Topic: Expressions & Operations AII · Topic: Expressions & Operations AII.3 AII.3 The student will...

1

Topic: Expressions & Operations AII.1

AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, complex numbers and matrices.

Notes and/or Formulas

Properties Identity Property

(+) 0a a (x) (1)a a

Inverse Property (+) ( ) 0a a

(x) 1

1aa

Associative Property (+) ( ) ( )a b c a b c

(x) ( ) ( )a bc ab c

Commutative Property

(+) a b b a

(x) ab ba Distributive Property ( )a b c ab ac

Axioms of Equality

Reflexive a a Symmetric

If a b , then b a Transitive

If a b , and b c then a c Order of Operations Parenthesis Exponent Multiply* Divide* Add** Subtract** * Multiply/Divide in order from left to right ** Add/Subtract in order from left to right

1. Which of the following is an example of the commutative property of addition?

A. a b a b B. ( ) ( )a b c a b c

C. ( ) ( )a b c a c b

D. ab c a bc

2. Which property justifies the statement ( )x a c xa xc ?

F. Associative Property of Multiplication G. Commutative Property of Multiplication H. Associative Property of Addition J. Distributive Property

3. For which of the following operations is the commutative property not valid? A. Multiplication of integers B. Multiplication of complex numbers C. Multiplication of matrices D. Multiplication of negative real numbers

4. Use the distributive property to simplify: 7(2 5 )x y

F. 2 5x y

G. 14 35x y

H. 14 5x y

J. 14 35x y

5. Use the distributive property to simplify: 9(6 4 )x y

A. 54 36x y

B. 54 4x y

C. 6 36x y

D. 6 4x y

6. Simplify: (54 3 48 3 48) 9

F. 3 G. 6 H. 2700 J. 106

7. Simplify: 10 5 3(8 2)

A. 36 B. 48 C. 20 D. 14

2

Notes and/or Formulas 8. Solve the equation by using the properties of equalities: 3( 2) 8x

F. 14

3

G. 14

3

H. 10

3

J. 10

3

9. Solve the equation by using the properties of equalities: 7 5( 3) 2 7x x

A. 5x

B. 5x C. No Solution D. Identity

10. If 4a , which of the following statement is true?

F. 3 7a

G. 4 16a

H. 4 14a

J. 5 10a

3


AII.5 The student will identify and factor completely polynomials representing the difference of squares, perfect trinomials, the sum and difference of cubes and general trinomials.


Ways to Factor: 1) Greatest Common Factor ( )xy xw x y w

2) Difference of Squares

2 2 ( )( )a b a b a b

3) Sum of Cubes*

3 3 2 2( )( )a b a b a ab b

4) Difference of Cubes*

3 3 2 2( )( )a b a b a ab b

*Square-Multiply-Square-Opposite –Plus 5) Trinomials 6) Factor by Grouping (leading coefficient) 7) Completing the square

1. Which is the factored form of 364 1x ?

A. 2(4 1)(4 4 1)x x x

B. 2(4 1)(16 1)x x

C. 2(4 1)(16 4 1)x x x

D. 2(4 1)(16 4 1)x x x

2. Factor: 2 100x F. ( 50)( 50)x x

G. ( 10)( 10)x x

H. ( 10)( 10)x x

J. ( 25)( 4)x x

3. Factor: 6 99 21x x

A. 6 93(3 7 )x x

B. 5 83 (3 7 )x x x

C. 6 33 (3 7 )x x

D. 6 3(9 21 )x x

4. Factor: 22 7 4x x F. (2 1)( 4)x x

G. (2 4)( 1)x x

H. (2 1)( 4)x x

J. (2 1)( 4)x x

5. Factor: 2 3 2x x A. ( 1)( 2)x x

B. ( 1)( 2)x x

C. ( 1)( 2)x x

D. ( 3)( 2)x x

6. Factor: 218 3 1u u F. (6 1)(3 1)u u

G. (6 1)(3 1)u u

H. (6 1)(3 1)u u

J. (6 1)(3 1)u u

4


7. 22 5 12x x represents the area of a rectangle. Which of the following could represent the length of one side of the rectangle?

