Upcoming SlideShare
Loading in...5
×

# Hprec7 4

170

Published on

Published in: Technology, Art & Photos
0 Comments
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Be the first to comment

• Be the first to like this

No Downloads
Views
Total Views
170
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
4
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

### 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.
1. #### A particular slide catching your eye?

Clipping is a handy way to collect important slides you want to go back to later.