Skelley/CORBIS StockMarket; 529 Jeff Zaruba/CORBIS Stock Market; 536 (l)MarkJones/Minden Pictures, (r)Jane Burton/Bruce Coleman, Inc.;537 David Weintraub/Photo Researchers; 542 SuperStock;545 Bettman/CORBIS; 558 Jim Craigmyle/Masterfile;561 Richard T. Nowitz/Photo Researchers; 564 KarlWeatherly/CORBIS; 574 Amanda Kaye; 574–575 Bryan Barr;576–577 Christine Osborne/CORBIS; 579 SuperStock;583 Michele Wigginton; 584 Michelle Bridwell/PhotoEdit;595 Hank Morgan/Photo Researchers; 599 IN THEBLEACHERS ©1997 Steve Moore. Reprinted with permissionof Universal Press Syndicate. All rights reserved; 603 JennyHager/ImageState; 609 Jeff Greenberg/Visuals Unlimited;612 SPL/Photo Researchers; 630 Steve Liss/Timepix;630–631 Greg Mathieson/Timepix; 632 D.F. Harris; 635©1979 United Feature Syndicate, Inc.; 636 (l)MitchKezar/Getty Images, (r)Jim Erickson/CORBIS Stock Market;638 Mark C. Burnett/Photo Researchers; 642 Matt Meadows;644 CORBIS; 648 Food & Drug Administration/SPL/PhotoResearchers; 651 Chris Trotman/DUOMO; 656 CharlesGupton/CORBIS Stock Market; 660 The Born Loser reprintedby permission of Newspaper Enterprise Association, Inc.;662 Bob Daemmrich/Stock Boston; 667 Greg Fiume/New Sport/CORBIS; 668 SuperStock; 671 AFP/CORBIS;675 Will & Deni McIntyre/Photo Researchers; 677 GreggForwerck/SportsChrome USA; 679 Steve Chenn/CORBIS;683 Aaron Haupt; 685 HMS Images/Getty Images;686 Aaron Haupt; 696 Photofest; 696–697 Ed and ChrisKumler; 698–699 Bill Ross/CORBIS; 705 John P. Kelly/Getty Images; 707 SuperStock; 709 L. Clarke/CORBIS;713 Ray Juno/CORBIS Stock Market; 716 Aaron Haupt;717 courtesy Skycoaster of Florida; 721 Reuters NewMediaInc./CORBIS; 723 Otto Greule/Allsport; 729 PeterMiller/Photo Researchers; 731 SuperStock; 735 RoyOoms/Masterfile; 737 John T. Carbone/Photonica;744 R. Ian Lloyd/Masterfile; 746 Doug Plummer/Photonica; 748 SuperStock; 750 Steven E. Sutton/DUOMO;760–761 Boden/Ledingham/Masterfile; 766 Larry Hamill;773 Ben Edwards/Getty Images; 780 James Schot/Martha’sVineyard Preservation Trust; 789 Cosmo Condina/GettyImages; 795 SuperStock; 799 SuperStock; 803 (l)GettyImages, (r)Frank Wiewandt/Image Finders.

About the Cover: Alexander Calder (1898–1976) was one of America’s most acclaimed sculptors. Renowned for hisinvention of the mobile, or movable sculpture, Calder also created sculptures called stabiles, or immovable sculptures.The cover photograph illustrates his Grand Stabile Rouge, located in Paris. One of Calder’s last great public works, thissculpture is reminiscent of another of his stabiles, Flamingo, in Chicago. Both stabiles feature large red arches thatresemble parabolas.

IndexIndex

Index R87

Index

Absolute value equations, 28, 53graphing, 299solving, 28–32, 39, 49

Absolute value functions, 90, 91,92, 104, 115, 247, 272, 370, 499,502, 503, 515, 599, 831, 848

Absolute value inequalities, 40–46,86, 829graphing, 97, 335multi-step, 42solving, 50

AdditionAssociative Property, 15, 162, 166,

828Commutative Property, 15, 162complex numbers, 270, 272Distributive Property, 221functions, 383, 403matrices, 160polynomials, 229, 277probabilities, 658–663, 689–690properties, 25radicals, 252, 253rational expressions, 480, 514signs for, 46solving inequality, 34

Addition Property of Equality, 21

Addition Property of Inequality, 33

Additive identity, 15, 32, 162, 828

Additive inverses, 13, 15, 16, 18,153, 828

Algebra ActivityAdding Radicals, 252Area Diagrams, 651Arithmetic Sequences, 580Completing the Square, 308Conic Sections, 453–454Distributive Property, 13Factoring Trinomials, 240Fractals, 611Graphing Equations in Three

Variables, 136–137Head versus Height, 83Inverses of Functions, 392Investigating Ellipses, 432Investigating Exponential

Functions, 522Investigating Polygons and

Patterns, 19Investigating Regular Polygons

using Trigonometry, 716

Locating Foci, 437Midpoint and Distance Formulas

in Three Dimensions, 417–418Multiplying Binomials, 230Parabolas, 421Rational Functions, 487Simulations, 681Special Sequences, 607Testing Hypotheses, 686

Algebraic expressions, 828evaluating, 7, 8, 9, 18, 27, 30, 53,

109fraction bar, 7simplifying, 14, 15, 16, 27, 48, 53,

62verbal expressions, 20, 24, 115

Algebra tiles. See also Modelingbinomials, 230complete the square, 308modeling binomials, 230polynomials, 240

Algorithms, division, 233–234

Alternative hypothesis, 686

Alternative method, 580, 590, 652

Alternative representations, 726

Amortization, 605schedule, 605

Amplitude, 763, 764, 765, 766, 767,771, 774, 775, 776, 781, 785, 805,806, 859

“And” compound inequalities, 40

Angles, 709–716, 734, 753coterminal, 711, 738depression, 705elevation, 705finding, 721general, 754inclination, 779measurement, 709, 711, 712, 745,

748, 753quadrantal, 718reference, 718–719, 722, 776trigonometric function of

general, 717–724vertex, 192vertices, 113, 192

Angles formulasdifferences, 786–790, 807sum, 786–790, 807

Angular velocity, 714

Antinodes, 791

Apothem, 716

Applications. See also Cross-Curriculum Connections; MoreAboutacidity, 550activities, 510advertising, 459, 668aeronautics, 732aerospace, 425, 429, 587aerospace engineering, 266agriculture, 565, 863, 865, 871airports, 122altitude, 557amusement parks, 380ancient cultures, 72animal control, 528animals, 319, 827archery, 298architecture, 497, 503, 749art, 490, 865, 872astronomy, 226, 238, 262, 310,

438, 439, 440, 445–447, 459, 478,498, 550, 712, 862, 869

auto maintenance, 517automobiles, 380automotive engineering, 255auto safety, 489aviation, 450, 706, 737, 790, 795,

869babysitting, 39baking, 16, 127ballooning, 341band boosters, 15banking, 9, 538, 608baseball, 88, 333, 722, 779, 827basketball, 17, 95, 143, 490, 874boating, 298, 768bowling, 25bridge construction, 705bridges, 424building design, 550bulbs, 745business, 26, 79, 80, 89, 97, 158,

165, 174, 181, 194, 237, 256, 334,352, 565, 570, 670

cable cars, 874cable TV, 356caffeine, 560camera supplies, 174card games, 642, 649car expenses, 26carousels, 723car rental, 51cars, 679, 713car sales, 38cartography, 637charity, 823

A

Index

R88 Index

child’s play, 602clocks, 602, 617clubs, 872coffee, 31coins, 571communications, 423, 559, 767,

789, 869, 874community service, 299computers, 563, 582construction, 266, 292, 586, 643,

821, 862, 865, 867contest, 46cooking, 142crafts, 863, 865cryptography, 199, 200cycling, 510deliveries, 36dentistry, 227design, 363, 627, 868diet, 549dining, 143dining out, 157diving, 304, 327drama, 98driving, 827earthquake, 797earthquakes, 458, 545, 547, 871ecology, 79, 207economics, 66, 114, 261, 564, 610,

684education, 328, 648, 660, 667, 863,

864, 866, 870, 873elections, 655electricity, 18, 122, 273, 274, 483, 517electronics, 389, 780employment, 357, 863, 868energy, 869engineering, 369entertainment, 73, 237, 399, 581,

598, 635, 713e-sales, 231exercise, 121, 707, 864extreme spots, 296family, 26farming, 134figure skating, 638finance, 61, 85, 173, 388financial planning, 405firefighting, 398fish, 248flagpoles, 821flooring, 44food, 30, 632, 674football, 318, 668footprints, 180forestry, 731fountains, 326, 750framing, 311fund-raising, 67, 173, 334furniture, 275games, 193, 616, 825, 872

gardening, 607, 802gardens, 484genealogy, 648genetics, 648golf, 809, 822, 823government, 61, 88, 641gymnastics, 180health, 45, 84, 95, 267, 425, 452,

503, 597, 824, 863, 873highway safety, 319hobbies, 62hockey, 84home security, 636hotels, 157housing, 81, 82, 121hurricanes, 126insurance, 94interior design, 193, 438Internet, 80intramurals, 616inventory, 121investing, 192investments, 140kennel, 312landscaping, 180, 243, 334, 412,

430, 597laughter, 497law enforcement, 254, 298, 335, 866lawn care, 327life expectancy, 865light, 803lighting, 780loans, 609lotteries, 642, 648mail, 45, 503manufacturing, 31, 132, 147, 149,

424, 674, 868, 870marriage, 440, 824measurement, 863media, 684medicine, 10, 84, 237, 376, 488,

544, 563, 592meteorology, 675mirrors, 459models, 867money, 10, 27, 529, 551, 824, 826movies, 641movie screens, 310music, 775navigation, 507, 723, 732newspapers, 291noise ordinance, 537nuclear power, 426number games, 394nursing, 9nutrition, 94oceanography, 201, 249optics, 248, 750ownership, 564packaging, 26, 134, 363pagers, 448

paleontology, 561, 563paper, 144parking, 93parties, 620part-time jobs, 37, 126passwords, 635personal finance, 231, 369pets, 341photography, 244, 304, 431, 447,

