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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