1. 7.4: Periodic Graphs &
Phase Shifts
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•State period, amplitude,
vertical shift and phase
shift of sine or cosine.
•Graph using differences
from a parent function

2. Try This
Find the
amplitude
and period
for
( ) 3cos2f t t= −
Amplitude=3; Period = π

3. Important Idea
A common mistake…
•a is not amplitude;
is amplitude.
a
•a may be positive or
negative; amplitude is
always positive.

4. Definition
The standard forms for sine
and cosine functions are:
( ) sin( )f t a bt c d= + +
( ) cos( )g t a bt c d= + +
where a,b,c and d are
constants.

5. Important Idea
In the standard form:
( ) sin( )f t a bt c d= + +
( ) cos( )g t a bt c d= + +
•a controls amplitude
•b controls period
•c controls phase shift
•d controls vertical shift
Sketchpad

6. Try This
What is the value of a, b, c,
and d in the following trig
equation:
cos( )y a bt c d= + +
2cos(2 3) 6y t= − + +

7. Try This
What is the value of a, b, c,
and d in the following trig
equation:
sin( )y a bt c d= + +
1sin( 2 3) 6y t= − − +

8. Example
Without using a calculator,
describe and sketch the
graph of
( ) 3sin 4f t t= − −

9. Example
Without using a calculator,
describe and sketch the
graph of
( ) 2cos 4g t t= +

10. Try This
Without using a calculator,
describe and sketch the
graph of
( ) 2cos 3k t t= − −

11. Solution
The graph of is the same
as the graph of the parent
function, except:
( )k t
cost
( )k t• is reflected across the
horizontal axis
• It is vertically stretched 2
units
•It is shifted down 3 units

12. Solution
Parent: cost
( ) 2cos 3k t t= − −

13. Definition
The phase shift of a
trigonometric function
results in a horizontal shift
of the graph. It is controlled
by the constant c in the
standard form.

14. Example
Factor: 2 3t +
Re-write:
3 2
2
2
t +
g

15. Try This
Factor: 4
3
t
π
+
4
12
t
π 
+ ÷
 

16. Example
Find the phase shift of
( ) sin 2
2
g t t
π 
= + ÷
 
Re-write as:
( ) sin 2
4
g t t
π 
= + ÷
 

17. Example
Find the phase shift of
( )( ) 3sin 3 5f t t= +
Re-write as:

18. Try This
Find the phase shift of
( )( ) 2cos 2p t t π= − +
Re-write
as:
( ) 2cos2
2
p t t
π 
= − + ÷
 
Phase
shift: 2
π
to left

19. Try This
2 2cos2
2
y x
π 
= − + ÷
 
Using your calculator, graph:
1 2cos2y x= −
Be sure you are in radian
mode.

20. Solution
y1
y2
2
π
to left

21. Try This
State the phase shift of:
( ) sin( 2)f t t= −
then use a graphing
calculator to graph the
function and its parent on the
same set of axes.

22. Solution
The phase shift of:
sin( 2)y t= −
is 2 units
to right
sin( 2)y t= −
siny t=
2

23. Important Idea
Changes in phase shift
move the graph left and
right. Phase shift is a
horizontal translation.

24. Definition
The vertical shift of
sin( )y a bt c d= + +
is d. If d >0, the graph is
translated up. If d <0, the
graph is translated down.
This definition applies to all
the trig functions.

25. Try This
Graph ( ) sin 2
6
f t t
π 
= + + ÷
 
and ( ) sin
6
g t t
π 
= + ÷
 
sin( 6) 2y x π= + +
sin( 6)y x π= +
on the same axes.

26. Example
Identify the amplitude,
period, phase shift and
vertical shift of:
( ) 3cos(2 1) 4f t t= − − +

27. Try This
Identify the amplitude,
period, phase shift and
vertical shift of:
( ) 3sin(3 1) 1g t t= + −
Amplitude=3, Period= 2 3π
Phase shift=1/3 unit to left
Vertical shift=-1

28. Example
As you ride a ferris wheel,
the height you are above the
ground varies periodically.
Consider the height of the
center of the wheel to be an
equilibrium point. A
particular wheel has a
diameter of 38 ft. and travels
at 4 revolutions per minute.

29. 1. Write an
equation
describing the
change in
2. Find the height of the seat
after 22 seconds, after 60
seconds and after 90
seconds
height of the last seat filled.
Example

30. Lesson Close
Because of the repeating or
periodic nature of
trigonometric graphs, they
are used to model a variety
of phenomena that involve
cyclic behavior.