Graphing Sinusoidal Functions

y=sin x

y= sin x

•Recall from the unit circle that:

sin .yr

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

y

6

1

2

4

1

2

3

3

2

2

Quadrant I Quadrant 2

0

1

0

y .5 .707 .866 1

0

y

2

3

3

2

y .866 .707 .5

1

23

4

5

6

1

2

00

Table of Values

y

Quadrant III Quadrant IV

5

4

4

3

7

6

3

2

1

2

1.707

2

3.866

2

1

Table of Values

y5

3

7

4

11

6

1.5

2

2 0

3.866

2

1.707

2

3

6

4

2

5

6

4

3

7

6

5

4

4

3

3

2

5

3

7

4

3

4

11

6

2

Parent Functiony=sin x

0

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 sin 1.

1

1

•Thus the range of this function is [-1 , 1 ]

Period

•One complete cycle occurs between 0 and 2.

•The period is 2.

How many periods are shown?

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

Critical Points

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

2

3

2

•Between 0 and 2, there are three zeros at (0,0), (,0) and (2,0).

Parent FunctionKey Points

2

3

2

2

0

1

1

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

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

•a = amplitude, the distance from the center to the maximum or minimum.

• If |a| > 1, vertical stretch • If 0<|a|<1, vertical shrink • If a is negative, the graph

reflects about the x-axis.

y = 3 sin x

2

3

2

20

1

1

What changed?

y= sin x1

4

1

1

y= -2 sin x

1

1

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

•b= horizontal stretch or shrink2b

• Period =

•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 dividing the new period by 4.

y = sin 2x

3

2

20

2

1

1

What is theperiod ofthis function?If we wanted to graph only one period, what would the tick marks need to be?

4

3

4

y = sin x 1

3

1

1

2

3

2

2

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

• c = horizontal 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))• In trigonometric functions, these horizontal shifts are called phase shifts.

y = sin(x- )2

1

0

12

3

2

2

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

y = sin (x + )

2

3

2

2

1

1

y=a sin 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.• In trigonometric functions, these vertical shifts are called the vertical displacement.

y = sin x +2

1

0

12

3

2

2

What changed?

Which way did the graph shift?

By how many units?

y = sin x - 31

12

3

2

2

y = 3 sin(2(x-)) - 2

1

0

1 2

3

2

2

Can you list all thetransformations?

y=-2sin(2x-) +1

23

2

2

1

1