451, 657, 819, 870photos, 113physics, 767pilot training, 206pool, 363population, 227, 529, 558, 562,

570, 862, 863, 864, 866, 868, 871population growth, 388, 563pricing, 193produce, 172production, 133puzzles, 620quality control, 673radio, 430, 731radioactivity, 587real estate, 415, 563recreation, 88, 105, 165, 604, 627retail, 377retail sales, 215rides, 867, 874, 875rockets, 458, 875roller coasters, 398rumors, 558running, 188safety, 84, 868salaries, 591sales, 77, 99, 868satellites, 869savings, 556, 557school, 634schools, 26, 37, 51, 74, 98, 135,

206, 615, 641, 661, 662, 668, 674,823, 825

school shopping, 17school trip, 26scrapbooks, 166sculpting, 376shadows, 819shopping, 39, 62, 98, 125, 149,

387, 685skiing, 120skycoasting, 723skydiving, 281slope, 744soft drinks, 827sound, 535, 542, 545space, 563space exploration, 124speed limits, 45, 873speed skating, 663sports, 164, 171, 248, 255, 358,

425, 451, 678, 872

Index

Index R89

stamps, 144state fair, 37statistics, 511structural design, 446surveying, 707, 737, 819, 874surveys, 873swimming, 495taxes, 67, 386telecommunications, 497telephones, 80television, 83, 875temperature, 388tennis, 298test grades, 38theatre, 84thinking, 564tides, 775, 875tourism, 292toys, 586transportation, 88, 93, 228, 487,

636travel, 69, 73, 113, 143, 249, 415,

707, 750, 862, 870tunnels, 467utilities, 656vending, 674water, 451water supply, 496weather, 60, 72, 156, 597, 823, 862,

867, 868, 871, 874, 875weekly pay, 103White House, 439woodworking, 416, 730work, 16, 496, 509world cultures, 586world records, 536writing, 649

Arcsine function, 747

Areacircles, 9, 415, 502diagrams, 651hexagons, 707parallelograms, 477polygons, 187rectangles, 255, 334trapezoids, 8, 67, 865triangles, 32, 184, 185, 186, 187,

231, 281, 866

Area tiles. See Algebra tiles;Modeling

Arithmetic expressions,simplifying, 6

Arithmetic means, 580, 582, 590,592, 622, 623, 851

Arithmetic operations, 383–384

Arithmetic sequences, 578–582,579, 583, 622–623, 768, 851

modeling, 580nth term, 579, 591, 851

Arithmetic series, 583–587, 620,623, 851sum, 583, 584, 586, 592

AssessmentPractice Chapter Test, 51, 105,

149, 215, 281, 341, 405, 467, 517,571, 627, 693, 757, 809

Practice Quiz, 18, 74, 95, 122, 135,174, 194, 238, 256, 328, 364, 382,431, 448, 484, 498, 538, 559, 592,617, 650, 670, 715, 738, 781, 797

Prerequisite Skills, 5, 10, 18, 27,32, 39, 55, 62, 67, 74, 80, 86, 95,109, 115, 122, 127, 135, 153, 158,166, 174, 181, 188, 194, 201, 221,228, 232, 238, 244, 249, 256, 262,267, 285, 293, 299, 305, 312, 319,328, 345, 352, 358, 364, 370, 377,382, 389, 394, 411, 416, 425, 431,440, 448, 452, 471, 478, 484, 490,498, 504, 521, 530, 538, 546, 551,559, 577, 582, 587, 592, 598, 604,610, 617, 631, 637, 643, 657, 663,670, 675, 680, 699, 708, 715, 724,732, 738, 745, 761, 768, 776, 781,785, 790, 797, 814–827

Standardized Test Practice, 10, 17,23, 24, 27, 31, 39, 46, 51, 67, 74, 76,78, 80, 86, 95, 99, 105, 115, 117,120, 122, 127, 134, 144, 149, 158,166, 173, 176, 179, 181, 187, 194,201, 207, 215, 228, 232, 236, 238,243, 244, 249, 255, 267, 281, 292,299, 302, 303, 305, 312, 319, 327,335, 341, 352, 358, 364, 370, 374,375, 377, 382, 389, 394, 399, 405,413, 414, 416, 425, 431, 439, 447,452, 459, 467, 473, 476, 478, 484,490, 498, 503, 511, 517, 530, 537,546, 559, 562, 563, 564, 582, 587,588, 591, 592, 598, 603, 610, 616,621, 627, 634, 636, 642, 649, 657,662, 669, 675, 685, 693, 706, 708,724, 732, 737, 745, 757, 768, 776,781, 784, 785, 790, 796, 804, 809

Extended Response, 53, 107, 151,217, 343, 407, 469, 519, 551, 573,621, 629, 695, 714, 759, 811

Grid In, 680, 751Multiple Choice, 52, 106, 150,

216, 342, 406, 468, 518, 572, 628,633, 694, 702, 758, 783, 810

Open Ended, See ExtendedResponse

Short Response/Grid In, 53, 107,151, 217, 343, 407, 469, 519, 573,629, 695, 759, 811

Test-Taking Tips, 23, 52, 76, 106, 117,151, 176, 217, 234, 282, 302, 342,407, 468, 473, 519, 562, 572, 588,628, 633, 695, 702, 758, 783, 811See also Preparing forStandardized Tests

Associative PropertyAddition, 15, 162, 166, 828Multiplication, 15, 171

Asymptotes, 491, 530determining, 471hyperbola, 846, 848vertical, 617, 763

Augmented matrices, 208

Axisconjugate, 442minor, 434symmetry, 287–288, 290, 291, 299,

339, 839transverse, 442

Bar graphs, 824

Base e equations, 555, 569

Base e inequalities, 556

Base e logarithms, 554–559inverse property, 555

Base formula, 548, 549

Bias, 682

Biased sample, 682

Binomials, 229, 366, 368, 382expansions, 631, 676experiments, 676–681, 677,

691–692

Binomial Theorem, 612–617,625–626factorial form, 614

Bivariate data, 81

Boundary, 96

Bounded region, 129–130

Box-and-whisker plots, 631,826–827

Break-even point analysis, 110, 111

Calculator. See Graphing calcuator; Graphing Calculator Investigation

Career Choicesarchaeologist, 187atmospheric scientist, 126chemist, 511

B

C

Index

R90 Index

cost analyst, 237designer, 363electrical engineering, 274finance, 85forester, 446interior design, 193landscape architect, 334paleontologist, 561physician, 685real estate agent, 609sound technician, 542surveyor, 707travel agent, 496veterinary medicine, 131

Cartesian coordinate plane, 56

Centercircles, 426, 845ellipses, 434

Central tendency measures of, 664,822–823

Change of Base Formula, 548, 569

Checking solutions, 13, 22, 24, 25,29, 30, 31, 34, 39, 46, 49, 51, 62,110, 115, 117, 197, 207, 263, 264,294, 302, 309, 314, 315, 325, 361,362, 367, 379, 481, 506, 509, 516,526, 527, 528, 530, 533, 534, 535,536, 538, 542, 543, 544, 546, 548,551, 555, 559, 580, 604, 621, 643,657, 801, 849, 850

Circles, 426–431, 450, 451, 460, 463,467, 565, 617area, 9, 415, 502center, 426, 845circumference, 496, 710connecting points, 352eccentricity, 440equations, 426, 846graphing, 428, 429radius, 845sectors, 713unit, 710, 740, 742, 743

Circular functions, 739–745, 756, 761

Circular permutation, 642

Circumference, 496, 710

Closure Property, 18

Coefficients, 222, 448integral, 376leading, 379least, 389

Column matrix, 155, 156

Combinations, 638–643, 640, 641,650, 688, 715

Combined variation, 497

Common difference, 578

Common logarithms, 547–553, 559,617

Common Misconception, 7, 12, 29,118, 130, 289, 308, 523, 659, 703,782. See also Find the Error

Common ratio, 588, 603

Communication, 633compare and contrast, 178, 673,

742copy, 60decide, 71, 242, 273, 590define, 297, 706, 712, 774describe, 8, 156, 163, 185, 192,

317, 350, 362, 397, 445, 535, 619,683, 749, 779, 784, 788

determine, 8, 24, 171, 226, 247,254, 273, 310, 332, 386, 393, 445,476, 495, 641, 722, 729, 788

disprove, 14draw, 660evaluate, 92examine, 332explain, 14, 30, 37, 65, 78, 98, 112,

142, 156, 171, 185, 198, 236, 247,260, 265, 297, 310, 317, 325, 350,356, 362, 375, 380, 393, 397, 414,450, 476, 549, 563, 580, 586, 596,602, 619, 634, 673, 678, 722, 736,749, 766, 779, 784, 794, 802

graph, 458identify, 78, 231–232, 290, 414,

423, 437, 527, 615, 774list, 615, 634make, 119name, 65, 544, 549, 557, 712show, 185, 265, 641sketch, 458state, 78, 125, 290, 356, 368, 375,