A. 2 3x

B. 2 3x

C. 4x

D. 12x

8. 23 5 2x x represents the area of a rectangle. Which of the following could represent the length of one side of the rectangle?

F. 3 2x

G. 1x

H. 3 2x

J. 2x

9. Factor: 3 64b

A. 3( 4)b

B. ( 4)( 4)( 4)b b b

C. 2( 4)( 4 16)b b b

D. 2( 4)( 4 16)b b b

10. Find the term that must be added to both sides of the equation so that the

equation can be solved by the method of completing the square 2 8 13x x F. 16 G. 64 H. – 13 J. 32

11. Find the term that must be added to both sides of the equation so that the

equation can be solved by the method of completing the square 2 6 9x x A. 18 B. 9 C. 36 D. -9

5


AII.2 The student will add, subtract, multiply, divide, and simplify rational expressions, including complex fractions.


Rules for fractions: 1) Always factor any

squared terms completely – watch for greatest common factors

2) Addition & Subtraction of fractions require a common denominator

3) When dividing fractions – flip the second fraction and multiply

4) Complex Fractions – simplify numeration, simplify the denominator, then Divide

1. Simplify: 2

2 1

2

a a

a a

A. 2

2

2

2

a

a

B. 2

3

2

2

a

a

C. 1a

a

D. 2

2

a

a

2. Simplify:

1

1 1

x

y

x y

F. 1

y

G. 1

x

H. y x

xy

J. x

3. Simplify: 2

2

3 2 1

1

m m

m

A. 3 2m

B. 3 2m

C. 3 1

1

m

m

D. 3 1

1

m

m

6


4. Multiply: 2 2

2

1 16

4 2 5 3

x x y

x y x x

F. 2 24

2 3

x y

x

G. 4

2 3

x y

x

H. 4

5 5

x y

x

J. 4

5

x y

5. Multiply: 2 2

2

4 9 16

3 4 2 13 20

x x y

x y x x

A. 3 4

2 5

x y

x

B. 2 23 4

2 5

x y

x

C. 3 4

7

x y

D. 3 4

7 13

x y

x

6. Divide: 2 2

2 2

2 15 25 7 10

2 13 20 4 24 32

x x x x

x x x x

F. – 4 G. – 3

H. 2x

J. 5x

7. Divide: 2 2

2 2

2 3 2 3

2 7 15 4 8 60

x x x x

x x x x

A. 1x B. – 3 C. – 4

D. 3x

7


8. Simplify:

2 6 9

203

4

x x

xx

x

F. 9

5

x

x

G. 3

5

x

H. 7 3x

J. 3

5

x

9. Simplify:

2 16 64

248

4

x x

xx

x

A. 64

6

x

x

B. 8

6

x

C. 8

6

x

D. 15 8x

8


AII.17 The students will perform operations on complex numbers and express the results in simplest form. Simplifying the results will involve using patterns of the powers of i.


Calculator (TI-83 or TI-84) Use i button on the calculator, however remember to use the parentheses to separate operations

EX. 2

3

i

i

(2 ) (3 )i i

Don’t Forget: 2 1i

1. Simplify: 3 4

2

i

i

A. 2 i B. – 2

C. 2 11

3

i

D. 2 5

3

i

2. Simplify: 3 5

7 4

i

i

F. 1 47

65 65i

G. 1 47

65 65i

H. 1 47

65 65i

J. 1 47

65 65i

3. Simplify: 7

8

i

i

A. 57

65

i

B. 57

65

i

C. 57

65

i

D. 57

65

i

4. Simplify: ( 2 5 )(8 3 )i i

F. 31 46i

G. 1 34i

H. 31 34i

J. 1 46i

5. Simplify: (1 7 )( 9 4 )i i

A. 37 59i

B. 19 59i

C. 19 67i

D. 37 67i

9

Notes and/or Formulas To simplify powers of i

Change to 2( )Poweri then

change 2( )i to (- 1)

Example 23 2 11

11

( )

