1.1 Lines Increments If a particle moves from the point (x 1,y 1 ) to the point (x 2,y 2 ), the...

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1.1 Lines

Increments

If a particle moves from the point (x1,y1) to the point (x2,y2), the increments in its coordinates are

1212 ΔandΔ yyyxxx

1.1 Lines

Let P1= (x1,y1) and P2= (x2,y2) be points on a nonverticalline L. The slope of L is

P1(x1,y1)

P2(x2,y2)

Q(x2,y1)

1.1 Lines

Theorem: If two lines are parallel, then they have the same slope and if they have the same slope, then they are parallel.

slope m1 slope m2

θ1 θ2

Proof: If L1 || L2, then θ1= θ2

and m1= m2. Conversely, ifm1 = m2, then θ1= θ2 and L1 || L2.

1.1 Lines

Theorem: If two non vertical lines L1and L2 are perpendicular, then their slopes satisfy m1m2 = -1 and conversely.

Slope m2Slope m1

Proof: Δ ADC ~ ΔCDB

m1 = tan θ1 = a/hm2 = tan θ2 = -h/a

so m1m2 =(a/h)(-h/a) = -1

1.1 Lines

Equations of lines• Point-Slope Formula y = m(x – x1) + y1

• Slope-Intercept form y = mx + b• Standard form Ax + By = C• y = a Horizontal line slope of zero• x =a Vertical line no slope

1.1 Lines

Regression Analysis1. Plot the data2. Find the regression equation y = mx + b3. Superimpose the graph on the data points.4. Use the regression equation to predict y-values.

1.1 Lines

Coordinate Proofs1. State given and prove.2. Draw a picture.3. Label coordinates, use (0,0) if possible.4. Fill in missing coordinates.5. Use algebra to prove

• parallel/perpendicular-slope• equidistant-distance formula• bisect-midpoint

1.1 Lines

Prove the midpoint of the hypotenuseof a right triangle is equidistantfrom the three vertices.

A(0,0) C(b,0)

B(0,a)

Given: ΔBAC is a right triangleProve: AM = BM = CM

M(b/2,a/2)

Since AM = BM = CM, themidpoint of the hypotenuseof a right triangle isequidistant from the three vertices

1.2 Functions and Graphs

Function

A function from a set D to a set R is a rule that assigns a unique element R to each element D.

y = f(x) y is a function of x

Domain All possible x values

Range All possible y values

x ),( 0

ax ),( aa

a bbxa ),( ba

a bbxa ],[ ba

closed

a b bxa

],( ba

),[ ba

half opened

•y = mx

•Domain (-∞ , ∞)•Range (-∞ , ∞)

•y = x2

•Domain (-∞ , ∞)•Range [0, ∞)

•y = x3

•Domain (-∞ , ∞)•Range (-∞ , ∞)

•y = 1/x

•Domain x ≠ 0•Range y ≠ 0

•Domain [0, ∞)•Range [0, ∞)

Function Domain Range

y = x ),( ),(

y = x2 ),( )0,[

y = |x| ),( )0,[

29 xy [-3,3] [0,3]

2 xy ),- 2[ )0,[

Definitions Even Function, Odd Function

A function y = f(x) is aneven function of x if f(-x) = f(x) odd function of x if f(-x) = -f(x)

for every x in the function’s domain.

Even Function – symmetrical about the y-axis.Odd Function - symmetrical about the origin.

Odd Function symmetrical about the origin.

Even Function symmetrical about the y-axis.

(-x,-y)

(-x,y) (x,y)

Transformations

h(x) = af(x) vertical stretch or shrink

h(x) = f(ax) horizontal stretch or shrink

h(x) = f(x) + k vertical shift

h(x) = f(x + h) horizontal shift

h(x) = -f(x) reflection in the x-axis

h(x) = f(-x) reflection in the y-axis

Piece Functions

1.2 Functions and GraphsPiece Functions

Composite Functions f(g(x))

f(x) = x2, g(x) = 3x - 1

Find:1. f(g(2))2. g(f(-1))3. g(f(x))4. f(g(x))

•25•2•3x2 – 1•(3x – 1)2 = 9x2 – 6x + 1

1.3 Exponential Functions

Definition Exponential Function

Let a be a positive real number other than 1,the function f(x) = ax is the exponential function with base a.

Rules For ExponentsIf a > 0 and b > 0, the following hold true for all real numbers x and y.

yxyx aa. a 1

xyyx aa. 3

xxx (ab)b. a 4

16 0 . a

a. a 8

Use the rules for exponents tosolve for x.