509, 608, 742tell, 125, 350, 802verify, 647write, 30, 37, 43, 98, 163, 178, 192,

198, 205, 303, 325, 368, 428, 450,563, 608, 647, 654

Commutative PropertyAddition, 15, 162Multiplication, 15, 32, 166, 170, 828

Comparisonquantitative, 117, 120real numbers, 5, 814

Completing the square, 306–312,328, 338, 352, 411, 490, 587, 840

Complex conjugates, 273, 374–375

Complex fractions, 475, 481

Complex numbers, 270–275, 280, 370addition, 270, 272division, 272–273

multiplication, 272–273subtraction, 270, 272

Complex roots, 315

Composition, functions, 384–386,530, 532

Compound event, 658

Compound inequalities, 40–46, 50and, 40or, 41

Concept Summary, 47, 48, 49, 57, 69,92, 100, 101, 102, 103, 104, 112,146, 162, 171, 177, 178, 209, 210,211, 212, 213, 214, 239, 246, 251,260, 265, 276, 277, 278, 279, 280,317, 323, 336, 337, 338, 339, 340,349, 371, 400, 401, 402, 403, 404,422, 449, 450, 461, 462, 463, 464,465, 466, 499, 513, 514, 515, 516,566, 567, 568, 569, 570, 622, 623,624, 625, 626, 634, 664, 687, 688,689, 690, 691, 692, 735, 747, 752,753, 754, 755, 756, 772, 805, 806,807, 808

Conditional probability, 653

Cones, surface areas, 22, 266

Congruent angles, 817

Conic sections, 419, 449–452,453–454, 465–466, 869

Conjectures, 19, 32, 83, 119, 240,252, 432, 437, 489, 522, 558, 585,607, 681, 686, 716

Conjugate axis, 442

Conjugates, 253

Conjunctions, 42

Constant functions, 90, 92, 115, 370,499, 502, 515, 831

Constants, 104, 222, 530variation, 492

Constraints, 129

Constructed Response, SeePreparing for Standardized Tests

Continuous functions, 62, 524

Continuously compoundedinterest, 556

Continuous probabilitydistribution, 671

Convergent series, 599, 622

Coordinate matrix, 175

Coordinate plane, 110

Coordinates, finding, 721

Index

Index R91

Coordinate system, 56

Corollary, 372

Corresponding elements, 156

Cosecant function, 701

Cosine function, 701, 706, 707, 740,747, 767, 770, 771definition, 739value, 747

Cotangent function, 701

Coterminal angles, 711, 712, 738

Counterexamples, 14, 16, 32, 92,185, 242, 580, 619, 620, 621, 643,666, 706, 794, 853

Counting Principle, 632–637, 644,687–688

Cramer’s Rule, 189–194, 207, 213,724, 835solving systems of equations, 670three variables, 191two variables, 189

Critical Thinking, 10, 17, 27, 31, 38,45, 62, 66, 73, 80, 85, 94, 99, 114,121, 127, 133, 143, 157, 166, 172,173, 181, 187, 193, 200, 207, 227,232, 237, 243, 249, 255, 262, 267,275, 292, 298, 304, 311, 319, 327,334, 357, 364, 369, 376, 377, 380,389, 394, 398, 416, 425, 430, 439,446, 452, 459, 477, 483, 489, 497,503, 511, 529, 537, 545, 546, 550,558, 582, 587, 592, 598, 603, 610,616, 621, 635, 642, 649, 656, 662,669, 675, 679, 685, 708, 714, 723,732, 737, 744, 750, 767, 776, 780,785, 789, 796, 804

Cross-Curriculum Connections. Seealso Applications; More Aboutanthropology, 563biology, 62, 227, 262, 350, 497, 529,

545, 564, 570, 621, 744, 767, 872chemistry, 203, 205, 206, 312, 460,

496, 511, 570geography, 58, 85, 187, 415, 451,

647, 796, 825geology, 67, 581, 708, 757history, 489literature, 656, 724physical science, 779physics, 66, 237, 267, 292, 318,

370, 393, 510, 546, 557, 604, 743,751, 774, 784, 788, 789, 796, 802,866, 867, 870, 874

physiology, 357, 672science, 80, 83spelling, 656zoology, 775

Cross products, 181

Cube root equation, 264

Cubes, volumes, 615

Curve Fitting, 300, 359, 539

Cylinders, surface areas, 25, 862

Dashed boundary, 96–97

Dataanalyzing, 522, 681, 716box-and-whisker plots, 631,

826–827collecting, 522, 681, 716distribution, 672graphs of polynomial functions,

353, 357modeling real-world, 359organizing, 154, 159scatter plots, 81–86, 87, 95, 99,

103, 598, 831skew, 856stem-and-leaf plots, 667, 825

Decayexponential, 524, 525, 528,

560–565, 561, 567, 570, 849rate of, 560

Decimals, 838, 850approximations for irrational

numbers, 247repeating, 601, 602, 603, 852

Degrees, 222, 724, 753, 757, 802, 803converting radian measures

between, 711measurement, 711polynomials, 229, 346, 350, 400,

837, 842

Denominatorsmonomials, 480polynomials, 475, 480

Dependent events, 633–634, 635,653, 654, 655, 687, 689, 854, 855

Dependent variable, 59

Depressed polynomial, 366

Depression, angle of, 705

Descartes, René, 372

Descartes’ Rule of Signs, 372–373,379

Determinants, 182–188, 212evaluating

using diagonals, 835using expansion by minors, 835

finding value, 186, 835second-order, 182third-order, 182, 183

3 3 matrices, 182, 1832 2 matrices, 182value, 185, 194

Deviation, mean, 669

Diagonals, 19, 182, 183, 184, 201, 642in decagon, 776evaluating determinants, 186, 835

Differencesrewriting as sums, 221squares, 816

Dilations, 175, 176, 177

Dimensional analysis, 225, 708

Dimensions, 155

Directrix, 419

Direct substitution, 366, 368

Direct variation, 492–493, 495, 496,499, 502, 515, 559, 650, 848

Discrete function, 62

Discrete probability distributions,671

Discriminant, 328quadratic formula, 313–319, 339

Disjunctions, 42

Distance Formulas, 413–414, 415,416, 417–418, 425, 441, 461–462,467

Distributionscontinuous probability, 671discrete probability, 671normal, 671–675, 680probability, 646skewed, 671

Distributive Property, 12, 13, 14, 15,17, 32, 162, 166Addition, 221Multiplication, 170, 171, 228, 828

Divisibility, 619

Divisionalgorithm, 233–234complex numbers, 272–273functions, 384, 403polynomials, 233, 277, 364,

365–366properties of equality, 21rational expressions, 474, 513simplifying expressions, 223solving inequality, 35synthetic, 345, 745, 837

Division Property of Equality, 21

Division Property of Inequality, 34

Domain, 56, 57, 58, 61, 93, 94, 95, 99,100, 101, 104, 181, 397, 398, 416,523, 527, 528, 530, 830, 831, 844, 849range, 67

D

Index

R92 Index

Double-angle formulas, 791–798, 808

Double root, 302

Doubling time, 558

Eccentricity, 440, 452

Elements, 155

Elements, 155corresponding, 156

Elevation, angle of, 705

Elimination, 146, 149, 153, 504simplifying rational expressions,

473solving systems of equations,

118–119, 120, 122, 135, 166, 832

Ellipses, 432, 433–440, 450, 451, 452,460, 464, 467, 565, 617center, 846equations, 433–435, 643graphing, 435–437major axes, 846minor axes, 846writing equations, 846

Empty set, 29

End behavior, 349

Endpoints, 418

Energy, 530

Enrichment. See Critical Thinking;Extending the Lesson

Equal matrices, 209

Equate complex numbers, 271

Equations, 23. See also Quadraticequations; Systems of equationsabsolute value, 28–32, 39, 49,

299circles, 426complex solutions, 309cube root, 264ellipses, 433–435equivalent exponential, 565, 850exponential, 526, 548, 570

solving, with logarithms, 548writing equivalent, 570

forms, 75–80graphing, 471hyperbolas, 441–443imaginary solutions, 271irrational roots, 307linear, 63–67, 75–80, 86, 101, 102,

109, 189, 191, 452, 830logarithmic, 533, 543, 546, 551,

565, 570, 850

matrix, 202–203, 358, 370, 834, 836

solving, 205, 834, 836writing, 202–203, 836

midline, 771, 774, 775, 781multi-step, 22, 201for nth term, 579, 589one-step, 21parabolas, 419–420polynomial, 360–364, 401, 837prediction, 81–82, 83, 84, 95, 99,

598quadratic, 604radical, 263–269, 280, 362rational roots, 306, 505–509, 516regression, 87rounding, 776solving, 20–27, 25, 48–49, 153,

157, 174, 535, 536, 538, 544, 546,549, 550, 557, 558, 559, 565, 568,569, 570, 577, 582, 604, 621, 637,643, 657, 708, 747, 768, 828, 829,839, 849, 850, 862

involving matrices, 155–156,202

with inverses, 746using Properties of

Logarithms, 543with rational numbers, 471

trigonometric, 799–804, 808two-variable matrix, 202

Equilateral triangles, 869

Equivalent exponential equations,565, 850

Equivalent expressions, 555

Error, 692measurement, 704, 738sampling, 682–686, 714

Error Analysis. See Find The Error;Common Misconceptions

Estimating, 225, 296

Events, 632compound, 658dependent, 633–634, 634, 635,

653, 654, 655, 687, 689, 854, 855inclusive, 659, 660, 661, 670, 689,

690, 855independent, 632–633, 634, 651,

652, 654, 687, 689, 854, 855multiple, 640mutually exclusive, 658–659, 661,

670, 689, 690, 855odds, 854

Excluded values, 472

Exclusive events, mutually,658–659, 661, 670, 689, 690, 855

Expansion by minors, 182, 183, 186,201evaluating determinants, 186, 835

Expansions, binomials, 631, 676

Expected value, 681

Experimental probability, 649

Exponential decay, 524, 525, 528,560–565, 561, 567, 570, 849

Exponential equations, 526solving, with logarithms, 548writing equivalent, 570

Exponential form, 257, 532, 535,536, 568, 849

Exponential functions, 520,566–567graphing, 523property of equality, 526property of inequality, 527solving, 526writing, 525, 528

Exponential growth, 524, 525, 528,560–565, 562, 567, 570, 849

Exponential inequalities, solving,527with logarithms, 548

Exponential relations, 871

Exponentsinverse property, 533irrational, 526negative, 222radical, 279rational, 257–262, 361–362, 838