( 1)

( 1)

i i i

i

i

i

Don’t Forget: 1 i

6. Simplify: (3 8 ) (7 6 )i i

F. 10 2i

G. 10 2i

H. 69 38i

J. 4 14i

7. Simplify: (6 6 ) (1 2 )i i

A. 7 8i

B. 7 8i

C. 6 18i

D. 5 4i

8. Simplify: (3 4 ) 2(5 6)i i

F. 24 13i

G. 9 14i

H. 18 14i

J. 5i

9. Simplify: 44i A. 1 B. – 1 C. i D. i

10. Simplify: 27i F. 1 G. i H. i J. – 1

11. Write the given expression in terms of i 8

A. 8i

B. 2 2i

C. 8i

D. 2 2i

12. Write the given expression in terms of i 64

F. 8

G. 8i

H. 8i

J. 8

10


AII.3 The student will add, subtract, multiply, divide, and simplify radical expressions containing radical expressions containing positive rational numbers and variables and expressions containing rational exponents; and write radical expressions as expressions containing rational exponents, and vise versa.


Multiplying Radicals n n na b ab

n

nn

a a

bb

1. Simplify: 93 27y

A. 3 B. 3y

C. 33y

D. 63y

2. Simplify: 18 225 x y

F. 3 4 3 25x y x y

G. 3 2 3 45x y x y

H. 13 17x y xy

J. 3 4 3 2x y x y

3. Simplify: 5 46 4x y xy

A. 34 6x y y

B. 2 6xy

C. 3 22 6x y y

D. 6 54 6x y

4. Simplify: 5

6

6

3

x

x

F. 2x

G. 2x

H. 2x

x

J. 2x

x

5. Rationalize the denominator: 4

11

A. 4

11

B. 4 11

C. 4 11

121

D. 4 11

11

11

Notes and/or Formulas Only radical expressions with like radicands (stuff under the radical) can be added or subtracted.

6. Rationalize the denominator: 2

x

F. 2

4

x

G. 2

x

H. 2

x

J. 2

2

x

7. Simplify: 5 2 49 2 20

A. 11 5

B. 3 5 14 2 20

C. 3 5 14

D. 74

8. Simplify: 8 7 4 6 63

F. 24 7

G. 26 7 2

H. 13 74

J. 26 7 2 6 63

9. Multiply: 4 3 4 3

A. 19 8 3

B. 19

C. 13 8 3

D. 13

10. Divide: 3

6 5

F. 18 3 5

G. 18 5

31

H. 18 3 5

31

J. 6 5

12

Notes and/or formulas Don’t Forget:

a a

b a b bx x x

Properties of Rational Exponent

1. m n m na a a

2. ( )m n mna a

3. ( )m m mab a b

4. 1m

ma

a

5. m

m n

n

aa

a

6.

m m

m

a a

b b

11. Rewrite 5

6 x using rational exponents.

A. 6

5x

B. 6

5x

C. 5

6x

D. 5

6x

12. Rewrite 8 7x using rational exponents

F. 8

7x

G. 7

8x

H. 7

8x

J. 8

7x

13. Simplify and write in simplest radical form: 5 1

52x x

A. x

B. 102 7x x

C. 2x

D. 27 10x

14. Simplify and write in simplest radical form: 51

32x x

F. 2 6x x

G. 13 6x

H. 6 5x

J. 5x x

15. Simplify: 3 4 3 2( 4 )( 2 )x y x y

A. 9 88x y

B. 3 28x y

C. 6 68x y

D. 6 68x y

16. Simplify: 2 2 2 4(2 )( 6 )x y x y

F. 2 412x y

G. 4 612x y

H. 4 612x y

J. 4 612x y

13

Notes and/or Formulas 17. Simplify:

7 3

4 6

27

9

x y

x y

A. 3

33

x

y

B. 3

3

3x

y

C. 3

3

3x

y

D. 11

9

3x

y

18. Simplify: 7 2

7

32

8

x y

xy

F. 6

5

4x

y

G. 6

5

4x

y

H. 6

54

x

y

J. 8

9

4x

y

14

Topic: Equations & Inequalities AII.4

AII.4 The student will solve absolute value equations and inequalities graphically and algebraically. Graphing calculators will be used both as a primary method of solution and to verify algebraic solutions.


