Graphs of Cosine

Section 4-5

2

Objectives

• I can determine all key values for 6 trig functions

• I can graph cosine functions

• I can determine amplitude, period, and phase shifts of cosine functions

3

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.

10-101cos x

0x2

2

32

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

2

3

2

22

32

2

5

1

1

x

y = cos x

4

6. The cycle repeats itself indefinitely in both directions of the x-axis.

Properties of Sine and Cosine Functions

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

3. The maximum value is 1 and the minimum value is –1.

4. The graph is a smooth curve.

1. The domain is the set of real numbers. (-∞, ∞)

5. Each function cycles through all the values of the range over an x-interval of .2

2. The range is the set of y values such that . [ 1,1]

5

Graphing Trig Functions

• Find the period of the function• We will use the following to graph:

– “x” represents radians– “” represents degrees– Example: sin x versus sin

• Determine the key points (Make data table)– Intercepts– Maximum points– Minimum points

y = cos x

Start HIGH, middle, low, middle, high

7

y

1

123

2

x 32 4

Example: Sketch the graph of y = 3 cos x on the interval [–, 4].

Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.

maxx-intminx-intmax

30-303y = 3 cos x20x 2

2

3

(0, 3)

2

3( , 0)( , 0)2

2( , 3)

( , –3)

8

2cos( 45 )y 2A360

: 3601

Period θ 0 90 180 270 360

Cos 1 0 -1 0 1

2 Cos 2 0 -2 0 2

Rx -2 0 2 0 -2

45 degrees

Left

-45 45 135 225 315

90 180 270 360-90-180-270-360

White Board

9

2 cos 2( 90 )y

White Board

10

11 3cos ( )

2 2y x

11

Homework

• Trig Value Table

• WS 9-2