•4x = 128•(2)2x = 27

•2x = 7•x = 7/2

•2x = 1/32•2x = 2-5

•x = -5

•(x3y2/3)1/2

•x3/2y1/3

•27x = 9-x+1

•(33)x = (32)-x+1

•33x = 3-2x+2

•3x = -2x+ 2•5x = 2•x = 2/5

Domain: Range:Increasing for:Decreasing for:Point Shared On All Graphs:Asymptote:

(-∞, ∞)(0, ∞)

a > 10 < a < 1

(0, 1)y = 0

Properties of f (x) = ax

xexf )(

Natural Exponential Function where e is the natural base and e 2.718…

Function f(x) = 2x h(x) = (0.5)x g(x) = ex

Domain

Increasing or Decreasing

Point Shared On All Graphs

(-∞, ∞) (-∞, ∞) (-∞, ∞)

(0, ∞) (0, ∞) (0, ∞)

Inc. Dec. Inc.

(0, 1)

Use translation of functions to graph the following. Determine the domain and range of each.

1. f(x) = -5(x + 2) – 3

2. g(x) = (1/3)(x – 1) + 2

Definitions Exponential Growth, Exponential Decay

The function y = k ax, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1.

Obyy y new amountyo original amountb baset timeh half life

An isotope of sodium, 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.

(a) Find the amount remaining after t hours.

(b) Find the amount remaining after 60 hours.

(c) Estimate the amount remaining after 4 days.

(d) Use a graph to estimate the time required for the mass to be reduced to 0.1 g.

An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.

(a) Find the amount remaining after t hours.

(b) Find the amount remaining after 60 hours.

• a. y = yobt/h

• y = 2 (1/2)(t/15)

• b. y = yobt/h

• y = 2 (1/2)(60/15)

• y = 2(1/2)4

• y = .125 g

1.3 Exponential FunctionsAn isotope of sodium, 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.

(c.) Estimate the amount remaining after 4 days.

(d.) Use a graph to estimate the time required for the mass to be reduced to 0.01 g.

• c. y = yobt/h

• y = 2 (1/2)(96/15)

• y = 2(1/2)6.4

• y = .023 g

A bacteria double every three days. There are 50 bacteria initially present

(a) Find the amount after 2 weeks.

(b) When will there be 3000 bacteria?

• a. y = yobt/h

• y = 50 (2)(14/3)

• y = 1269 bacteria

A bacteria double every three days. There are 50 bacteria initially present

When will there be 3000 bacteria?

• b. y = yobt/h

• 3000 = 50 (2)(t/3)

• 60 = 2t/3

Equations where x and y are functions of a third variable, such as t. That is,

x = f(t) and y = g(t).

The graph of parametric equations are called parametric curves and are defined by (x, y) = (f(t), g(t)).

1.4 Parametric Equations

Equations defined in terms of x and y. These may or may not be functions. Some examples include:

x2 + y2 = 4y = x2 + 3x + 2

Sketch the graph of the parametric equation for tin the interval [0,3]

1 -1 3

2 -3 6

3 -5 9

Eliminate the parameter t from the curve

Circle:If we let t = the angle, then:

cos sin 0 2x t y t t

Since: 2 2sin cos 1t t

2 2 1y x

2 2 1x y We could identify the parametric equations as a circle.

Ellipse: 3cos 4sinx t y t

cos sin3 4

x yt t

2 22 2cos sin

x yt t

This is the equation of an ellipse.

The path of a particle in two-dimensional space can be modeled by the parametric equations x = 2 + cos t and y = 3 + sin t. Sketch a graph of the path of the particle for 0 t 2.

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

How is t represented

on this graph?

Graphing calculators and other mathematical software can plot parametric equations much more efficiently then we can. Put your graphing calculator and plot the following equations. In what direction is t increasing?

(a) x = t2, y = t3 (b)

(c) x = sec θ, y = tan θ; -/2 < θ < /2

1;,ln ttytx

Parametric equations can easily be converted to Cartesian equations by solving one of the equations for t and substituting the result into the other equation.

(a) x = t2, y = t3

xxy,xt

2ytty 2lnln ytx

0;ln 22 xeyeyeyyx xxx

1,ln tfortytx(b)

(c) x = sec t, y = tan twhere -/2 < t < /2

Hint: sec2 θ – tan2 θ = 1

2222 tansec tyt,x

1tansec 2222 tt-yx

122 yx

Find a parametrization for the line segment with endpoints(2,1) and (-4,5).

x = 2 + at y = 1 + bt

when t = 1, a = -6when t = 1, b = 4

x = 2 – 6t and y = 1 + 4t

Cartesian Equationm = (5 – 1)/(-4 – 2) = -2/3

y = mx + b1 = (-2/3)(2) + bb = 7/3

y = (-2/3)x + 7/3

1.5 Functions and Logarithms

A function is one-to-one if two domain values do not have the same range value.

Algebraically, a function is one-to-one if f (x1) ≠ f (x2) for all x1 ≠ x2.

Graphically, a function is one-to-one if its graph passes the horizontal line test. That is, if any horizontal line drawn through the graph of a function crosses more than once, it is not one-to-one.

To be one-to-one, a function must pass the horizontal line test as well as the vertical line test.