To Solve Absolute Value Equations Set = to positive value Set = to negative value To Solve Absolute Value Inequalities 1. Write equation as is 2. Write equation, switch inequality symbol, change to negative value To Graph Absolute Value Inequalities GreatOR Than Less ThAND OR….

(Open – Left & Right)

(Closed – Left & Right)

(Open – Between)

(Closed – Between)

1. Which graph represents the solution of 3 6 9x ?

A.

-2 -1 0 1 2 3 4 5 6

B.

-2 -1 0 1 2 3 4 5 6

C.

-2 -1 0 1 2 3 4 5 6

D.

-6 -5 -4 -3 -2 -1 0 1 2

2. Solve: 2 2 4x

F. 0x and 4x

G. 4x

H. 0x J. No Solution

3. Solve: 4 1 5x

A. 3

,12

B. 5 3

,2 2

C. 3 3

,2 2

D. 3

, 12

4. Solve and Graph: 2 5 9x

F.

8 7 6 5 4 3 2 1 0 1 2 3

G.

8 7 6 5 4 3 2 1 0 1 2 3

H.

8 7 6 5 4 3 2 1 0 1 2 3

J.

8 7 6 5 4 3 2 1 0 1 2 3

15


AII.6 The student will select, justify and apply a technique to solve a quadratic equation over the set of complex numbers. Graphing calculators will be used for solving and for confirming the algebraic solutions.


Methods for Solving Quadratics 1. Factor 2. Complete the Square 3. Square Root Method 4. Quadratic Formula

2 4

2

b b acx

a

negative number use i

Calculator Hints: To Find Roots, Solutions, Zeros 1. Graph quadratic in y = 2. Press 2nd Trace 3. Press #2 for zeros 4. Left Enter 5. Right Enter 6. (Guess) Enter

1. Solve: 23 4 4 0x x

A. 2

, 23

B. 2

, 23

C. 4, 12

D. 2, 3

2. Solve: 23 14 8x F. 2, 2

G. 2 . 2i i

H. 2, 2

J. 2, 2i i

3. Solve: 2 2 0x x A. 1,2

B. 1,2

C. 2,1

D. 1. 2

4. Solve: 22( 1) 8x

F. 3, 1

G. 2, 2

H. 1 2,1 2

J. No Solution

5. Solve: 24 13 12 0x x

A. 4

4,3

B. 3

4,4

C. 3

4,4

D. 4

4,3

16


Describe nature of roots

1. You can graph the quadratic in the calculator look for the number of times the graph touches the x-axis. Touches once – One real solution Touches twice – Two real solutions Does not touch – Two Imaginary solutions 2. Use the discriminant

2 4b ac Discriminant > 0 Two real solutions Discriminant < 0 Two imaginary solutions Discriminant = 0 One real solution

6. Solve: 2 3 10 0x x F. 5,1

G. 1,5

H. 5,1

J. 5, 1

7. Solve: 26 2 3x x

A. 3 15

2

B. 3 3

2

i

C. 3 15

2

D. 3 3

2

8. Which statement is true for the quadratic equation 20 2 4 48x x F. The product of the roots is 24. G. The product of the roots is – 24. H. The sum of the roots is – 24. J. The sum of the roots is – 2.

9. Find the quadratic equation with roots – 3 and 2

5

A. 25 17 6 0x x

B. 25 17 6 0x x

C. 25 17 6 0x x

D. 25 17 6 0x x

10. Find the quadratic equation with roots – 1 and 5

3

F. 23 2 5 0x x

G. 23 2 5 0x x

H. 23 2 5 0x x

J. 23 2 5 0x x

11. Describe the nature of the roots of the equation 23 2 2 0x x A. One real root B. Two imaginary roots C. One real root and one imaginary root D. Two real roots

17

Notes and/or Formulas 12. Describe the nature of the roots of the equation 24 4 5 0x x F. Two imaginary roots G. Two real roots H. One real root J. One real root and one imaginary root

18


AII.7 The student will solve equations containing rational expressions and equations containing radical expressions algebraically and graphically. Graphing calculators will be used for solving and confirming algebraic solutions.


