1. Further Transformations
Chapter 4 Trigonometric Functions
4.4
Further Transformations
of Sine and Cosine
Functions
MATHPOWERTM 12, WESTERN EDITION
4.4.1

2. Transformations of Functions
The principles of transformations of functions apply to
trigonometric 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 < 0
Horizontal 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
4.4.2

3. 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. Transforming a Trigonometric Function
y = sin x
A horizontal translation of a trigonometric function
is called a phase shift.
4.4.4

5. Transforming a Trigonometric Function
Sketch the graph of
y = sin x
y = 3sin 2x
y = 3sin 2x
4.4.5

6. Analyzing a Sine Functionp 2p
y- intercept:
x = 0
Domain:
Range:
Amplitude:
Vertical Displacement:
Period:
Phase Shift:
the set of all real numbers
-5 ≤ y ≤ 1 3
2 units down p
units to the left
4.4.6

7. Analyzing a Sine Function In the equation of y = asin[b(x + c)] + d:
a = 4, b = 3, d = -3, and
Compare the graph of this function to the graph
of y = sin x with respect to the following:
a) domain and range
b) amplitude
Domain:
Amplitude: 4
Range:
-7 ≤ y ≤ 1
c) period
d) x- and y-intercepts
x-intercepts:
Period:
0.02, 0.5, 2.12, 2.80
y-intercept:
e) phase shift
f) vertical displacement
right
down
3 units
g) equation
4.4.7

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 .
4.4.8

9. Determining an Equation From a Graph
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 .
4.4.9

10. Determining an Equation From a Graph
A partial graph of a sine function is shown.
Determine the equation as a function of sine.
Amplitude:
Vertical Displacement:
Period:
The equation as a
function of sine is 3 2 p
4.4.10

11. Graphing Sine as a Function of Time
The motion of a weight on a spring can be described by
the equation
Sketch this function.
y = sin t
The period is 2.
The amplitude is 2.
The phase shift is
indicating a shift to the right.
4.4.11

12. Assignment Suggested Questions: Pages 218 and 219 1-23 odd,
25-33, 34 (graphing calculator)
4.4.12