Expressions, 47–48, 53, 779. See alsoAlgebraic expressions;Arithmetic expressions; Radicalexpressions; Rationalexpressions; Verbal expressionsevaluating, 158, 201, 394, 535,

536, 546, 557, 558, 568, 570, 577,582, 610, 615, 617, 631, 637, 641,643, 650, 779, 780, 790, 828, 829,838, 853, 854

simplifying, 223–224, 528, 538,546, 604, 637, 776, 778, 779, 780,790, 828, 838, 847

Extended Response, 364. See alsoPreparing for StandardizedTests

Extending the Lesson, 18, 32, 62, 80,86, 275, 299, 335, 416, 440, 447,452, 636, 642, 649, 669, 738

Extraneous solutions, 263–264, 534

Extra Practice, 828–861

E

Index

Index R93

Factorial, 613, 614, 637

Factoring, 367, 460polynomials, 239–241, 358, 377,

815–816, 837solving quadratic equations by,

301–305, 338, 840solving system of equations, 643

Factors, polynomials, 366

Factor Theorem, 365–370, 402

Failure, 644probability, 644

Families of graphs, 70absolute value graphs, 91parabolas, 320–321

Feasible region, 129, 134, 833

Fibonacci sequence, 606, 609, 610

Field, 12

Figurescongruent, 817–819similar, 817–819translating, 175

Find the Error, 24, 43, 60, 71, 119,142, 185, 205, 226, 236, 303, 310,325, 380, 386, 423, 428, 481, 509,535, 544, 557, 590, 602, 654, 660,730, 735, 766. See also CommonMisconceptions

Finite graph, 636

Focusellipse, 432parabola, 419

FOIL Method, 230, 240

Foldables™ Study Organizers, 5,53, 55, 109, 153, 221, 285, 345, 411,471, 521, 577, 631, 699, 761

Forms of equations, 75–80

Formulas, 6–10, 25, 47–48, 122angles, 786–790area, 184base, 548, 549change of base, 548, 569differences, 786–787, 790distance, 413–414, 415, 416,

417–418, 425, 441, 461–462, 467double-angle, 791, 808half-angle, 791–798, 792, 793, 794,

795, 797, 808, 861midpoint, 412, 414, 416, 417–418,

461–462, 467quadratic, 313–319, 339, 345, 370,

460, 841

recursive, 606, 607, 608summation, 618sums, 596, 600, 786–787, 790

45°-45°-90° triangles, 699, 703, 707

Fourth term, 589

Fractals, 611

Fraction bar, 7

Fractionscomplex, 475, 481repeating decimals, 601, 602, 603,

852

Free Response, See Preparing forStandardized Tests

Function notation, 59

Function values, 348, 604

Functions, 57, 100–102, 830absolute value, 91, 92, 115, 370,

499, 502, 503, 515, 831, 848addition, 383, 403circular, 739–745,756, 761classes, 499–504, 515composition, 384–386, 521constant, 370, 831division, 384, 403equations, 58–62exponential, 523–530graphing, 577, 768, 863inverse, 390–394, 404, 405, 521,

617, 699, 749inverse trigonometric, 746–751, 756iterating, 608multiplication, 384, 403operations, 383–389, 403periodic, 741piecewise, 89–95, 104, 370, 831step, 89–95, 370, 831subtraction, 383, 403zero, 376

Fundamental Counting Principle,633, 644, 687

Fundamental Theorem of Algebra,344, 371–372

General angles, 717, 754

Geometric means, 590, 591, 598,623, 852

Geometric sequences, 588–593, 594,623–624, 852limits, 593nth term, 589, 852sums, 852terms, 594

Geometric series, 594–598, 617, 620,624, 781infinite, 599–605, 624–625, 745,

852sum, 595, 597, 610

Geometry, 186areas

circles, 9, 415, 502hexagons, 707parallelograms, 477polygons, 187rectangles, 255, 334trapezoids, 8, 67, 865triangles, 32, 185, 186, 187,

231, 281, 866arrays of numbers, 582circumferences of circles, 496degrees in convex polygon, 79diagonals in decagons, 776dimensions of inscribed

rectangle, 292equilateral triangles, 869exact coordinates, 744factoring, 243height of parallelogram, 477isosceles triangles, 869leg of right triangle, 243matrix multiplication, 200measures of diagonals, 737midpoint, 414ordered pairs, 390perimeters

octagons, 26quadrilaterals, 415, 482rectangles, 255right triangles, 382squares, 603triangles, 592

perpendicular lines, 73slope of a line, 481squares, 609surface areas

cones, 22, 266cylinders, 25, 862pyramids, 27rectangular prisms, 18spheres, 862

triangular numbers, 609vertices

angles, 113, 192parallelograms, 121, 192triangles, 113, 415

volumescubes, 615rectangles, 866rectangular prism, 367rectangular solid, 379, 380

widthrectangle, 242rectangular prism, 363

G

F

Index

R94 Index

Golden ratio, 311

Golden rectangle, 311

Graph functions, 285

Graphingabsolute value equations, 299absolute value inequalities, 335circles, 428, 429ellipses, 435–437exponential function, 523horizontal translations, 770hyperbola, 846inequalities, 657, 680, 832, 841, 844linear inequalities, 329linear relations and functions, 863parabolas, 420–423polynomial functions, 348–349,

353–358, 401polynomial model, 355quadratic equations, 294–299, 337quadratic functions, 286–293,

336–337quadratic inequalities, 329–333,

340rational functions, 514, 848square root functions, 395–396,

404square root inequalities, 397–399,

404systems of equations, 110–115,

194, 832systems of inequalities, 123–127,

135, 484, 833, 847table of values, 352, 356, 364transformations, 772trigonometric functions, 762–768,

772vertical translation, 771

Graphing calculator, 39, 431, 444,455, 456, 460, 585, 613addition of trigonometric

inverses, 751approximating value, 247, 248binomial distribution, 680check factoring, 244family of graphs, 74families of graphs, 530intersect feature, 115inverse functions, 201inverse matrices, 207logic menu, 46matrix function, 188maxima, 293, 358minima, 293, 358shade command, 99sum of each arithmetic series, 587sum of geometric series, 598verifying trigonometric

identities, 785Zero function, 296, 307

Graphing Calculator Investigationaugmented matrices, 208factoring polynomials, 241families of absolute value graphs,

91families of exponential functions,

524families of parabolas, 320–321graphing rational functions, 491horizontal translations, 769limits, 593lines of regression, 87–88lines with same slope, 70matrix operations, 163maximum and minimum points,

355–356modeling real-world data, 300,

359, 539–540one-variable statistics, 666order of operations, 7point discontinuity, 491quadratic systems, 457sine and cosine on unit circle,

740solving exponential and

logarithmic equations andinequalities, 552–553

solving inequalities, 36solving radical equations and

inequalities, 268–269solving rational equations by

graphing, 512solving trigonometric equations,

798square root functions, 396sums of series, 585systems of linear inequalities, 128systems of three equations in

three variables, 205

Graphing functions, 577

Graph relations, 56–62

Graphsbar, 824finite, 636line, 824

Greatest common factor, 239, 302

Greatest integer function, 89, 104,499, 503, 515, 517, 530

Grid In, 530, 708. See alsoAssessment

Gridded Response, See Preparingfor Standardized Tests

Grouping, 240symbols, 6

Growth, exponential, 524, 525, 528,560–565, 562, 567, 570, 849rate of, 562

Half-angle formulas, 791–798, 792, 793, 794, 795, 797, 808, 861

Harmonics, 791

Hexagons, area, 707

Histogram, 669, 671relative-frequency, 646, 647

Homework Help, 15, 24, 31, 37, 44,60, 66, 72, 78, 84, 93, 98, 113, 120,126, 132, 142, 156, 164, 172, 179,186, 192, 199, 206, 226, 231, 237,243, 248, 254, 261, 266, 274, 291,304, 310, 318, 326, 333, 350, 356,368, 375, 380, 387, 393, 398, 414,424, 429, 458, 476, 482, 489, 496,502, 510, 528, 536, 544, 550, 557,563, 581, 586, 591, 597, 602, 609,615, 620, 635, 641, 648, 655, 661,667, 674, 678, 684, 706, 713, 722,730, 736, 743, 749, 767, 775, 780,784, 789, 795, 803

Horizontal lines, 65, 70

Horizontal line test, 392

Horizontal translations, 769–770graphing, 770

Hyperbolas, 441–448, 450, 451, 452,460, 464–465, 467, 565, 617, 670equations, 441–443, 846graphing, 443–444, 846

Hypothesis, 686

Identify functions, 92

Identify matrices, 213

Identify properties, real numbers, 13

Identities, 12, 861additive, 15, 32, 162, 828multiplicative, 15, 199Pythagorean, 777quotient, 777reciprocal, 777trigonometric, 777, 785, 806verifying, 784, 788, 794

Identity function, 90, 391–392, 499,515

Identity matrices, 195

Image, 175

Imaginary unit, 270

Imaginary zeros, 375, 402, 843

Inclination, angle of, 779

H

I

Index

Index R95

Included angle, 734

Inclusive events, 659, 660, 661, 670,689, 690, 855

Independent events, 632–633, 634,651, 652, 654, 687, 689, 854, 855

Independent variable, 59

Index of summation, 585

Indicated sum, 583

Indicated terms of expansion, 853

Indirect measurement, 705

Induction, mathematical, 618–621,626

Inductive hypothesis, 618

Inequalities, 95, 122absolute value, 829graphing, 96–99, 104, 109, 115,

657, 680, 832, 841, 844logarithmic functions property, 534solving, 33–39, 39, 49–50, 62, 67,

