Period and Amplitude Changes

Objectives:1. Evaluate sine and cosine functions with

amplitude and period changes

Vocabulary:Amplitude, period

Graph of the Sine Function

To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts.

0-1010sin x

0x2

2

32

Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

y

2

3

2

22

32

2

5

1

1

x

y = sin x

Remember that y = sin x is an odd function whose graph is symmetry to origin.

Transformations of Trig Functions

sin ( )y A B x h k

Amplitude |A|: A controls vertical stretching/shrinkingPeriod p can be in radians or degrees: p is the horizontal length to complete one cycleB can be in radians or degrees: B controls horizontal stretching/shrinking

pB

2

pB

360

-6 -4 -2 2 4 6 8

4

3

2

1

-1

-2

-3

-4

f x = sin x

Period =2π Amplitude = 1

4

3

2

1

-1

-2

-3

-4

-8 -6 -4 -2 2 4 6 8

f x = cos x

Period =2π Amplitude = 1

Vertical Stretching/Shrinking of Sine Functions

KEY TAKE-AWAY: x-intercepts are unchanged; multiply y-value of max/min by A.

y = sin(x)

y = sin(2x)

y = sin(x)

y = sin(x/2)

y

x

2y = cos (–x)

Sketch the graph of y = sin (–x).

The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis.

Sketch the graph of y = cos (–x).

The graph of y = cos (–x) is identical to the graph of y = cos x.

y

x

2y = sin x

y = sin (–x)

y = cos (–x)

Give the amplitude and period of the function

4sin 3 . Then sketch at least one cycle of its graph.y x

amplitude = 4 4 2 2

period = 3B

Key points:

x sin3x0

2

3

3

6

2

0

1

0

1

0

x 4sin 3x0

6

3

22

3

0

4

0

4

0

x 4sin3x0

6

3

22

3

0

4

0

4

0

Assignment:

p. 305 #1-9 odd

Period and Amplitude Changes

Objectives:1. Evaluate sine and cosine functions with

amplitude and period changes2. Identify Period and Amplitude from a

graph3. Write the equation or a trig function

from a graph

Vocabulary:Amplitude, period

3cos ( )3

y x

-6 -4 -2 2 4 6 8

4

3

2

1

-1

-2

-3

-4

q x = -3cos

3 x

Use the cosine curve, cos , with amplitude 3 and period 8.y A Bx2 2

88 4

BB

So the equation is 3cos .4

y x

2sin 4y x

-10 -8 -6 -4 -2 2 4 6 8 10

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

3sin2

y x

-10 -8 -6 -4 -2 2 4 6 8 10

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

12cos

2y x

Assignment:

p. 305 #11-17 odd, 27, 29