Graphing Sinusoidal Functions

Y=cos x

y = cos x

•Recall from the unit circle that:

– Using the special triangles and quadrantal angles, we can complete a table.

cosx

r

Y Y Y 0

1 0

-1

0 1

6

3.866

2

1.707

2

3.866

2 3

2

2

3

4

1.5

2

1.5

2

3

4

1.707

2

5

6

3.866

2

7

6

5

4

4

3

1.5

2

3

2

5

3

1.5

2

1.707

2

11

6

7

4

1.707

2

3.866

2

2

Table of Values

y = cos x

3

6

4

2

5

6

2

3

7

6

5

4

4

3

3

2

5

3

7

4

3

4

11

6

2

Parent Functiony = cos x

Domain

•Recall that we can rotate around the circle in either direction an infinite number of times.

•Thus, the domain is (- , )

Range

•Recall that –1 cos 1.•Thus the range of this function is [-1 , 1 ]

1

1

Period

•One complete cycle occurs between 0 and 2.

•The period is 2.

How many periods are shown?

Critical Points

•Between 0 and 2, there are two maximum points at (0, 1) and (2,1).

•Between 0 and 2, there is one minimum point at (,-1).

•Between o and 2, there are two zeros at

30 0

2 2, and , .

Parent FunctionKey Points

2

3

2

20

1

1

* Notice that the key points of the graph separate the graph into 4 parts.

y = a cos b(x-c)+d

• a = amplitude, the distance between the center of the graph and the maximum or minimum point.

• If |a| > 1, vertical stretch • If 0<|a|<1, vertical shrink • If a is negative, reflection about

the x-axis

y = 3 cos x

2

3

2

20

1

1

What changed?

y = - cos x

0

1

2

3

2

2

1

y = a cos b(x - c)+d

•b= horizontal stretch or shrink

•Period = .2b

•If |b| > 1, horizontal shrink •If 0 < |b|< 1, horizontal stretch•If b < 0, the graph reflects about the y-axis.

Tick Marks

•Recall that the key points separate the graph into 4 parts.•If we alter the period, we need to alter the x-scale.•This can be done by diving the new period by 4.

y = cos 3x

3

2

2

0

2

1

1

What is theperiod ofthis function?

1

2

3

2

2

0

1

1cos

2y x

y = a cos b(x - c ) + d

•c= phase shift•If c is negative, the graph shifts left c units (x+c)=(x-(-c))

•If c is positive, the graph shifts right c units (x-c)=(x-)+c))

1

0

12

3

2

2

What changed?Which way did the graph shift?By how many units?

3cos

2y x

0

1

2

3

2

2

1

cosy x

y = a cos b(x-c) + d

•d= vertical shift•If d is positive, graph shifts

up d units•If d is negative, graph shifts

down d units

y = cos x - 2

1

0

1 2

3

2

2What changed?

Which way did the graph shift?

By how many units?

1

0

1

2

3

2

2

1cos

2y x

y = -2 cos(3(x-)) +1

1

0

1 2

3

2

2

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