4.4 Graphs of Sine and Cosine Functions

These will be key points on the graphs of y = sin x and y = cos x Fill in the chart. x Sin x Cos x These will be key points on the graphs of y = sin x and y = cos x

Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. -1 1 sin x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x

Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. -1 1 sin x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x

Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1 cos x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x

Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1 cos x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x

The graphs of y = sin x and y = cos x have similar properties:

The graphs of y = sin x and y = cos x have similar properties:

The graphs of y = sin x and y = cos x have similar properties:

The graphs of y = sin x and y = cos x have similar properties:

The graphs of y = sin x and y = cos x have similar properties:

The graphs of y = sin x and y = cos x have similar properties:

The graphs of y = sin x and y = cos x have similar properties:

The graphs of y = sin x and y = cos x have similar properties:

The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.

Amplitude – Constant that gives vertical stretch or shrink. The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis.

Amplitude – Constant that gives vertical stretch or shrink. y x y = 2sin x y = sin x y = sin x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x

Amplitude – Constant that gives vertical stretch or shrink. The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis.

Example 1: Sketch the graph of y = 3 cos x on the interval [0, 2]. Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. max x-int min y = 3 cos x 2  x y x (0, 3) ( , 3) ( , 0) ( , 0) ( , –3) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: y = 3 cos x

Example 1: Sketch the graph of y = 3 cos x on the interval [0, 2]. Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. max x-int min y = 3 cos x 2  x y x (0, 3) ( , 3) ( , 0) ( , 0) ( , –3)

Example 2: Sketch the graph of y = 3 cos x on the interval [0, 2]. max x-int min y = -3 cos x 2  x ( , –3) y x ( , 0) ( , 0) ( , 3) (0, -3)

Example 2: Sketch the graph of y = 3 cos x on the interval [0, 2]. Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. max x-int min y = -3 cos x 2  x y x (0, 3) ( , 3) ( , 0) ( , 0) ( , –3)

Example: Sketch the graph of y = 1/3 cos x on the interval [0, 2]. max x-int min y = 1/3 cos x 2  x y x ( , 0) ( , 0) ( , 1/3) (0, 1/3) ( , –1/3) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: y = 3 cos x

HOMEWORK Pgs. 264-265 1-4 ALL

For b  0, the period of y = a sin bx is . The period of a function is the x interval needed for the function to complete one cycle. For b  0, the period of y = a sin bx is . For b  0, the period of y = a cos bx is also . Interval – Divide period by 4 Critical points – You need 5.(max., min., intercepts.)

Amplitudes and Periods The graph of y = a sin bx or a cos bx amplitude = | a | period = To get your critical points (max, min, and intercepts) just take your period and divide by 4. Example: Interval

If b > 1, the graph of the function is shrunk horizontally. The period of a function is the x interval needed for the function to complete one cycle. If b > 1, the graph of the function is shrunk horizontally. y x period: 2 period:

The period of a function is the x interval needed for the function to complete one cycle. If 0 < b < 1, the graph of the function is stretched horizontally. y x period: 4 period: 2

Example 1   x

Sketch the graph of y = sin 2x Example 2 Sketch the graph of y = sin 2x y = sin 2x x

Example 3     x

Example: Sketch the graph of y = -2 sin (3x). period: 2 3 = amplitude: |a| = |-2| = 2 Calculate the five key points. 2 -2 y = 2 sin 3x x y x ( , 2) (0, 0) ( , 0) ( , 0) ( , -2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: y = 2 sin(-3x)

HOMEWORK Pgs. 264-265 5-8 ALL

Example 4

Example 6 y = Sin x Amplitude PERIOD /INT CRITICAL POINTS Phase shift   Amplitude PERIOD /INT CRITICAL POINTS  Phase shift ADD to x Vertical Shift ADD to y

Example 6

Example 6 y = Cos x 1 Amplitude PERIOD /INT CRITICAL POINTS   Amplitude PERIOD /INT CRITICAL POINTS  Phase shift ADD to x Vertical Shift ADD to y