Chp. 4.5 Graphs of Sine and Cosine

Functions

p. 323

In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are positive constants and x is in radian measure. The graphs of all sine and cosine functions are related to the graphs of y = sin x and y = cos xwhich are shown below.

y = sin x

y = cos x

x

Sin x

Cos x

Fill in the chart.

2

2

3 20

These will be key points on the graphs of y = sin x and y = cos x.

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 . 11 y

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.

0-1010sin 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 = 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.

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

Before sketching a graph, you need to know:

• Amplitude – Constant that gives vertical stretch or shrink.

• Period –

• Interval – Divide period by 4• Critical points – You need 5.(max., min.,

intercepts.)

.2

b

Amplitudes and PeriodsThe graph of y = A sin Bx has

amplitude = | A|period =

The graph of y = A sin Bx hasamplitude = | A|period =

To get your critical points (max, min, and intercepts) just

take your period and divide by 4.

Example: xy cos3

2

1

2Period

24

2

4

Period

2 ,

2

3 , ,

2 0,at come

willpoints critical So

2

2

2

2

B

2

Interval

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.

2

32

4

y

x

4

2

y = – 4 sin xreflection of y = 4 sin x y = 4 sin x

y = 2sin x

2

1y = sin x

y = sin x

Notice that since

all these graphs

have B=1, so the

period doesn’t

change.

y

x

2

sin xy period: 2 2sin xy

period:

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 .b

2

For b 0, the period of y = a cos bx is also .b

2

If 0 < b < 1, the graph of the function is stretched horizontally.

If b > 1, the graph of the function is shrunk horizontally.

y

x 2 3 4

cos xy period: 2

2

1cos xy

period: 4

y

1

123

2

x 32 4

Example 1: 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)

Determine the amplitude of y = 1/2 sin x. Then graph y = sin x and y = 1/2 sin x for 0 < x < 2.

Example 2

2

3x

y

2˝

˝

-1

1

y = sin x

y = 1/2sinx

2

2

2

1

2

1

Example 3.

2sin

xyofgraphtheSketch

Example 3.

2sin

xyofgraphtheSketch

For the equations y = a sin(bx-c)+d and y = a cos(bx-c)+d

• a represents the amplitude. This constant acts as a scaling factor – a vertical stretch or shrink of the original function.

• Amplitude =• The period is the sin/cos curve making one complete

cycle. • Period =• c makes a horizontal shift.• d makes a vertical shift.• The left and right endpoints of a one-cycle interval can

be determined by solving the equations bx-c=0 and bx-c=

.a

.2

b

.2

Example 4.

3sin

2

1

xyofgraphtheSketch

Example 4.

3sin

2

1

xyofgraphtheSketch

Example 6

.2cos32 xyofgraphtheSketch

Example 6

.2cos32 xyofgraphtheSketch