http://www.wku.edu/~tom.richmond/Cosine.html

Animation of the Cosine curve:

http://www.wku.edu/~tom.richmond/Sine.html

Animation of the Sine curve:

Phase Phase ShiftShift

The only transformation of the function is a phase shift. The amount of the shift can be calculated by: In this case, c = π & k = 1

k

c

left) the toisshift the0,c (since 1

right) the toisshift the0,c (since

In this case,

421

22

2

2

2&

kc

MidlinMidline:e:

A new horizontal axis that results from an upward or downward shift. For the graph of

Since h = -5, the graph will shift down 5 units. The midline of the graph is y = -5.

The amplitude is 2, so the graph will stretch vertically.

The midline is the equation: 4

Graph it.

- 6

The amplitude is:

Use dashed lines to mark max & minsThe

period is:

y = -6

k = ½

2122 kPeriod 4

1 2 3 4

This is the graph before the horizontal (phase) shift

The phase shift is:k

c 2

21

C is positive; shift to left.

Example 4Example 4

12.4

hckAy )sin(

8 k

c

Solve for k (criss-cross)

k 2 Divide both sides by πk2

28 c

Use the phase shift to solve for c

482 c

6)2sin(4 4 y

Use the period to solve for K BEFORE you can find c

(replace k with 2 & criss-cross)82

c

First, calculate the cos x, then add it to x for your y values to graph.

1

9996.0 6.29996.02

9985.0 1.49985.0 9966.0 2.59966.02

3

9940.0 3.7994.02

9906.0 8.89906.025

9865.0 4.109865.03

110

Then graph the ordered pairs:

3 2

1

2

3

4

5

6

7

8

9

10

At this point, freehand the

curve along y = x

xy cos

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