7.4: Periodic Graphs &
Phase Shifts
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•State period, amplitude,
vertic...
Try This
Find the
amplitude
and period
for
( ) 3cos2f t t= −
Amplitude=3; Period = π
Important Idea
A common mistake…
•a is not amplitude;
is amplitude.
a
•a may be positive or
negative; amplitude is
always ...
Definition
The standard forms for sine
and cosine functions are:
( ) sin( )f t a bt c d= + +
( ) cos( )g t a bt c d= + +
w...
Important Idea
In the standard form:
( ) sin( )f t a bt c d= + +
( ) cos( )g t a bt c d= + +
•a controls amplitude
•b cont...
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= − + +
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= − − +
Example
Without using a calculator,
describe and sketch the
graph of
( ) 3sin 4f t t= − −
Example
Without using a calculator,
describe and sketch the
graph of
( ) 2cos 4g t t= +
Try This
Without using a calculator,
describe and sketch the
graph of
( ) 2cos 3k t t= − −
Solution
The graph of is the same
as the graph of the parent
function, except:
( )k t
cost
( )k t• is reflected across the...
Solution
Parent: cost
( ) 2cos 3k t t= − −
Definition
The phase shift of a
trigonometric function
results in a horizontal shift
of the graph. It is controlled
by the...
Example
Factor: 2 3t +
Re-write:
3 2
2
2
t +
g
Try This
Factor: 4
3
t
π
+
4
12
t
π 
+ ÷
 
Example
Find the phase shift of
( ) sin 2
2
g t t
π 
= + ÷
 
Re-write as:
( ) sin 2
4
g t t
π 
= + ÷
 
Example
Find the phase shift of
( )( ) 3sin 3 5f t t= +
Re-write as:
Try This
Find the phase shift of
( )( ) 2cos 2p t t π= − +
Re-write
as:
( ) 2cos2
2
p t t
π 
= − + ÷
 
Phase
shift: 2...
Try This
2 2cos2
2
y x
π 
= − + ÷
 
Using your calculator, graph:
1 2cos2y x= −
Be sure you are in radian
mode.
Solution
y1
y2
2
π
to left
Try This
State the phase shift of:
( ) sin( 2)f t t= −
then use a graphing
calculator to graph the
function and its parent...
Solution
The phase shift of:
sin( 2)y t= −
is 2 units
to right
sin( 2)y t= −
siny t=
2
Important Idea
Changes in phase shift
move the graph left and
right. Phase shift is a
horizontal translation.
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 tr...
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= +...
Example
Identify the amplitude,
period, phase shift and
vertical shift of:
( ) 3cos(2 1) 4f t t= − − +
Try This
Identify the amplitude,
period, phase shift and
vertical shift of:
( ) 3sin(3 1) 1g t t= + −
Amplitude=3, Period=...
Example
As you ride a ferris wheel,
the height you are above the
ground varies periodically.
Consider the height of the
ce...
1. Write an
equation
describing the
change in
2. Find the height of the seat
after 22 seconds, after 60
seconds and after ...
Lesson Close
Because of the repeating or
periodic nature of
trigonometric graphs, they
are used to model a variety
of phen...
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Hprec7 4

  1. 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. 2. Try This Find the amplitude and period for ( ) 3cos2f t t= − Amplitude=3; Period = π
  3. 3. Important Idea A common mistake… •a is not amplitude; is amplitude. a •a may be positive or negative; amplitude is always positive.
  4. 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. 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. 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. 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. 8. Example Without using a calculator, describe and sketch the graph of ( ) 3sin 4f t t= − −
  9. 9. Example Without using a calculator, describe and sketch the graph of ( ) 2cos 4g t t= +
  10. 10. Try This Without using a calculator, describe and sketch the graph of ( ) 2cos 3k t t= − −
  11. 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. 12. Solution Parent: cost ( ) 2cos 3k t t= − −
  13. 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. 14. Example Factor: 2 3t + Re-write: 3 2 2 2 t + g
  15. 15. Try This Factor: 4 3 t π + 4 12 t π  + ÷  
  16. 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. 17. Example Find the phase shift of ( )( ) 3sin 3 5f t t= + Re-write as:
  18. 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. 19. Try This 2 2cos2 2 y x π  = − + ÷   Using your calculator, graph: 1 2cos2y x= − Be sure you are in radian mode.
  20. 20. Solution y1 y2 2 π to left
  21. 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. 22. Solution The phase shift of: sin( 2)y t= − is 2 units to right sin( 2)y t= − siny t= 2
  23. 23. Important Idea Changes in phase shift move the graph left and right. Phase shift is a horizontal translation.
  24. 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. 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. 26. Example Identify the amplitude, period, phase shift and vertical shift of: ( ) 3cos(2 1) 4f t t= − − +
  27. 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. 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. 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. 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.
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