74, 80, 352, 358, 521, 533, 534,535, 536, 538, 546, 549, 550, 557,558, 559, 565, 568, 569, 570, 604,643, 829, 839, 849, 850, 862

writing, 36

Infinite geometric series, 599–605,624–625, 745, 852sigma notation, 601sum, 600, 610

Infinity symbol, 601

Initial side, 709

Integers, 11, 32, 48positive, 620, 853

Integral coefficients, 376, 389

Integral Zero Theorem, 378, 403

Intercept form, 80

Internal notation, 829

Internet Connectionswww.algebra2.com/careers, 26,

85, 121, 126, 187, 193, 237, 274,334, 363, 446, 496, 511, 542, 561,609, 685, 707

www.algebra2.com/chapter_test,51, 105, 149, 215, 281, 341, 405,467, 517, 571, 627, 693, 757, 809

www.algebra2.com/data_update,10, 66, 143, 165, 255, 318, 357,440, 477, 558, 598, 667, 723, 775

www.algebra2.com/extra_examples, 7, 13, 21, 23, 29, 35,41, 59, 65, 69, 77, 83, 91, 97, 111,117, 125, 131, 139, 155, 161, 169,177, 182, 183, 191, 197, 203, 223,

229, 235, 241, 247, 251, 259, 265,271, 289, 295, 303, 307, 315, 323,331, 347, 355, 361, 379, 385, 391,397, 413, 421, 427, 435, 443, 449,457, 473, 481, 487, 493, 501, 507,525, 533, 543, 549, 555, 561, 579,585, 589, 595, 601, 607, 613, 619,633, 639, 645, 653, 659, 665, 673,677, 683, 685, 703, 711, 719, 727,735, 741, 747, 765, 771, 779, 783,787, 793, 801

www.algebra2.com/other_calculator_keystrokes, 86, 128,208, 268, 320, 359, 491, 512, 539,552, 593, 798

www.algebra2.com/self_check_quiz, 9, 15, 17, 31, 37, 45, 61, 73,79, 85, 93, 99, 113, 121, 133, 143,157, 165, 173, 179, 187, 193, 199,207, 227, 231, 243, 249, 255, 261,267, 275, 291, 297, 305, 311, 319,327, 333, 351, 357, 363, 369, 375,379, 381, 387, 393, 399, 415, 425,429, 439, 445, 451, 459, 477, 483,489, 497, 503, 511, 529, 537, 545,551, 557, 563, 581, 587, 591, 597,603, 609, 615, 621, 635, 641, 649,655, 661, 667, 675, 679, 707, 713,723, 731, 737, 743, 749, 767, 775,781, 785, 789, 795, 803

www.algebra2.com/standardized_test, 53, 107, 151, 217, 283, 343,407, 469, 519, 573, 629, 695, 759,811

www.algebra2.com/usa_today, 69www.algebra2.com/vocabulary_

review, 47–50, 145, 209, 276, 400,461, 513, 566, 622, 687, 752, 805

www.algebra2.com/webquest,3, 27, 84, 120, 192, 207, 219, 227,328, 369, 399, 409, 429, 504, 529,565, 575, 616, 635, 697, 708, 775,804

Interquartile range, 827

Intersecting lines, 111

Intervals, 803, 808notation, 35, 37, 40, 41, 51

Inverse Cosine, 747

Inverse functions, 390–394, 399,404, 405, 521, 531, 617, 699, 749,844, 859

Inverse matrices, 195, 196, 201, 205,206, 207, 213, 214, 228, 312, 358,637

Inverse propertyexponents, 533logarithms, 533

Inverse relations, 390–394, 399, 404,405, 844

Inverses, 195, 836additive, 13, 15, 16, 18, 153, 828multiplicative, 13, 14, 15, 16, 32,

153, 199, 828verifying, 196

Inverse Sine, 747

Inverse Tangent, 747

Inverse trigonometric functions,746–751, 756

Inverse variation, 493–495, 496,500, 515, 517, 559, 848

Irrational numbers, 11, 32

Irrational roots, 315

Isometry, 175

Isosceles triangles, 869

Iteration, 608, 853

Joint variation, 492–493, 496, 515, 559, 848

Key Concept, 6, 11, 12, 21, 28, 33, 34, 40, 41, 42, 57, 64, 68, 70, 75, 76,130, 138, 160, 161, 162, 168, 182,183, 184, 189, 191, 195, 196, 222,223, 224, 230, 245, 250, 251, 257,258, 271, 287, 288, 295, 301, 306,307, 313, 316, 346, 347, 354, 360,365, 374, 378, 383, 384, 390, 391,412, 413, 420, 426, 434, 435, 442,443, 474, 485, 492, 493, 494, 524,526, 532, 533, 534, 541, 542, 543,548, 579, 583, 589, 595, 600, 613,614, 618, 633, 638, 639, 640, 644,645, 652, 653, 658, 660, 665, 672,677, 682, 701, 703, 711, 717, 718,719, 725, 726, 727, 733, 739, 741,747, 764, 770, 771, 777, 787, 791,793

Keystrokes. See GraphingCalculator; Graphing CalculatorInvestigations; InternetConnections

Law of Cosines, 733–738, 755, 858

Law of Large Numbers, 682

Law of Sines, 725–732, 726, 736,754–755, 858

K

J

L

http://www.algebra2.com





http://www.algebra2.com/webquest

http://www.algebra2.com/careers

http://www.algebra2.com/chapter_test

http://www.algebra2.com/data_update


Index

R96 Index

Leading coefficient, 346, 350, 379

Leaf, 667

Least common denominator (LCD),505–506, 516

Least common multiple (LCM)monomials, 479polynomials, 479, 480, 482, 504, 847

Like radical expressions, 252

Like terms, 229

Limits, 593

Linear correlation coefficient, 87

Linear equations, 63–67, 86, 101, 830graphing, 109identifying, 63solving systems, 452standard form, 64systems of three, 191systems of two, 189writing, 75–80, 102

Linear function, 64, 830

Linear inequalities, graphing, 96,329, 411

Linear permutations, 638

Linear programming, 129–135, 147

Linear-quadratic system, 455–456

Linear relations, graphing, 863

Line graphs, 824

Line of best fit, 87

Line of fit, 81–86

Lineshorizontal, 65intersecting, 111parallel, 70, 77–78, 101, 112perpendicular, 70–71, 77–78, 101slope, 68–74, 80, 82, 101–102, 201,

643, 830, 831vertical, 65

Line segment, midpoint, 845

Loans, amortization, 605

Location Principle, 353, 354

Logarithmic equations, 551, 850solving, 533, 534, 543, 546writing, 565, 570

Logarithmic expressions,evaluating, 532

Logarithmic form, 532, 535, 536,568, 849

Logarithmic functions, 531–540, 532

Logarithmic inequalities, solving,546

Logarithmic relations, 871

Logarithmic to exponential form,532

Logarithmic to exponentialinequality, 533

Logarithms, 520, 531–540base b, 532base e, 554–559common, 547–553, 569, 617functions, 567inverse property, 533natural, 554–559, 569power property, 543properties, 541–546, 568using, 548

Logical reasoning. See CriticalThinking

Lower quartile, 826

Major axis, 434

Mapping, 57

Margin of error, 683

Margin of sampling error, 682, 684

Mathematical induction, 618–621,620, 626

Matrices, 152–217, 865addition, 160column, 155, 156coordinate, 175determinants of 3 3, 183dimensions, 155, 156, 166, 834equal, 209identity, 195–201, 213inverse, 195–201, 205, 206, 207,

213, 214, 358, 637, 836modeling real-world data, 161multiplication, 167–174, 210, 211

different dimensions, 169scalar, 162square, 168

operations, 160–166, 210, 834organizing data, 154reflection, 177rotation, 178row, 155, 156solving systems of equations,

155–156, 202–208, 214square, 155, 156, 198subtraction, 161transformations, 175–181, 211translation, 175, 176zero, 155, 156

Matrix multiplication, AssociativeProperty, 171

Matrix operations, 163combination, 163properties, 162

Matrix products, 167

Maximum points, 354–356, 358, 364

Maximum values, 129, 158,288–289, 290, 291, 293, 337, 377,663, 839

Mean deviation, 669

Mean, 663, 664, 667, 668, 669,822–823, 855arithmetic, 580, 582, 590, 592, 622,

623, 851geometric, 590, 591, 598, 623, 852

Measurementangles, 709, 711, 712, 713, 745,

748, 753conversions, 390, 394tendency, 664variation, 665

Measures of central tendency, 664,822–823

Median, 82, 663, 664, 667, 668, 669,822–823, 855

Median-fit line, 86

Midline, 771

Midpoint, 414formula, 412, 414, 416, 417–418,

461–462, 467line segments, 845

Minimum points, 354–356, 358, 364

Minimum values, 129, 158, 288–289,290, 291, 293, 337, 377, 663, 839

Minor axis, 434

Mixed Problem Solving, 862–875

Mixed Review. See Review

Mode, 663, 664, 667, 668, 822–823,855

Modelingabsolute value, 28algebra tiles, 308area diagrams, 651arithmetic sequences, 580circular functions, 739complex numbers, 272conic sections, 453–454data, 159distance formula, 413distributive property, 13ellipses, 432

M

Index

Index R97

fractals, 611irrational numbers, 252location principle, 354midpoint formula, 412parabolas, 421parallel lines, 70perpendicular line, 71point discontinuity, 485–487polynomials, 240quadratic equations, 295quadratic functions, 287radicals, 252Real-World Data, 103real-world data, 81–86, 300, 359,

539–540slope-intercept form, 75solving inequalities, 36special sequences, 607vertical asymptotes, 485–487vertical line test, 57

Monomials, 222–228, 276–277denominators, 480division, 233, 521, 538least common multiple, 479multiplication, 521, 538

