4.5 Graphs of Sine and Cosine

Functions

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

23 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

23

2

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

23

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

0x 2

23

2

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

23

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.)

.2b

Amplitudes and PeriodsThe graph of y = A sin Bx has

amplitude = | A|period =

To get your critical points (max, min, and intercepts) just take your period and divide by 4. Example: xy cos3

21

2Period

242

4Period

2 ,2

3 , ,2

0,at come

willpoints critical So

2

2

2

2

B2

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.

23

2

4

y

x

4

2

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

y = 2sin x

21y = 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 .

b2

For b 0, the period of y = a cos bx is also .b2

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

21cos 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 x

20x 2

23

(0, 3)

23( , 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

23

x

y

2˝

˝

-1

1

y = sin x

y = 1/2sinx

2

2

21

21

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

.2b

.2

Example 4.

3sin

21

xyofgraphtheSketch

Example 4.

3sin

21

xyofgraphtheSketch

Example 6

.2cos32 xyofgraphtheSketch

Example 6

.2cos32 xyofgraphtheSketch

Tides• Throughout the day, the

depth of the water at the end of a dock in Bangor, Washington varies with the tides. The tables shows the depths (in feet) at various times during the morning.

(a) Use a trig function to model the data.

(b) A boat needs at least 10 feet of water to moor at the dock. During what times in the evening can it safely dock?

Time Depth, yMidnight 3.12 a.m. 7.84 a.m 11.36 a.m. 10.98 a.m. 6.6

10 a.m. 1.7Noon 0.9

cos( )y a bt c d

Homework

• Page 304-305• 3-13 odd, 15-21 odd, 31 – 34 all, 47-57

odd, 65, 67