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MATHPOWERTM 12, WESTERN EDITION

Chapter 4 Trigonometric Functions4.4

4.4.1

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4.4.2

The principles of transformations of functions apply to

trigonometric functions and can be summarized as follows:

Vertical Stretchy =af(x) y =a sinx changes the amplitude

to |a |Horizontal Stretch

y =f(bx) y = sinbx changes the period

Vertical Translation

y =f(x) +k y = sinx +k shifts the curve vertically

k units upward whenk > 0

andk units downward

whenk < 0Horizontal Translation

y =f(x +h) y = sin (x +h) shifts the curve horizontally

h units to the left whenh > 0

andh units to the right

whenh < 0

Transformations of Functions

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Transforming a Trigonometric Function

Graphy = sinx + 2 andy = sinx - 3.

y = sinx + 2

y = sinx - 3

The range fory = sinx + 2 is 1 y 3 .

The range fory = sinx - 3 is -4 y -2 .

4.4.3

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4.4.4

Transforming a Trigonometric Function

A horizontal translation of a trigonometric function

is called a phase shift.

y = sinx

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Transforming a Trigonometric Function

Sketch the graph of

y = sinx

y = 3sin 2x

4.4.5

y = 3sin 2x

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4.4.6

Analyzing a Sine Function

2

Domain:

Range:

Amplitude:

Vertical Displacement:

Period:

Phase Shift: units to the left

2 units down

3-5 y 1

the set of all real numbers

y- intercept: x = 0

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4.4.7

Analyzing a Sine Function

In the equation ofy =asin[b(x +c)] +d:

a = 4,b = 3,d= -3, and

Compare the graph of this function to the graph

ofy = sinx with respect to the following:

a) domain and range b) amplitude

c) period d)x- andy-intercepts

e) phase shift f) vertical displacement

Domain:

Range: -7 y 1

Amplitude:

Period: x-intercepts: 0.02, 0.5, 2.12, 2.80

y-intercept:

right down

g) equation

4

3 units

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4.4.8

Determining an Equation From a Graph

A partial graph of a sine function is shown.

Determine the equation as a function of sine.

a = 2

d= 1

b = 2

Therefore, the equation is .

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Determining an Equation From a Graph

4.4.9

A partial graph of a cosine function is shown.

Determine the equation as a function of cosine.

a = 2

d= -1

b = 2

Therefore, the equation is .

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4.4.10

Determining an Equation From a Graph

Amplitude:

Vertical Displacement:

Period:

3

2

The equation as a

function of sine is

A partial graph of a sine function is shown.

Determine the equation as a function of sine.

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Pages 218 and 219

1-23 odd,

25-33, 34 (graphing calculator)

Suggested Questions:

4.4.12