Solve a radical equation Isolate the radical Square or cube both sides

1. Solve: 2 4y y

A. 0y

B. 5y

C. 0y and 5y

D. No Solution

2. Solve: 9 3x x

F. 7x

G. 7x

H. 7x and 7x

J. 0x and 7x

3. Solve: 3 4 1 3x

A. 7x

B. 3 7x

C. 13

7x

D. 7x and 13

2x

4. Use the quadratic formula to solve: 2 10 41 0x x

F. 5 4i

G. 5 8i

H. 5 8i

J. 5 4i

19

Topic: Relations & Functions AII.8

AII.8 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step and exponential functions) and convert between a graph, table and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators.


To find the x-intercept, plug in zero for y

To find the y-intercept, plug

in zero for x.

1. Graph 7 5 35x y by determining its x- and y-intercepts

A. B.

C. D.

2. Graph 5 3 15x y by determining its x- and y-intercepts.

F. G. H. J.

20


Slope = rise

run

For Positive Slope

Count , Count OR

Count , Count

For Negative Slope

Count , Count OR

Count , Count

If slope is a whole number put a 1 underneath to

make it a fraction.

3. A line goes through the point (1, 5) and has a slope 1

2. Graph this line.

A. B. C. D.

4. A line goes through the point (5, 2) and has slope 1

3. Graph this line.

F. G. H. J.

21


5. Identify the function ( ) 4f x x .

A. Greatest Integer B. Absolute Value C. Direct Variation D. Constant

6. Which is an identity function?

F. ( ) 1f x

G. ( )f x x

H. ( )f x x

J. ( )f x x

7. Identify the type of function for the graph below.

A. Greatest Integer B. Constant C. Identity D. Absolute Value

8. Graph: 2 1y x

F. G. H. J.

22

Notes and /or Formulas

Degree Name

X Linear

X2 Quadratic

X3 Cubic

X4 Quartic

X5 Quintic

The degree of a polynomial is the highest power of x.

9. Graph: 2 3y x

A. B. C. D.

10. Write the following equation in the form 2( )y a x h k and graph.

2 2 4y x x

F. G. H. J.

11. Identify the polynomial function 2 3 4( ) 5 6 9 3f x x x x x .

A. Quartic B. Cubic C. Quadratic D. Quintic

23

Notes and/or Formula Real zeroes occur where the graph crosses the x-axis.

12. Identify the polynomial function, give the degree and the maximum number of

real zeros : 5 4 2( ) 6 7 4 2 13.5g x x x x x .

F. Quintic, 4, 5 G. Quartic, 4, 4 H. Quintic, 5, 5 J. Not a polynomial function

13. Determine whether the degree of the function below is odd or even. How many

real zeroes does the function have? A. Odd; 4 Zeroes B. Even; 3 Zeroes C. Even; 4 Zeroes D. Odd; 3 Zeroes

14. Which the following represents the graph of 3 2( ) 12 47 60f x x x x ?

F. G. H. J.

24


AII.9 The student will find the domain, range, zeros, and inverse of a function, the value of a function for a given element in its domain, and the composition of multiple functions. Functions will include those that have domains and ranges that are limited and/or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions, including exponential and logarithmic.


