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

Try ThisFind the amplitude and period for

( ) 3cos2f t t

Amplitude=3; Period =

Important IdeaA common mistake…

•a is not amplitude; is amplitude.

a

•a may be positive or negative; amplitude is always positive.

DefinitionThe 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.

Important IdeaIn 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

Try ThisWhat 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 ThisWhat is the value of a, b, c, and d in the following trig equation:

sin( )y a bt c d

1 sin( 2 3) 6y t

ExampleWithout using a calculator, describe and sketch the graph of

( ) 3sin 4f t t

ExampleWithout using a calculator, describe and sketch the graph of

( ) 2cos 4g t t

Try ThisWithout using a calculator, describe and sketch the graph of

( ) 2cos 3k t t

SolutionThe graph of is the same as the graph of the parent function, except:

( )k t

cos t( )k t• is reflected across

the horizontal axis• It is vertically stretched 2 units•It is shifted down 3 units

Solution

Parent: cos t

( ) 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 constant c in the standard form.

Example

Factor:

2 3t Re-write:

3 22

2t

Try This

Factor:

43

t

412

t

ExampleFind the phase shift of

( ) sin 22

g t t

Re-write as:

( ) sin 24

g t t

ExampleFind the phase shift of

( ) 3sin 3 5f t t

Re-write as:

Try ThisFind the phase shift of

( ) 2cos 2p t t

Re-write as:

( ) 2cos22

p t t

Phase shift: 2

to left

Try This

2 2cos22

y x

Using your calculator, graph:

1 2cos2y x

Be sure you are in radian mode.

Solution

y1

y2

2

to left

Try ThisState 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.

SolutionThe phase shift of:

sin( 2)y t is 2 units to right

sin( 2)y t siny t

2

Important IdeaChanges in phase shift

move the graph left and right. Phase shift is a horizontal translation.

DefinitionThe 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.

Try ThisGraph ( ) sin 2

6f t t

and ( ) sin6

g t t

sin( 6) 2y x sin( 6)y x

on the same axes.

Example

Identify the amplitude, period, phase shift and vertical shift of:

( ) 3cos(2 1) 4f t t

Try ThisIdentify the amplitude, period, phase shift and vertical shift of:

( ) 3sin(3 1) 1g t t

Amplitude=3, Period=2 3Phase shift=1/3 unit to leftVertical shift=-1

ExampleAs 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.

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

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.