1. 7.3: Periodic Graphs and
Amplitude
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Understand period and
amplitude of a Trigonometric
function.
•Introduce real world
applications of periodic
graphs.

2. Important Idea
Period and amplitude are
important in describing the
nature of many real world
phenomenon such as light
waves, sound waves and
radio waves.

3. Larger Period,
Smaller Frequency
Important Idea
Radio Visible X-Ray

4. Smaller Period
Greater Frequency
Important Idea
Radio Visible X-Ray

5. Important Idea
Radio
Waves

6. Important Idea
Medicine

7. Important Idea
Ocean Tides

8. Definition
A complete
waveform
of a
periodic
function
with no
repeats is a
cycle.

9. Important Idea
A cycle may
be measured
in different
ways but for
a given
function, the
period will be
the same .
Period

10. Definition
The
Period is
the time
required
for the
function
to
repeat.
t
( )f t
Go to Sketchpad

11. Definition
Frequency (in cycles per
second) is the reciprocal
of the period.

12. Start Movie

13. Try This
•What are your observations
about the movement of the
bridge?
•What do you think is the
connection between this
bridge and today’s lesson?

14. Important Idea
The x-axis
is now the
t-axis
(time).
The y-axis
is now the
f (t) axis.
t
( )f t

15. Important Idea
For the trigonometric
functions:
( ) sinf t bt=
( ) cosg t bt=
or
The constant b changes
the period.

16. Definition
The trigonometric functions:
( ) sinf t bt=
( ) cosg t bt=
have periods (repeat) every
2
b
π
radians.

17. Example
What is the period of:
( ) sinf t t=
0
2π
Period=
2
2
1
π
π=

18. Example
What is the period of:
( ) sin(2 )g t t=
0 π 2π
2
2
π
π=Period=

19. Example
What is the period of:
( ) sin(.5 )h t t=
Period=
2
4
.5
π
π=
0 4π 8π

20. Important Idea
( ) sin( )f t x=
( ) sin(2 )g t t=
( ) sin(.5 )h t t=
0 2π 4π
As b gets small, period gets
large.

21. Try This
What is the period (how
often does the function
repeat)? : ( ) cosp t t=
2
2
1
π
π=
0 2π 4π

22. Try This
What is the period (how
often does the function
repeat)? : ( ) cos1.2p t t=
2
1.67
1.2
π
π=
0 1.67π

23. Try This
Which is the
graph of
( ) sin(4 )f t t= 0
red
2
π π

24. Try This
What is the
equation of
the function
graphed in
blue?
( ) sin(2 )f t t=
0
2
π π

25. Important Idea
The
trigonometric
function:
( ) tanf t bt=
has a period
(repeats)
every
b
π
radians.
t
( )f t
b
π
b
π

26. Example
Determine how often the
function repeats (period):
( ) tan(2 )k t t=
( ) tan
3
t
j t
 
=  ÷
 

27. Try This
Determine how often the
function repeats (period):
( ) tan
6
t
p t
 
=  ÷
 
1
6
b = 6π; period=

28. Try This
Using your
calculator, graph:
1 tany t=
2 tan
6
t
y
 
=  ÷
 
Compare and
contrast.

29. Definition
Amplitude is
where is
the distance
from the
middle to the
top and
a
a+
a−
is the distance from the
middle to the bottom.
a
a−
Geom Sketchpad

30. Try This
Compar
e and
Contrast
( ) 5sinf t t=
( ) 2sing t t=

31. Try This
What is the
amplitude
of
( ) 6tanp t t=
Hint: this is a
trick question.
?

32. Important Idea
( ) sinf t a bt= ( ) cosg t a bt=
The constant a controls the
amplitude (vertical stretch).
The constant b controls the
horizontal stretch. Amplitude
is .
Period is .
a
2
b
π

33. Try This
Find the
amplitude
and period
for
( ) sinf t t=
Amplitude=1; Period = 2π

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

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

36. Try This
Find the
amplitude
and period
for
( ) 3tan(.5 )f t t= −
Amplitude: None;
Period=2π

37. Example
For what value(s) of t
between is
sin 2 1tπ =
0 1t≤ ≤

38. Example
For what value(s) of t
between [0, ] is
cos2 0t =
2π

39. Example
For what value(s) of t
between [0, ] is
cos3 0t =
2π

40. Try This
For what value(s) of t
between [0, ] is
sin3 0t =
π
0,
3
t t
π
= =

41. Try This
For what value(s) of t
between [0,1] is
sin3 0tπ =
1 2
0, , ,1
3 3
t =

42. Lesson Close
Period and amplitude will
be important when we
study modeling and
simple harmonic motion
in section 8.4.