Domain: x-values Range: y-values

To be a function: No 2 x’s may be the same

OR Passes the vertical line test

To find the value of a function, substitute the

value into the function for every x

3. Find the range of the relation {( 4, 1),( 2,1),( 5,4)} .

A. { 5, 4, 2}

B. { 5, 1,1}

C. { 4, 2,4}

D. { 1,1,4}

4. Find the domain of the relation {(2,5),(3, 6),(0, 3)}

F. { 6,0,5}

G. {0,2,3}

H. { 3,2,3}

J. { 6, 3,5}

5. What is the range of the function 2( ) 3f x x if the domain is { 4, 1,5} ?

A. { 2,13,22}

B. {4,19,28}

C. { 28, 4,13}

D. { 28, 19, 4}

6. Which of the following is NOT a function?

F. {(5, 1),( 4,4),(2, 1)}

G. 24 3x y

H. 24 3y x

J. 5y

7. Which of the following is NOT a function?

A. {( 2, 5),(3,0),(4, 5)}

B. 1y

C. 25 3y x

D. 25 3x y

8. Find (2)f given 2( ) 3 2 19f x x x

F. 29 G. 35 H. 39 J. 16

25

Notes and/or Formula

Composite Function f(g(x)) Substitute the entire

expression for g(x) into all of the x’s in the expression

for f(x)

9. If 2( ) 2Q x x x , find ( 4)Q .

A. -14 B. 18 C. 10 D. -10

10. Find (3)f given 2( ) 4 3 13f x x x

F. 35 G. 22 H. 41 J. 27

11. Find ( ( ))g f x where ( ) 1f x x and 8

( )5

xg x

.

A. 6 13

5

x

B. 13

5

x

C. 2 9 8

5

x x

D. 9

5

x

12. If 2( ) 3 5f x x and ( ) 2 6g x x , find ( ( ))g f x .

F. 22(3 5) 6x

G. 3(2 6) 5x

H. 23 2 1x x

J. 23(2 6) 5x

13. Find ( ( ))g f x where ( ) 7f x x and 5

( )2

xg x

A. 19

2

x

B. 3 19

2

x

C. 2 12 35

2

x x

D. 12

2

x

14. Solve for x by factoring. 2 2 0x x F. 2, 1 G. -2, 1 H. -2, -1 J. -1, 2

26


Rational Zero Theorem The factors of the leading

coefficient will be “Q”. The factors of the constant

will be “P”. To find all the possible

rational roots for a polynomial,

“every P over every Q”

15. Find all the real zeros. 2 2 15 0x x A. 5, 3 B. -3, 5 C. -5, -3 D. -5, 3

16. Find all real zeros of the function. 4 3 28 8 336y x x x

F. 0, 6, 8 G. 0, 6 H. -7, 0, 6 J. None of these answers

17. Find all real zeros of the function. 4 3 23 12 15y x x x

A. -5, 0 B. -5, 0, 1 C. -5, -3, 0 D. None of these answers

18. List all the possible rational zeros of the polynomial

4 3 2( ) 55 6 2 3 14f x x x x x according to the rational zero theorem.

F. 1, 2, 7, 14

G. 2 7 14 1 2 7 14

, , , , , ,5 5 5 11 11 11 55

H.

1 2 7 14 11, 2, 7, 14, , , , , ,

5 5 5 5 11

2 7 14 1 2 7 14, , , , , ,

11 11 11 55 55 55 55

J. 2 7 2 7

1, 2, , , ,5 5 11 11

19. Given that one zero is 4, which of the following is NOT a zero of P(x).

3( ) 13 12P x x x

A. -1 B. 4 C. -3 D. -6

20. Given that one zero is 3 2i , which of the following is NOT a zero of P(x). 3 2( ) 7 19 13P x x x x

F. 3 2i

G. 3 2i H. -1 J. 1

27


To find an inverse 1. Switch x and y

2. Solve for y

21. Which of the following is a zero of the function 4 3 2( ) 2 7 8 43 30f x x x x x ?

A. 5

2

B. 5

2

C. 2

5

D. 2

5

22. Find the inverse given the

function 3( ) 5 9f x x .

F. 3

1

5 9x

G. 39

5

x

H. 3( 9)

125

x

J. 15 9x

23. Given 2( ) 4 1f x x , find 1( )f x .

A. 2

1

4 1x

B. 14 1x

C. 2( 1)

64

x

D. 1

4

x

28


AII.15 The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions.

