The document discusses transformations of trigonometric functions including sine and cosine functions. It provides examples of how to graph functions involving stretches, compressions, phase shifts, and vertical and horizontal translations. Key aspects covered include identifying the amplitude, period, phase shift, and vertical shift based on the coefficients of the trigonometric function. Examples demonstrate applying these transformations to model real-world phenomena like sound waves and seasonal employment levels.

1. UNIT 9.7 TRANSLATIONSUNIT 9.7 TRANSLATIONS

4. You can use the parent functions to graph
transformations y = a sin bx and y = a cos bx.
Recall that a indicates a vertical stretch (|a|>1) or
compression (0 < |a| < 1), which changes the
amplitude. If a is less than 0, the graph is
reflected across the x-axis. The value of b
indicates a horizontal stretch or compression,
which changes the period.

6. Example 2: Stretching or Compressing Functions
Sine and Cosine Functions
Using f(x) = sin x as a guide, graph the
function g(x) = Identify the
amplitude and period.
Step 1 Identify the amplitude and period.
Because the amplitude is
Because the period is

7. Example 2 Continued
Step 2 Graph.
The curve is vertically
compressed by a factor of
horizontally stretched by a
factor of 2.
The parent function f has x-intercepts at
multiplies of π and g has x-intercepts at
multiplies of 4 π.
The maximum value of g is , and the minimum
value is .

8. Check It Out! Example 2
Using f(x) = cos x as a guide, graph the
function h(x) = Identify the amplitude
and period.
Step 1 Identify the amplitude and period.
Because the amplitude is
Because b = 2, the period is

9. Check It Out! Example 2 Continued
Step 2 Graph.
The curve is vertically
compressed by a factor of
and horizontally compressed
by a factor of 2.
The parent function f has x-intercepts at
multiplies of π and h has x-intercepts at
multiplies of π.
The maximum value of h is , and the minimum
value is .

10. Sine and cosine functions can be used to model
real-world phenomena, such as sound waves.
Different sounds create different waves. One way
to distinguish sounds is to measure frequency.
Frequency is the number of cycles in a given unit
of time, so it is the reciprocal of the period of a
function.
Hertz (Hz) is the standard measure of frequency
and represents one cycle per second. For example,
the sound wave made by a tuning fork for middle A
has a frequency of 440 Hz. This means that the
wave repeats 440 times in 1 second.

11. Example 3: Sound Application
Use a sine function to graph a sound wave
with a period of 0.002 s and an amplitude of
3 cm. Find the frequency in hertz for this
sound wave.
Use a horizontal scale
where one unit represents
0.002 s to complete one
full cycle. The maximum
and minimum values are
given by the amplitude.
The frequency of the sound wave is 500 Hz.
period
amplitude

12. Check It Out! Example 3
Use a sine function to graph a sound wave
with a period of 0.004 s and an amplitude of
3 cm. Find the frequency in hertz for this
sound wave.
Use a horizontal scale
where one unit represents
0.004 s to complete one
full cycle. The maximum
and minimum values are
given by the amplitude.
The frequency of the sound wave is 250 Hz.
period
amplitude

13. Sine and cosine can also be translated as y
= sin(x – h) + k and y = cos(x – h) + k.
Recall that a vertical translation by k units
moves the graph up (k > 0) or down
(k < 0).
A phase shift is a horizontal translation of a
periodic function. A phase shift of h units moves the
graph left (h < 0) or right (h > 0).

14. Example 4: Identifying Phase Shifts for Sine and
Cosine Functions
Using f(x) = sin x as a guide, graph g(x) =
Identify the x-intercepts and phase shift.
g(x) = sin
Step 1 Identify the amplitude and period.
Amplitude is |a| = |1| = 1.
The period is

15. Example 4 Continued
Step 2 Identify the phase shift.
Because h = the phase shift is
radians to the right.
Identify h.
All x-intercepts, maxima, and minima
of f(x) are shifted units to the right.

16. Step 3 Identify the x-intercepts.
The first x-intercept occurs at .
Because cos x has two x-intercepts in
each period of 2π, the x-intercepts
occur at + nπ, where n is an integer.
Example 4 Continued

17. Step 4 Identify the maximum and minimum values.
The maximum and minimum values occur
between the x-intercepts. The maxima occur
at + 2πn and have a value of 1. The
minima occur at + 2πn and have a value
of –1.
Example 4 Continued

18. sin x
sin
Step 5 Graph using all the information about the
function.
Example 4 Continued

19. Check It Out! Example 4
Using f(x) = cos x as a guide, graph
g(x) = cos(x – π). Identify the x-intercepts
and phase shift.
Step 1 Identify the amplitude and period.
Amplitude is |a| = |1| = 1.
The period is

20. Check It Out! Example 4 Continued
Step 2 Identify the phase shift.
Identify h.x – π = x – (–π)
Because h = –π, the phase shift is π
radians to the right.
All x-intercepts, maxima, and minima
of f(x) are shifted π units to the right.

21. Step 3 Identify the x-intercepts.
Check It Out! Example 4 Continued
The first x-intercept occurs at .
Because sin x has two x-intercepts in
each period of 2π, the x-intercepts
occur at + nπ, where n is an integer.

22. Check It Out! Example 4 Continued
Step 4 Identify the maximum and minimum values.
The maximum and minimum values occur
between the x-intercepts. The maxima occur
at π + 2πn and have a value of 1. The
minima occur at 2πn and have a value of –1.

23. cos x
cos (x–π)
π–π
y
x
Check It Out! Example 4 Continued
Step 5 Graph using all the information about the
function.

24. You can combine the transformations of
trigonometric functions. Use the values of a, b,
h, and k to identify the important features of a
sine or cosine function.
y = asinb(x – h) + k
Amplitude Phase shift
Period
Vertical shift

25. Example 5: Employment Application
A. Graph the number of people employed in the
town for one complete period.
Step 1 Identify the important features of the graph.
month of the year.
The number of people, in thousands, employed
in a resort town can be modeled by,
where x is the
a = 1.5, b = , h = –2,
k = 5.2

26. Example 4 Continued
Amplitude: 1.5
Period:
The period is equal to 12 months or 1 full year.
Phase shift: 2 months left
Vertical shift: 5.2
Maxima: 5.2 + 1.5 = 6.7 at 1
Minima: 5.2 – 1.5 = 3.7 at 7

27. Example 4 Continued
Step 2 Graph using all the information about
the function.
B. What is the maximum number of people
employed?
The maximum number of people employed is
1000(5.2 + 1.5) = 6700.

28. Check It Out! Example 5
a. Graph the height of a cabin for two complete
periods.
H(t) = –16cos + 24 a = –16, b = , k = 24
Step 1 Identify the important features of the
graph.
What if…? Suppose that the height H of a
Ferris wheel can be modeled by,
, where t is the
time in seconds.

29. Check It Out! Example 5 Continued
Amplitude: –16
The period is equal to the time required for one
full rotation.
Vertical shift: 24
Maxima: 24 + 16 = 40
Minima: 24 – 16 = 8
Period:

30. Check It Out! Example 5 Continued
Time (min)
Height(ft)

31. Lesson Quiz: Part I
1. Using f(x) = cos x as a guide, graph
g(x) = 1.5 cos 2x.

32. Lesson Quiz: Part II
Suppose that the height, in feet, above ground
of one of the cabins of a Ferris wheel at t
minutes is modeled by
2. Graph the height of the cabin for two complete
revolutions.
3. What is the radius of this
Ferris wheel? 30 ft