2y x 21

2y x 2x y

one-to-one not one-to-one not a function

(also not one-to-one)

Determine if the following functions are one-to-one.

(a) f (x) = 1 + 3x – 2x 4

(b) g(x) = cos x + 3x 2

xx eexh

xxf 5)(

The inverse of a one-to-one function is obtained by exchanging the domain and range of the function. The inverse of a one-to-one function f is denoted with f -1.

Domain of f = Range of f -1

Range of f = Domain of f -1

f −1(x) = y <=> f (y) = x

To prove functions areinverses show that

f(f-1(x)) = f-1(f(x)) = x

To obtain the formula for the inverse of a function, do the following:

1. Let f (x) = y.2. Exchange y and x.3. Solve for y.4. Let y = f −1(x).

Inverse functions:

2f x x

Given an x value, we can find a y value.

Switch x and y:

1 2 2f x x

Inverse functions are reflections about y = x.

Solve for y:

11 22 xy

2f x x 1 2 2f x x

Prove f(x) and f-1(x) are inverses.

xxxxfxff 111)22(2

1)22())(( 1

xxxxfxff

1))(( 11

You can obtain the graph of the inverse of a one-to-one function by reflecting the graph of the original function through the line y = x.

Sketch a graph of f (x) = 2x and sketch a graph of its inverse. What is the domain and range of the inverse of f.

Domain: (0, ∞)Range: (-∞, ∞)

Determine the formula for the inverse of the following one-to-one functions.

32)( 3 xxf2

xxg 3)(

The inverse of an exponential function is called a logarithmic function.

Definition: x = a y if and only if y = log a x

The function f (x) = log a x is called a logarithmic function.

Domain: (0, ∞)Range: (-∞, ∞)

Asymptote: x = 0 Increasing for a > 1

Decreasing for 0 < a < 1 Common Point: (1, 0)

Find the inverse of g(x) = 3x.

Definition: x = a y if and only if y = log a x

xxg 31 log)(

1. log a (ax) = x for all x 2. alog ax = x for all x > 03. log a (xy) = log a x + log a y4. log a (x/y) = log a x – log a y5. log a xn = n log a x

Common Logarithm: log 10 x = log xNatural Logarithm: log e x = ln x

All the above properties hold.

The natural and common logarithms can be found on your calculator. Logarithms of other bases are not. You need the change of base formula.

ba log

loglog

where b is any other appropriate base.

$1000 is invested at 5.25 % interest compounded annually.How long will it take to reach $2500?

1000 1.0525 2500t

1.0525 2.5t We use logs when we have an

unknown exponent.

ln 1.0525 ln 2.5t

ln 2.5

ln 1.0525t 17.9

17.9 years

In real life you would have to wait 18 years.

Example 7: Indonesian Oil Production (million barrels per year):

1960 20.56

1970 42.10

1990 70.10

Use the natural logarithm regression equation to estimate oil production in 1982 and 2000.

How do we know that a logarithmic equation is appropriate?

In real life, we would need more points or past experience.

1. Determine the exact value of log 8 2.2. Determine the exact value of ln e 2.3.3. Evaluate log 7.3 5 to four decimal places.4. Write as a single logarithm: ln x + 2ln y – 3ln z.5. Solve 2x + 5 = 3 for x.

1.6 Trigonometric Functions

The Radian measure of angle ACBat the center of the unit circle equalsthe length of the arc that ACB cutsfrom the unit circle.

1 circle,unit for thebut

terminal ray

initial ray

rP(x,y)

secsecantcsccosecant

cotcotangenttantangent

coscosinesinsine

607590105

240255 270 285

(2,/4)

(5,5 /6)

(4, 11/6)

(-4, /2)

Let a point P have rectangular coordinates (x,y)and polar coordinates (r,). Then

(-1,0)

(0,-1)

60° 1

2 3 /3

Even and Odd Trig Functions:

“Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change.

Cosine is an even function because: cos cos

Secant is also an even function, because it is the reciprocal of cosine.

Even functions are symmetric about the y - axis.

Even and Odd Trig Functions:

“Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the sign of the y value also changes.

Sine is an odd function because: sin sin

Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function.

Odd functions have origin symmetry.

Definition Periodic Function, Period

A function f(x) is periodic if there is a positive number p such that f(x + p) = f(x) for every value of x. The smallestsuch value of p is the period of p.

y a f b x c d

Vertical stretch or shrink;reflection about x-axis

Horizontal stretch or shrink;reflection about y-axis

Horizontal shift

Vertical shift

Positive c moves left.

Positive d moves up.is a stretch.1a

is a shrink.1b

2sinf x A x C D

Horizontal shift

Vertical shift

is the amplitude.

is the period.B

D 21.5sin 1 2

1.1 Lines Increments If a particle moves from the point (x 1,y 1 ) to the point (x 2,y 2 ), the...

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