Notes and/or Formulas 1. Identify the polynomial function 2 3( ) 5 6 5 9f x x x x

A. Cubic B. Quadratic C. Quartic D. Quintic

2. Identify the polynomial function, give the degree and the maximum number of

real zeros. 5 4 2( ) 4 7 5 12g x x x x

F. Not a polynomial function G. Quintic, 4, 5 H. Quartic, 4, 4 J. Quintic, 5, 5

3. Use synthetic division to perform the following 4 2(5 6 7) ( 1)x x x

A. 3 2 185 11

1x x

x

B. 3 2 185 5 11 11

1x x x

x

C. 3 2 185 5 11 11

1x x x

x

D. 3 2 65

1x x

x

29


AII.16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term and evaluating summation formulas.

Notation will include and na .


Term Formulas

A: 1 ( 1)na a n d

G: 1

1

n

na a r

Sum Formulas

A: 1( )

2

nn a aS

G: 1(1 )

1

na rS

r

IG: 1

1

aS

r

If you forget the formulas, just list series out and

count terms or add terms on calculator.

1. Insert two arithmetic means between -2 and 13. A. 1, 10 B. 3, 8 C. 2, 9 D. 4, 7

2. Evaluate: 25

1

(3 2)n

n

F. 984 G. 1000 H. 1025 J. 2050

3. Which is an arithmetic sequence?

A. 2, 5, 9, 14... B. 100, 50, 12.5, 1.6... C. 3, 10, 17, 24... D. -8, -4, -2, -1...

4. If 3(2)na n which of the following represents 3a

F. 12 G. 18 H. 24 J. 27

5. Which of the following represents 8

2

(3 2)n

n

A. 26 B. 69 C. 91 D. 92

6. Find the next term in the sequence 8, 5, 2, -1.

F. -2 G. -3 H. -4 J. -5

30


AII.20 The student will identify, create, and solve practical problems involving inverse variation and a combination of direct and inverse variation.


Inverse Variation

Function k

yx

k y x

Direct Variation

Function y kx

yk

x

1. The area (A) of a circle varies directly as the square of the radius (r). If k is the constant of proportionality, which is the formula for this relationship?

A. 2

kA

r

B. A kr

C. 2A kr

D. 2r kA

2. The frequency of a radio signal varies inversely as the wave length. A signal of frequency 1200 kilohertz (kHz), which might be the frequency of an AM radio station, has a wave length 250 m. What frequency has a signal wave length of 400m? F. 83 kHz G. 750 kHz H. 1350 kHz J. 1920 kHz

31

Topic: Analytical Geometry AII.10

AII.10 The student will investigate and describe through the useof graphs the relationships between the solution of an equation, zero of a function, x-intercept of a graph, and factors of a polynomial expression.


Zeros are where the graph crosses the x-axis.

Factored form (x – zero)

Ex: If zero at 2, then factor is (x – 2)

If zero at -5, then factor is (x + 5)

1. Which of the following functions has x-intercepts at 1 and -2?

A. 2 2y x x

B. 2 2y x x

C. 2 2 1y x x

D. 2 1y x

2. Find all real zeros. 20 3 10x x F. -2, -5 G. 2, 5 H. -2, 5 J. -5, 2

3. Use the graph to determine the roots of the equation.

A. 1 and 4 B. 1 and -4 C. -3 D. None of these answers

4. Find all the real zeros of the function. 4 3 27 56 84y x x x

F. 0, 2, 6 G. 0, 2 H. 0, 2, 7 J. None of these answers

5. Find all the real zeros of the function. 22 3 0x x

A. 3

, 12

B. 3

,12

C. 3, 2

D. 6, 4

32


6. Find all the real zeros of the function. 23 4 15 0x x F. 5, 3

G. 5

,33

H. 5

, 33

J. 10, 18

7. Which of the following could not be a factor of the function?

A. ( 5)x

B. ( 5)x

C. ( 3)x

D. ( 3)x

8. What are the factors of the given graph? F. (2 1)( 3)( 4)x x x

G. (2 1)( 4)( 3)x x x

H. (2 1)( 4)( 3)x x x

J. (2 1)( 4)( 3)x x x

9. What type of polynomial function is illustrated in the graph? A. Linear B. Quadratic C. Cubic D. Quartic

33

Topic: Systems of Equations & Inequalities AII.13

AII.13 The student will solve practical problems using systems of linear inequalities and linear programming, and describe the results both orally and in writing. A graphing calculator may be used to facilitate solutions.


y mx b

dotted line, shade above

y mx b

dotted line, shade below

y mx b

solid line, shade above

y mx b

solid line, shade below

1. Choose the system of linear inequalities shown by the graph.

A.