More Aboutaerospace, 327, 398amusement parks, 255, 780animals, 161architecture, 291, 503area codes, 636astronomy, 225, 459aviation, 603ballooning, 731baseball, 723basketball, 143, 477, 667betta fish, 44bicycling, 483bridges, 318building, 243card games, 642child care, 38child development, 357computers, 529construction, 579cryptography, 197dinosaurs, 737drawbridges, 748driving, 713earthquakes, 537elections, 190emergency medicine, 735Empire State Building, 298energy, 355engineering, 311entrance tests, 648farming, 525finance, 99fireworks, 10food, 380

food service, 14football, 331forestry, 304genealogy, 595genetics, 232guitar, 744health, 267, 675, 683, 773home improvement, 23Internet, 679investments, 140job hunting, 43lighthouses, 729magnets, 483math history, 16meteorology, 31military, 64money, 558movies, 157museums, 435music, 111, 262navigation, 443nutrition, 94oceanography, 766Olympics, 564optics, 795Pascal’s triangle, 612population, 114radio, 584railroads, 26recycling, 662René Descartes, 372robotics, 721satellite TV, 422shopping, 388, 668skiing, 705space, 494space exploration, 124, 376space science, 249spelling, 656sports, 61, 677star light, 545submarines, 396technology, 180temperature, 394theater, 351tourism, 292track and field, 169, 750tunnels, 507veterinary medicine, 131waves, 803weather, 165weight lifting, 259White House, 439world cultures, 661

Multiple Choice. See Assessment

Multiple events, 640

Multiple Representations, 11, 12,21, 28, 40, 42, 57, 68, 71, 75, 160,161, 162, 168, 182, 195, 223, 245,250, 251, 257, 258, 271, 287, 295,

301, 307, 346, 347, 378, 390, 391,412, 413, 474, 485, 526, 527, 532,533, 534, 541, 543, 548, 633, 634,658, 660, 725, 764

Multiplication, 781, 828Associative Property, 15, 171Commutative Property, 15, 32,

166, 170, 828complex numbers, 272–273Distributive Property, 170, 171,

228, 828functions, 384, 403matrices, 167–174, 168, 210, 211polynomials, 230, 277probabilities, 651–657, 689pure imaginary numbers, 270, 272radicals, 252rational expressions, 474, 513scalar, 162, 163, 211scientific notation, 225simplifying expressions, 222–223

Multiplication Property ofEquality, 21

Multiplication Property ofInequality, 34, 35

Multiplicative identities, 15, 199

Multiplicative inverses, 13, 14, 15,16, 32, 153, 199, 828

Multi-step equations, solving, 22,201

Multi-step inequality, solving, 35

Mutually exclusive events,658–659, 661, 670, 689, 690, 855

Natural base, e, 554

Natural base exponential function,554

Natural base expressions,evaluating, 554

Natural logarithmic equations,solving, 556

Natural logarithmic expressions,evaluating, 555

Natural logarithmic function, 554

Natural logarithmic inequalities,solving, 556

Natural logarithms, 554–559, 569inverse property, 555

Natural numbers, 11, 17, 32, 48

Negative angle, 709, 712

N

Index

R98 Index

Negative base, 258

Negative exponents, 222

Negative measure, 713, 732, 754

Negative numbers, square roots of,270

Negative zeros, 373, 375, 402, 843

Nodes, 791

Normal distribution, 671–675, 672,680, 685, 691

Notationfunction, 59internal, 829intervals, 35, 37, 40, 41, 51scientific, 225, 226, 227, 836set-builder, 34, 37, 51, 829sigma, 585, 595, 601, 602standard, 225

nth root, 245, 246

nth termarithmetic sequences, 579, 591, 851geometric sequences, 852

Null hypothesis, 686

Number line, 44, 46

Numbersclassification, 221complex, 270–275, 280, 370irrational, 11, 32natural, 11, 17, 32, 48pure imaginary, 270, 272rational, 5, 11, 32, 48, 471real, 5, 11–18, 32, 48, 245–249,

278, 814triangular, 609whole, 11, 18, 48

Number theory, 15, 295, 297, 298,304, 510, 866, 872, 873

Numerators, polynomials, 475

Oblique triangle, 735

Octagons, perimeter, 26

Octants, 136

Odds, 644, 645–646, 647, 648, 663, 854

One-step equations, solving, 21

One-to-one functions, 57, 392, 524

Online Research, See also InternetConnectionscareer choices, 121, 187, 193, 237,

274, 334, 363, 446, 496, 511, 542,561, 609, 685, 707

data update, 10, 66, 143, 165, 255,318, 357, 440, 477, 558, 598, 667,723, 775

Open Ended, 8, 14, 24, 30, 37, 43, 60,65, 71, 78, 83, 92, 98, 112, 119, 125,132, 142, 156, 171, 178, 185, 192,198, 205, 226, 231, 236, 242, 247,254, 260, 265, 273, 290, 297, 303,317, 325, 332, 350, 356, 362, 368,375, 380, 382, 386, 393, 397, 414,423, 428, 437, 445, 450, 458, 476,481, 488, 495, 501, 509, 527, 535,544, 549, 557, 563, 580, 586, 590,596, 602, 608, 615, 634, 647, 654,660, 666, 673, 678, 683, 706, 712,722, 729, 736, 742, 749, 766, 774,779, 784, 788, 794, 802

Open Response, See Preparing forStandardized Tests

Open sentences, 20

Operationsarithmetic, 383–384functions, 383–389, 403radicals, 252

Or compound inequalities, 41

Ordered array, 154

Ordered pairs, 56, 78, 83, 84, 153,387–388, 522, 831, 844

Ordered triples, 136, 139, 833

Ordering real numbers, 814

Order of operations, 6–7

Outcomes, 632, 854

Outliers, 83, 827

Parabolas, 419–425, 450, 451, 460, 462–463, 467, 565, 617, 637equations, 419–420, 841, 845, 846graphing, 420–423

Parallel lines, 70, 77–78, 101, 112

Parallelograms, 192area, 477vertices, 121, 192

Parent graph, 70

Partial sum, 599

Pascal’s triangle, 612, 613, 625–626,872

Patterns, 352

Perfect square trinomials, 310, 816,840

Perimeteroctagons, 26

quadrilaterals, 415, 482rectangles, 255right triangles, 382squares, 603triangles, 592

Period, 741, 762, 764, 765, 767, 771,774, 775, 781, 785, 805, 806, 859

Periodic functions, 741, 742, 743,762

Permutations, 638–643, 650, 688, 715circular, 642linear, 638repetition, 639

Perpendicular lines, 70–71, 77–78,101

Phase shift, 769, 770, 774, 785, 806,859

Piecewise functions, 90–91, 92, 104,115, 370, 831

Plotsbox-and-whisker, 631, 826–827stem-and-leaf, 667, 825

Point discontinuity, 485–487

Point-slope form, 76, 78, 102

Polygonal region, vertices, 124–125,126

Polygonsarea, 187finding areas, 187

Polynomial equationssimplifying, 837solving using quadratic

techniques, 360–364, 401

Polynomial functions, 344–407,400, 868end behavior, 349evaluating, 347even-degree, 349, 357graphing, 348–349, 353–358, 401odd-degree, 349, 357zero, 371

Polynomials, 229–232, 866addition, 229, 277degrees, 229, 346, 350, 400, 837, 842denominators, 475, 480depressed, 366division, 233, 277, 364, 365–366factoring, 239–241, 278, 358, 366,

377, 761, 815–816, 837least common multiple, 479, 480,

482, 504, 847multiplication, 230, 277, 285numerator, 475one variable, 346, 350

O

P

Index

Index R99

operations, 382simplifying, 244subtraction, 229, 277

Positive angle, 709, 712

Positive integers, 620, 853

Positive measure, 713, 732, 754

Positive zeros, 373, 375, 402, 843

Power function, 347, 853

Power Property of Logarithms, 543

Powers, 5, 222expanding, 615, 617, 621simplifying expressions, 224

Practice Chapter Test. See Assessment

Practice Quiz. See Assessment

Prediction equations, 81–82, 83, 84,95, 99, 598

Preimage, 175

Preparing for Standardized Tests,877–892Constructed Response, 884Free Response, 884Grid In, 880Gridded Response, 880–883Multiple Choice, 878, 879Open Response, 884Selected Response, 884–887Student-Produced Questions, 884Student-Produced Response, 880Test Taking Tips, 877, 879, 883,

887, 891

Prerequisite Skills. See alsoAssessmentbar and line graphs, 824box-and-whisker plots, 826–827comparing and ordering real

numbers, 814congruent and similar figures,

817–819factoring polynomials, 815–816Getting Ready for the Next Lesson,

10, 18, 27, 32, 39, 62, 67, 74, 80, 86,95, 115, 122, 127, 135, 158, 166,174, 181, 188, 194, 201, 228, 232,238, 244, 249, 256, 262, 267, 293,299, 305, 312, 319, 328, 352, 358,364, 370, 377, 382, 389, 394, 416,425, 431, 440, 448, 452, 478, 484,490, 498, 504, 530, 538, 546, 551,559, 582, 587, 592, 598, 604, 610,617, 637, 643, 657, 663, 670, 675,680, 708, 715, 724, 732, 738, 745,768, 776, 781, 785, 790, 797

Getting Started, 5, 53, 55, 109,153, 221, 285, 345, 411, 471, 521,

577, 631, 699, 761mean, median, and mode,

822–823Pythagorean Theorem, 820–821stem-and-leaf plots, 825

Prime, 239, 242

Principal root, 246

Principal values, 746

Probability, 644–650, 655, 660, 663,670, 688–689, 708, 732, 768, 785,854, 855, 856, 873addition, 658–663, 689–690combinations, 645conditional, 653distribution, 646events, 647, 648

dependent, 633–634, 634, 635,653, 654, 655, 687, 689, 854,855

inclusive, 659, 660, 661, 670,689, 690, 855

independent, 632–633, 634, 651,652, 654, 687, 689, 854, 855

mutually exclusive, 658–659,661, 670, 689, 690, 855

experimental, 649failure, 644multiplication, 651–657, 689odds, 644, 645–646, 647, 648, 663,

