7.3: Periodic Graphs and
Amplitude
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Understand period and
amplitude ...
Important Idea
Period and amplitude are
important in describing the
nature of many real world
phenomenon such as light
wav...
Larger Period,
Smaller Frequency
Important Idea
Radio Visible X-Ray
Smaller Period
Greater Frequency
Important Idea
Radio Visible X-Ray
Important Idea
Radio
Waves
Important Idea
Medicine
Important Idea
Ocean Tides
Definition
A complete
waveform
of a
periodic
function
with no
repeats is a
cycle.
Important Idea
A cycle may
be measured
in different
ways but for
a given
function, the
period will be
the same .
Period
Definition
The
Period is
the time
required
for the
function
to
repeat.
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per
second) is the reciprocal
of the period.
Start Movie
Try This
•What are your observations
about the movement of the
bridge?
•What do you think is the
connection between this
b...
Important Idea
The x-axis
is now the
t-axis
(time).
The y-axis
is now the
f (t) axis.
t
( )f t
Important Idea
For the trigonometric
functions:
( ) sinf t bt=
( ) cosg t bt=
or
The constant b changes
the period.
Definition
The trigonometric functions:
( ) sinf t bt=
( ) cosg t bt=
have periods (repeat) every
2
b
π
radians.
Example
What is the period of:
( ) sinf t t=
0
2π
Period=
2
2
1
π
π=
Example
What is the period of:
( ) sin(2 )g t t=
0 π 2π
2
2
π
π=Period=
Example
What is the period of:
( ) sin(.5 )h t t=
Period=
2
4
.5
π
π=
0 4π 8π
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.
Try This
What is the period (how
often does the function
repeat)? : ( ) cosp t t=
2
2
1
π
π=
0 2π 4π
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
Try This
Which is the
graph of
( ) sin(4 )f t t= 0
red
2
π π
Try This
What is the
equation of
the function
graphed in
blue?
( ) sin(2 )f t t=
0
2
π π
Important Idea
The
trigonometric
function:
( ) tanf t bt=
has a period
(repeats)
every
b
π
radians.
t
( )f t
b
π
b
π
Example
Determine how often the
function repeats (period):
( ) tan(2 )k t t=
( ) tan
3
t
j t
 
=  ÷
 
Try This
Determine how often the
function repeats (period):
( ) tan
6
t
p t
 
=  ÷
 
1
6
b = 6π; period=
Try This
Using your
calculator, graph:
1 tany t=
2 tan
6
t
y
 
=  ÷
 
Compare and
contrast.
Definition
Amplitude is
where is
the distance
from the
middle to the
top and
a
a+
a−
is the distance from the
middle to th...
Try This
Compar
e and
Contrast
( ) 5sinf t t=
( ) 2sing t t=
Try This
What is the
amplitude
of
( ) 6tanp t t=
Hint: this is a
trick question.
?
Important Idea
( ) sinf t a bt= ( ) cosg t a bt=
The constant a controls the
amplitude (vertical stretch).
The constant b ...
Try This
Find the
amplitude
and period
for
( ) sinf t t=
Amplitude=1; Period = 2π
Try This
Find the
amplitude
and period
for
( ) 3cos2f t t=
Amplitude=3; Period = π
Try This
Find the
amplitude
and period
for
( ) 3cos2f t t= −
Amplitude=3; Period = π
Try This
Find the
amplitude
and period
for
( ) 3tan(.5 )f t t= −
Amplitude: None;
Period=2π
Example
For what value(s) of t
between is
sin 2 1tπ =
0 1t≤ ≤
Example
For what value(s) of t
between [0, ] is
cos2 0t =
2π
Example
For what value(s) of t
between [0, ] is
cos3 0t =
2π
Try This
For what value(s) of t
between [0, ] is
sin3 0t =
π
0,
3
t t
π
= =
Try This
For what value(s) of t
between [0,1] is
sin3 0tπ =
1 2
0, , ,1
3 3
t =
Lesson Close
Period and amplitude will
be important when we
study modeling and
simple harmonic motion
in section 8.4.
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Hprec7.3

  1. 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. 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. 3. Larger Period, Smaller Frequency Important Idea Radio Visible X-Ray
  4. 4. Smaller Period Greater Frequency Important Idea Radio Visible X-Ray
  5. 5. Important Idea Radio Waves
  6. 6. Important Idea Medicine
  7. 7. Important Idea Ocean Tides
  8. 8. Definition A complete waveform of a periodic function with no repeats is a cycle.
  9. 9. Important Idea A cycle may be measured in different ways but for a given function, the period will be the same . Period
  10. 10. Definition The Period is the time required for the function to repeat. t ( )f t Go to Sketchpad
  11. 11. Definition Frequency (in cycles per second) is the reciprocal of the period.
  12. 12. Start Movie
  13. 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. 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. 15. Important Idea For the trigonometric functions: ( ) sinf t bt= ( ) cosg t bt= or The constant b changes the period.
  16. 16. Definition The trigonometric functions: ( ) sinf t bt= ( ) cosg t bt= have periods (repeat) every 2 b π radians.
  17. 17. Example What is the period of: ( ) sinf t t= 0 2π Period= 2 2 1 π π=
  18. 18. Example What is the period of: ( ) sin(2 )g t t= 0 π 2π 2 2 π π=Period=
  19. 19. Example What is the period of: ( ) sin(.5 )h t t= Period= 2 4 .5 π π= 0 4π 8π
  20. 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. 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. 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. 23. Try This Which is the graph of ( ) sin(4 )f t t= 0 red 2 π π
  24. 24. Try This What is the equation of the function graphed in blue? ( ) sin(2 )f t t= 0 2 π π
  25. 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. 26. Example Determine how often the function repeats (period): ( ) tan(2 )k t t= ( ) tan 3 t j t   =  ÷  
  27. 27. Try This Determine how often the function repeats (period): ( ) tan 6 t p t   =  ÷   1 6 b = 6π; period=
  28. 28. Try This Using your calculator, graph: 1 tany t= 2 tan 6 t y   =  ÷   Compare and contrast.
  29. 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. 30. Try This Compar e and Contrast ( ) 5sinf t t= ( ) 2sing t t=
  31. 31. Try This What is the amplitude of ( ) 6tanp t t= Hint: this is a trick question. ?
  32. 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. 33. Try This Find the amplitude and period for ( ) sinf t t= Amplitude=1; Period = 2π
  34. 34. Try This Find the amplitude and period for ( ) 3cos2f t t= Amplitude=3; Period = π
  35. 35. Try This Find the amplitude and period for ( ) 3cos2f t t= − Amplitude=3; Period = π
  36. 36. Try This Find the amplitude and period for ( ) 3tan(.5 )f t t= − Amplitude: None; Period=2π
  37. 37. Example For what value(s) of t between is sin 2 1tπ = 0 1t≤ ≤
  38. 38. Example For what value(s) of t between [0, ] is cos2 0t = 2π
  39. 39. Example For what value(s) of t between [0, ] is cos3 0t = 2π
  40. 40. Try This For what value(s) of t between [0, ] is sin3 0t = π 0, 3 t t π = =
  41. 41. Try This For what value(s) of t between [0,1] is sin3 0tπ = 1 2 0, , ,1 3 3 t =
  42. 42. Lesson Close Period and amplitude will be important when we study modeling and simple harmonic motion in section 8.4.

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