4-4 Graphing Sine and Cosine Functionst - North Hunterdon-Voorhees ...

4-4: Graphing Sine and Cosine Functions

CP Precalculus Mr. Gallo

Periodic Function Function which repeats a pattern of y-values at regular intervals.

Cycle One complete pattern Can begin at any point on the graph.

Period Horizontal length of one cycle

Cycle

Period

1

Examples of Periodic Functions

Sine

Period2

? Cosine

Period2

Maximum

Midline

Amplitude

Midline

Minimum

Horizontal line halfway between the min and max

y-values. Amplitude

midline

1 2

max

min

Measures amount of variation in function values

Half the distance between the min and max

values

amplitude

1 2

max

min

2

The Graph of sin

Graphing y = a sin(bx)

|a| = amplitude of function. b = number of complete cycles from 0 to 2

is the period of the function.

1. Calculate the amplitude, # of cycles, and period in order to graph a sine function. 2. For one period, identify the x-values for: zero, max, zero, min, zero by dividing the period by 4. 3. Above the max x-value, plot a point using the amplitude as your height. 4. Below the min x-value, plot a point using the amplitude

as your height. 5. Connect the points with a smooth curve. 6. Repeat the cycle across the rest of the graph.

3

Graph y 3sin 2

Amplitude: ___3____ # of cycles from 0 to 2: __2__

Period: __2b_____22_______

3

Zero: __0__ Max: __4__ Zero: __2__ Min: __4__ Zero: ___

GAmrapplhituyde:_s_in_1__2__ # of

1 cycles from 0 to 2: __2__

Period: _2_b____2_1_2___4___

Zero: __2__ Max: ____ Zero: __0__ Min: ____ Zero: _2__

4

(a) Determine the period and amplitude of each graph. (b) Determine the equation of the sine curve in each graph.

a). Period __2_4____2___

Amplitude1_.5___2__1._5___ 32

b). y a sin b

y

3 2

sin

4

a). Period ____3______

Amplitude _2__2__2____2_

b). y a sin b

y

2

sin

2 3

The Graph of y = cos(x)

5

Graphing acos

= amplitude of function. = number of complete cycles from 0 to 2

is the period of the function Follow the same steps for graphing cosine, except

the five values will be max, zero, min, zero, max. Note: negative "a" means that the graph starts

BELOW the x-axis first.

Graph 3 2

3 2 Amplitude: ____ # of cycles from 0 to 2: ____

Period: __2_2_______

0 Max: _______ 4 Zero: _______ 2 Min: ________ Zero: _3____4__

Max: _______

6

a). Period: __22________

Amplitude:_____4______

b) Equation:

y 4 cos(2 )

a). Period: __21_____2___

Amplitude:_____5______

b) Equation:

y 5cos( )

Properties of the Sine Function

7

Properties of the Cosine Function

Homework: 4-4 Homework WS1 #7-15 8

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