854simple, 631success, 644theoretical, 649

Problem solving, 854distributive property, 14inverses, 197matrix equation, 203mixed, 862–875right triangles, 703translations, 773

Product of powers, 223

Product Property, 542Logarithms, 541–542Radicals, 250

Proof, 618–621, 626

Properties of Equality, 21, 23, 566,781Logarithmic Functions, 567

Properties of Inequality, 566Logarithmic Functions, 567

Properties of Logarithms, solvingequations using, 543

Properties of MatrixMultiplication, 171

Properties of Order, 33

Properties of Powers, 224, 226, 526

Proportional sides, 817

Proportions, 181solving, 471, 490

Pure imaginary numbers, 270multiplication, 270, 272

Pyramid, surface area, 27

Pythagoras, 16

Pythagorean identities, 777, 779

Pythagorean Theorem, 699, 720,820–821

Quadrantal angle, 718

Quadrants, 56, 720

Quadratic equations, 328, 604, 841solving, 761

by completing the square,306–312, 328, 338, 352, 411,490, 587, 840

by factoring, 301–305, 338, 840by graphing, 294–299, 337,

345, 352for variables, 389

Quadratic form, 360, 363, 370, 842

Quadratic Formula, 345, 370, 460, 841discriminant, 313–319, 339

Quadratic functions, 286, 499, 502,503, 515, 839, 848, 867graphing, 286–293, 322–328,

336–337, 339–340

Quadratic identities, 375

Quadratic inequalities, 839, 867graphing, 329–333, 340solving, 329–333, 340

Quadratic-quadratic system,456–457

Quadratic solutions, 271

Quadratic systems, solving,455–460, 466

Quadratic techniques, 401solving polynomial equations

using, 360–364

Quadrilaterals, perimeter, 415, 482

Quartile, 826lower, 826upper, 826

Quotient identities, 777

Quotient of Powers, 223

Q

Index

R100 Index

Quotient Propertylogarithms, 542radicals, 251

Quotients, 328, 364simplifying, 242, 251trinomials, 242

Radian measure, 710, 711, 713conversion, 711

Radians, 710, 713, 724, 749, 753, 757,802, 803, 808, 857measuring, 711, 712

Radical equations, 263–269, 280solving, 263, 362

Radical exponents, 279

Radical expressions, 250–256, 255,279, 285

Radical form, 257, 838

Radical inequalities, solving,264–265

Radicalsaddition, 252, 253approximating, 247multiplication, 252simplifying, 245subtraction, 253

Radius, 426

Random, 645

Random sample, 682, 856

Random variable, 646

Range, 56, 57, 58, 61, 93, 94, 95, 99,101, 104, 181, 397, 398, 416, 523,527, 528, 530, 663, 823, 830, 831,844, 849

Rate of change, 69, 560

Rate of decay, 560

Rate of growth, 562

Rate problem, 507

Ratiocommon, 588, 603finding term given, 589

Rational equations, 505–509solving, 505–509, 516

Rational exponents, 257–262, 838solving equations, 361–362

Rational expressions, 472, 870addition, 480, 514

division, 474, 513multiplication, 474, 513simplifying, 472–475subtraction, 480, 514

Rational functions, 500, 502, 504, 515graphing, 485–490, 514, 848

Rational inequalities, solving,505–509, 508–509, 516

Rationalizing denominators, 251,253, 715

Rational numbers, 11, 32, 48operations, 5solving equations, 471

Rational zeros, 379, 381, 394, 403,675, 843

Rational Zero Theorem, 378–382, 403

Reading and Writing, 5, 53, 109,153, 221, 285, 345, 411, 471, 521,577, 631, 699, 761

Reading Math, 11, 12, 56, 59, 71, 82,154, 175, 182, 229, 252, 270, 271,272, 273, 306, 313, 316, 323, 442,449, 606, 619, 633, 638, 644, 646,665, 669, 709, 711, 718, 740, 786, 788

Real numbers, 11–18, 32comparing and ordering, 5, 814Identify Properties, 13properties, 48roots, 245–249, 278

Real-world applications. SeeApplications; More About

Real-world data, modeling, 81–86,103

Reciprocal identities, 777

Rectanglesarea, 255, 334golden, 311perimeter, 255volumes, 866width, 242

Rectangular prismssurface areas, 18volumes, 367width, 363

Rectangular solid, volumes, 379, 380

Recursion, 606–611, 625

Recursive formula, 606, 607, 608

Reference angles, 718–719, 722, 776finding trigonometric value, 720

Reflection, 177

Reflection matrices, 177

Reflexive Property of Equality, 21

Regression equation, 87

Regression line, 87

Relations, 56, 100–102

Relative-frequency histogram, 646,647

Relative maximum, 354, 356, 842

Relative minimum, 354, 356, 842

Remainder Theorem, 365–370, 402

Repeating decimals, as fractions,601, 602, 603, 852

Repetition, permutation, 639

Replacement sets, 377

Research, 85, 133, 200, 227, 311, 398,415, 497, 529, 545, 592, 636. Seealso Online Research

Residuals, 540

ReviewLesson-by-Lesson, 47–50,

100–104, 145–148, 209–214,276–280, 336–340, 400–404,461–466, 513, 566–570,622–626, 687–692, 752–756,805–808

Mixed Review, 18, 27, 32, 46, 62,67, 74, 80, 86, 95, 99, 115, 122,127, 135, 144, 158, 166, 174, 181,188, 194, 201, 207, 228, 232, 238,244, 249, 256, 262, 267, 275, 293,299, 305, 312, 319, 328, 335, 352,358, 364, 370, 377, 382, 389, 394,399, 416, 425, 431, 440, 447, 452,460, 478, 484, 490, 498, 504, 511,530, 538, 546, 551, 559, 565, 582,587, 592, 598, 604, 610, 617, 621,637, 643, 650, 657, 663, 670, 675,680, 685, 708, 714, 724, 732, 738,745, 751, 768, 776, 781, 785, 790,797, 804

Right triangles, 700, 704perimeter, 382

Right triangle trigonometry,701–708, 752

Roots, 296, 371–377, 376, 840, 843complex, 315double, 302irrational, 315nth, 245, 246principal, 246real number, 245–249, 278square, 245, 249, 362, 530, 650

R

Index

Index R101

Rotation matrices, 178

Rotations, 177, 178

Rounding, 358, 549, 550, 565, 569,663, 704, 706, 714, 724, 730, 731,732, 736, 738, 745, 748, 749, 751,753, 756, 821, 823, 855, 858, 859

Row matrix, 155, 156

Samplebias, 682random, 682, 856unbiased, 682

Sample space, 632

Sampling, 692

Sampling error, 682–686, 714margin, 682, 684

Scalar multiplication, 162, 163, 211Associative Property, 171

Scatter plots, 81–86, 87, 95, 99, 103,598, 831

Scientific notation, 225, 226, 227, 836

Secant, 701, 708

Second-order determinant, 182

Sector, 713

Selected Response, See Preparingfor Standardized Tests

Sequences, 578, 872arithmetic, 578–582, 583,

622–623, 768, 851Fibonacci, 606, 609, 610geometric, 588–593, 623–624, 852

Series, 583, 872arithmetic, 583–587, 623, 851geometric, 594–598, 624, 781infinite geometric, 599–605,

624–625

Set-builder notation, 34, 37, 51, 829

Sets, 18, 828empty, 29replacement, 377solution, 37, 41, 44, 46, 95, 829

Short Response, 546, 559, 564, 724,732, 745. See also Assessment andPreparing for Standardized Tests

Sides, 734initial, 709proportional, 817terminal, 709

Sigma notation, 585, 595, 601, 602

evaluating sum, 585, 595infinite series, 601

Similar figures, 817–819

Simple event, 658

Simple probability, 631

Simplify Powers of i, 270, 272

Simulations, 681

Sin�1, 747

Sine function, 701, 706, 707, 747,767, 770, 771definition, 739finding, 740value, 747

Skewed distributions, 671

Slope-intercept form, 75, 78, 79, 86,102, 188, 637, 831

Slope of line, 68–74, 80, 82,101–102, 201, 643, 830, 831

Solid boundary, 97

Solution, 20, 801

Solution set, 37, 41, 44, 46, 95, 829

Special angles, 703

Special functions, 89–95, 104

Special sequences, 606–611

Special values, 533

Spheres, surface areas, 862

Spreadsheet Investigationamortizing loans, 605organizing data, 159special right triangles, 700

Square matrix, 155, 156, 198

Square root, 245, 249, 362, 530, 650approximate, 247negative numbers, 270

Square root functions, 395–396,398, 399, 404, 500, 502, 503, 515,848

Square root inequalities, 397–399,404graphing, 404

Square Root Property, 250–251,306, 310, 313, 790

Squares, perimeter, 603

Standard deviation, 665, 666, 667,669, 670, 675, 685, 690, 855

Standard form, 64, 101, 122, 422, 424,428, 449, 460, 478, 830, 845, 846

Standardized Test Practice. SeeAssessment

Standard notation, 225

Standard position, 709

Statistics, 664–670, 690, 873

Stem, 667

Stem-and-leaf plots, 667, 825

Step functions, 89–90, 92, 115, 158,370, 831

Student-Produced Questions, SeePreparing for Standardized Tests

Student-Produced Response, SeePreparing for Standardized Tests

Study Organizer. See Foldables™Study Organizers

Study Tipsabsolute value, 90, 599absolute value inequalities, 42additive identity, 162A is acute, 728algebra tiles, 240alternative method, 77, 264, 474,

580, 590, 652, 728, 734alternative representations, 726amplitude and period, 764angle measure, 748area formula, 184checking solutions, 110, 265, 481,

