Do Now: If y = 2sin 2x, fill in the table below

xy

0 4

4

32

4

52

34

7

Aim: How do we sketch y = A(sin Bx) and y = A(cos Bx)?

2

0 2 0 2 0 2 0 02

HW: Handout

Math Composer 1. 1. 5http: / / www. mathcomposer. com

2 74 3

2 5

4 1 3

4 1

2 1

4 1

4 1

2 3

4 1 5

4 3

2 7

4 2

-2.5

-2.0

-1.5

-1.0

-0.5

0.5

1.0

1.5

2.0

2.5

x

y

Math Composer 1. 1. 5http: / / www. mathcomposer. com

Math Composer 1. 1. 5http: / / www. mathcomposer. com

y = 2sin2x

When we multiply x, the measure of the angle, by some value, B, we change the frequency of the curve.

Frequency: The number of complete curves in every 2π radians.

In the form of BxAy sin

This means we have more “complete” curves in the interval 0 2 x

So the frequency of y = sin 2x is 2

BxAy sin or BxAy cos

A implies the amplitude

To find the amplitude of sine or cosine function, we simply take absolute value of A ( )A

We only use positive number for amplitude. For example: the amplitude of y = 2 sin x and y = –2 sin x are both 2

2B

the amplitude of y = ½ sin x and y = -1/2 sin x are both ½

The rule to find the period is

In

For y = sin x, the frequency is 1 since the value of B is 1, and the period is 2. That means there is only one complete curve within 2 radians also means to have a complete curve, the angle measure is 2 radians.

xy2

1cos

The amplitude is 1 and the period is

4

21

2

xy2

3sin2

The amplitude is 2 and the period is 3

4

23

2

Determining the Period and Amplitude of y = a sin bx

Given the function y = 3sin 4x, determine the period and the amplitude.

The period of the function is2b

Therefore, the period is24

2

The amplitude of the function is | a |. Therefore, the amplitude is 3.

y = 3sin 4x

Writing the Equation of the Periodic Function

| maximum minimum|2

Amplitude

| 2 ( 2) |

2= 2

Period 2b

2b

b = 2

Therefore, the equation as a function of sine isy = 2sin 2x.

Writing the Equation of the Periodic Function

| maximum minimum|2

Amplitude Period 2b

| 3 ( 3) |2

= 3

4 2b

b = 0.5

Therefore, the equation as a function of cosine isy = 3cos 0.5x.

xy2

1cos3 0 4 x a) Sketch the graph of , over

xy2

1cos

b) Find the value(s) of x in the interval so that the

value of

(1) a maximum (2) a minimum (3) 0

Find the amplitude and period for the following functions:

xy3

1sin3

xy 3cos2

1

xy4

1sin4

a)

c)

b)