32

2

23

3

y x

y x

B.

32

2

23

3

y x

y x

C.

32

2

23

3

y x

y x

D.

32

2

23

3

y x

y x

2. Graph the system of inequalities: 6

0

y x

x y

F. G. H. J.

34


3. Graph the system of inequalities: 3

0

y x

x y

A. B. C. D.

4. Find the maximum and minimum values of the function subject to the given

constraints.

2

6 6 12

8 4 16

( , ) 3 7

x y

x y

y x

f x y x y

F. The maximum value of f is 76 at (9, 7). The minimum value of f is 9 at (3, 0). G. The maximum value of f is 56 at (7, 5). The minimum value of f is 3 at (1, 0). H. The maximum value of f is 72 at (10, 6). The minimum value of f is 0 at (0, 0). J. The maximum value of f is 66 at (8, 6). The minimum value of f is 6 at (2, 0).

5. Eleanor raises only free-range chickens and turkeys. She wants to raise no

more than 60 animals with no more than 20 turkeys. She spends $1 to raise a chicken and $4 to raise a turkey. She has at most $105 to spend on the animals. Find the maximum profit Eleanor can make if she makes a profit of $3 per chicken and $8 per turkey. How many chickens should she raise? A. 45 B. 25 C. 35 D. 15

35

Topic: Systems of Equations & Inequalities AII.14

AII.14 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically.

Notes and/or Formulas Number of solutions Number of points of intersection on graph

1. How many solutions are there for this system? A. 0 B. 1 C. 2 D. 3

2. Solve: 2 2 41

3 7

x y

y x

F. (-4, -4) G. (4, 5) H. (5, 8) J. (-4, -19)

3. Which is the solution to the system below?

A. {(0, 7),(0,7)}

B. C. {( 7,0),(7,0)}

D. {(0, 5),(0,5)}

4. Solve the system graphically:

2 2

2 2

49

164 81

x y

x y

F. {(0, 8),(0,8)}

G. {(0, 9),(0,9)}

H. J. {( 8,0),(8,0)}

36

Topic: Statistics AII.19

AII.19 The student will collect and analyze data to make prediction and solve practical problems. Graphing calculators will be used to investigate scatter plots and to determine the equation for the curve of best fit.

Notes and/or Formulas Negative slope & correlation Positive slope & correlation Line of best fit: Look for slope & y-int. Calculator: 1. Put data into lists Stat-edit-L1-L2 2. Stat-calc-4:Lin Reg-enter 3. Equation for line of best is y = ax + b and substitute the values given for a and b

1. Determine the correlation for the scatter plot.

A. Strong positive correlation. B. Strong negative correlation. C. No correlation D. Not enough information given.

2. Look at the given scatter plot. This data best fits what type of equation? F. Linear G. Exponential H. Logarithmic J. Quadratic

3. Which of the following equations represents the line of best fit for the following data?

X 23 26 26 33 34 44 44 45 64

Y 25 25 26 21 18 19 21 19 15

A. 4.8 32y x

B. 0.125 32.4y x

C. 0.324 32.4y x

D. 3.24 32y x

4. The table shows the number of students enrolled in the Honors Algebra-Trig

program at Menchville High School the first 5 years since its initiation. What is your prediction for the number of students in the eighth year? F. 120 G. 130 H. 140 J. 150

Year (X) Number of Students (Y)

1 55

2 71

3 84

4 97

5 108

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Topic: Expressions & Operations AII · Topic: Expressions & Operations AII.3 AII.3 The student will...

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