543choosing a committee, 659choosing the independent

variable, 81choosing the sign, 793coefficient, 116combinations, 640combining functions, 386common factors, 480common misconception, 7, 12, 29,

118, 130, 289, 308, 523, 659, 703,782

conditional probability, 653continuously compounded

interest, 556coterminal angles, 712deck of cards, 640depressed polynomial, 366Descartes’ Rule of Signs, 379element, 155elimination, 139equations with ln, 556equations with roots, 303error in measurement, 704exponential growth and decay, 524expressing solutions as multiples,

800extraneous solutions, 506, 534

S

Index

R102 Index

factor first, 475factoring, 367finding zeros, 374focus of parabola, 419formula for sum, 600graphing calculators, 225, 247,

436, 444, 456, 525, 585, 613graphing polynomial functions,

353graphing quadratic inequalities,

457graphing rational functions, 486graphs of piecewise functions, 92greatest integer function, 89horizontal lines, 70identity matrix, 204indicated sum, 583inequality phrases, 36interval notation, 40, 41inverse functions, 392Law of Large Numbers, 682location of roots, 296look back, 91, 97, 123, 189, 204,

273, 329, 361, 365, 371, 420, 485,508, 524, 526, 531, 532, 608, 634,664, 676, 720, 747, 762, 771, 772

matrix operations, 163memorize trigonometric ratios, 702message, 198midpoints, 412missing steps, 614multiplication and division

properties of equality, 22multiplying matrices, 168negative base, 258normal distribution, 672number of zeros, 349outliers, 83one real solution, 295parallel lines, 112permutations, 640power function, 347properties of equality, 21properties of Inequality, 33quadratic formula, 314quadratic solutions, 271radian measure, 710rate of change, 560rationalizing denominator, 251reading math, 11, 12, 34, 35, 56,

59, 71, 82, 124, 129, 154, 175,182, 229, 246, 252, 270, 271, 272,273, 294, 306, 313, 316, 323, 354,372, 384, 391, 442, 449, 606, 619,638, 644, 646, 665, 669, 701, 709,711, 718, 740, 786, 788

sequences, 578sides and angles, 734sigma notation, 585simplified expressions, 224skewed distributions, 671

slope, 68slope-intercept form, 75solutions to inequalities, 35solving quadratic inequalities

algebraically, 332solving quadratic inequalities by

graphing, 330special values, 533step 1, 618substitution, 361symmetry, 288technology, 547terms of geometric sequences, 594using the discriminant, 316using logarithms, 548using quadratic formula, 315verifying a graph, 770verifying inverses, 196vertical and horizontal lines, 65vertical line test, 58vertical method, 230vertices of ellipses, 434zero at origin, 372zero product property, 302

Substitution, 21, 146, 149, 153, 504,781, 828direct, 366, 368solving systems of equations,

116, 119, 120, 122, 135, 166, 832synthetic, 365–366, 368, 369, 377,

402, 715, 751, 843

Substitution Property of Equality, 25

Subtractioncomplex numbers, 270, 272functions, 383, 403matrices, 161polynomials, 229, 277radicals, 253rational expressions, 480, 514solving inequality, 34

Subtraction Property of Equality, 21

Subtraction Property of Inequality,33

Success, 644probability, 644

Sum and difference formulas,786–787

Summation formula, 618

Sums, 657, 787arithmetic series, 583, 584, 586,

587, 592, 598geometric series, 595, 596, 597, 610infinite geometric series, 602, 610partial, 599rewriting differences, 221series, 585, 663

sigma notation, 585, 595two cubes, 361

Surface areacones, 22, 266cylinders, 25, 862pyramids, 27rectangular prisms, 18spheres, 862

Symbols, infinity, 601

Symmetric Property of Equality,21, 25, 46, 781

Symmetry, 288, 767

Synthetic division, 234–236, 345,745, 837

Synthetic substitution, 365–366,368, 369, 372–373, 377, 402, 551,715, 751, 843

Systems of equations, 110, 158, 864consistent, 111, 112, 113, 122, 293Cramer’s Rule in solving, 835dependent, 111, 112, 113, 122,

293inconsistent, 111, 112, 113, 122,

293independent, 111, 112, 113, 122,

293solving, 166, 188, 203, 657, 724

algebraically, 116–122, 146elimination, 153, 832graphing, 110–115, 122, 145,

146, 147, 148, 194, 832matrices, 205, 206, 214substitution, 832three variables, 138–144, 148

Systems of inequalities, 663solving, graphing, 123–127, 135,

144, 147, 158, 484, 833, 847

Systems of quadratic inequalities,457

Systems of three linear equations,191

Systems of two linear equations,189

Table of values, 286, 288, 290, 291, 299, 352, 356, 364, 839

Tangent, 706, 707

Tangent function, 427, 701, 747, 770,771

Tangent ratio, 708

Terminal side, 709

T

Index

Index R103

Terms, 229, 578, 615finding, 578, 579, 588, 589like, 229series, 596

Testing hypotheses, 686

Test preparation. See Assessment

Test-taking tips. See Assessment

Theoretical probability, 649

Third-order determinant, 182, 183

30°-60°-90° triangles, 699, 703, 707

3 � 3 matrices, determinants, 183

Towers of Hanoi game, 607

Transformations, 175graphing, 772matrices, 175–181, 211verifying, 783

Transitive, 21, 46, 828

Transitive Property of Equality, 25

Translation matrix, 175, 176

Translations, 175horizontal, 769–770trigonometric graphs, 769–776vertical, 771–772

Transverse axis, 442

Trapezoid, area, 8, 67, 865

Triangle Inequality Theorem, 45

Trianglesarea, 32, 184, 185, 186, 187, 231,

281, 725, 866equilateral, 86945°-45°-90°, 699, 703, 707isosceles, 869Pascal’s, 612, 625–626, 872perimeter, 592right, 382, 700, 70430°-60°-90°, 699, 703, 707vertices, 113, 415

Trichotomy Property, 33

Trigonometric identities, 875

Trigonometric equations, 799, 802solving, 799–804, 800, 801, 808

Trigonometric functions, 698–759,701, 717, 722, 723, 732, 738, 754,761, 790, 796, 857, 874evaluating, 717, 741, 742, 778general angles, 717–724graphing, 762–768, 765, 772, 805inverse, 746–751solving equations, 724, 732using, 766variations, 764

Trigonometric graphs, 875translations, 769–776, 806

Trigonometric identities, 777, 785,806basic, 777verifying, 782–785, 785, 807

Trigonometric inverses, addition,751

Trigonometric values, 703, 720, 761finding, 702, 748, 777

Trigonometry, 701, 875right triangle, 701–708, 752

Trinomials, 229, 310perfect square, 816, 840quotient of two, 242

2 � 2 matrices, determinants, 182

Two-variable matrix equation, 202

Unbiased sample, 682

Unbounded region, 130

Uniform distribution, 646

Union, 41

Unit circle, 710, 739, 742, 743

Univariate data, 664

Upper quartile, 826

USA TODAY, Snapshots, 3, 69, 84,135, 206, 219, 228, 328, 368, 409,448, 492, 535, 565, 575, 604, 697,715, 797

Valuesmaximum, 158, 663minimum, 158, 663

Variables, 7, 25dependent, 59functional values, 348independent, 59random, 646solving for, 22, 109, 389

systems of equations, 138–144,148

Variance, 665, 666, 667, 669, 670,675, 690

Variationsdirect, 496, 559, 650, 848inverse, 496, 559, 848joint, 496, 559, 848

Velocity, angular, 714

Venn diagram, 12, 271

Verbal expressions, 828algebraic expressions, 20, 24, 115

Vertex form, 322–328, 335

Vertex matrix, 175

Vertical asymptotes, 485–487, 617,763

Vertical lines, 65

Vertical Line Test, 57, 58

Vertical shift, 771, 774, 775, 781,806, 859

Vertical translations, 771–772

Vertices, 129, 287–288, 290, 291, 299,339, 636angles, 113, 192coordinates, 846exact coordinates, 744parallelograms, 121, 192polygonal region, 124–125, 126triangles, 113, 415

Volumescubes, 615rectangular prism, 367, 866rectangular solid, 379, 380

Von Koch snowflake, 611

WebQuest, 3, 27, 120, 192, 207, 219, 227, 328, 369, 399, 409, 430, 504,529, 565, 575, 616, 635, 685, 697,708, 775, 804

Whole numbers, 11, 18, 48

Work problem, 507

Writing in Math, 10, 17, 27, 31, 38,45, 62, 67, 73, 80, 86, 94, 99, 114,121, 127, 134, 144, 158, 166, 173,181, 187, 193, 200, 207, 227, 232,238, 243, 255, 262, 267, 275, 292,299, 305, 312, 319, 327, 334, 352,357, 364, 370, 377, 382, 389, 394,399, 416, 425, 430, 439, 447, 452,459, 477, 484, 490, 498, 503, 530,537, 546, 551, 559, 564, 582, 587,592, 598, 603, 610, 616, 621, 636,642, 649, 657, 662, 675, 679, 685,708, 714, 724, 732, 737, 744, 751,768, 776, 781, 785, 790, 796, 804

x-coordinate, 68, 290, 299, 348, 354, 356, 401, 839, 842

x-intercept, 65, 66, 70, 74, 101, 174,330, 830

U

V

W

X

Index

R104 Index

y-coordinate, 68

y-intercept, 65, 66, 70, 74, 78, 82,101, 174, 287–288, 291, 299, 530,830, 831

Zero matrix, 155, 156

Zero Product Property, 301, 302,305, 361, 362solving equations, 797

Zeros, 294, 371–377, 604function, 294, 348, 349, 354imaginary, 375, 402, 843negative, 373, 375, 402negative real, 843origin, 372positive, 373, 375, 402positive real, 843rational, 379, 381, 394, 403, 675,

843synthetic substitution, 373–374

Z

Y

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