mathpower tm 12, western edition chapter 4 trigonometric functions 4.4 4.4.1
TRANSCRIPT
MATHPOWERTM 12, WESTERN EDITION
Chapter 4 Trigonometric Functions4.4
4.4.1
4.4.2
The principles of transformations of functions apply totrigonometric functions and can be summarized as follows:
Vertical Stretch y = af(x) y = a sin x changes the amplitude to | a |Horizontal Stretch y = f(bx) y = sin bx changes the period
Vertical Translation y = f(x) + k y = sin x + k shifts the curve vertically k units upward when k > 0
and k units downward when k < 0Horizontal Translation y = f(x + h) y = sin (x + h) shifts the curve horizontally h units to the left when h > 0
and h units to the right when h < 0
Transformations of Functions
2b
Transforming a Trigonometric Function
Graph y = sin x + 2 and y = sin x - 3.
y = sin x + 2
y = sin x - 3
The range for y = sin x + 2 is 1 ≤ y ≤ 3.
The range for y = sin x - 3 is -4 ≤ y ≤ -2.
4.4.3
4.4.4
Transforming a Trigonometric Function
Graph y sin(x
4
) and y sin(x 4
).
A horizontal translation of a trigonometric functionis called a phase shift.
y sin(x
4
)
y sin(x
4
)y = sin x
Transforming a Trigonometric Function
Sketch the graph of y 3sin2(x
4
) 2 .
y = sin x
y = 3sin 2x
4.4.5 y 3sin2(x
4
) 2
y = 3sin 2x
4.4.6
Analyzing a Sine Function
2
y 3sin2(x
4
) 2
Domain:Range:Amplitude:Vertical Displacement:Period:Phase Shift:
4
units to the left
2 units down3
-5 ≤ y ≤ 1the set of all real numbers
y- intercept: x = 0
y 3sin2(x
4
) 2
y 3sin2(0
4
) 2
y 3sin(
2
) 2
y 1 y 3(1) 2
4.4.7
Analyzing a Sine Function
In the equation of y = asin[b(x + c)] + d:
a = 4, b = 3, d = -3, and c 34
.
Compare the graph of this function to the graphof y = sin x with respect to the following:
a) domain and range b) amplitude
c) period d) x- and y-intercepts
e) phase shift f) vertical displacement
Domain:
xR
Range: -7 ≤ y ≤ 1Amplitude:
Period:23
x-intercepts: 0.02, 0.5, 2.12, 2.80
y-intercept: 2 3 3
right
34
units down
y 4 sin3(x 34
) 3g) equation
4
3 units
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 = 2d = 1
c 6
period =2b
=
2b
b = 2
Therefore, the equation is y 2 sin2(x 6
) 1 .
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 = 2d = -1
c 4
period =2b
=
2b
b = 2
Therefore, the equation is y 2 cos2(x 4
) 1.
4.4.10
Determining an Equation From a Graph
y 3 sin2x 2.
Amplitude:
Vertical Displacement:
Period:
3
2
The equation as afunction of sine is
A partial graph of a sine function is shown. Determine the equation as a function of sine.
Graphing Sine as a Function of Time
The motion of a weight on a spring can be described by
the equation
y = sin t
Sketch this function.
y 2 sin(t 1
4)
y 2 sin( t
4
).
The period is 2.The amplitude is 2.
1
4The phase shift is
4.4.11indicating a shift to the right.
Pages 218 and 2191-23 odd,25-33, 34 (graphing calculator)
Suggested Questions:
4.4.12