MATH 107

Section 5.4

Graphs of the Sine and Cosine Functions

2© 2011 Pearson Education, Inc. All rights reserved

GRAPH OF THE SINE FUNCTIONPlotting y = sin x for common values of x and connecting the points with a smooth curve yields the following:

3© 2011 Pearson Education, Inc. All rights reserved

GRAPH OF THE COSINE FUNCTIONPlotting y = cos x for common values of x and connecting the points with a smooth curve yields the following:

4© 2011 Pearson Education, Inc. All rights reserved

KEY POINTS OF THE SINEAND COSINE GRAPHS

***Trigonometric Transformations***

All sine and cosine transformations can be written in the forms:

a* sin[b(x-c)] +d or a* cos[b(x-c)] +d ,

Where a, b, c, and d are numbers used as such:

The amplitude (vertical stretch) = |a|, the period (cycle) = 2 / b, the phase (horizontal) shift = c, and the vertical shift = d.

Example: 7sin(4x-4)-5 = 7sin[4(x-)]-5Amplitude = 7, period = /2, phase shift is units to the right,

vertical shift is 5 units down.

Example: -2cos[(x/3)+1] = -2cos[ (x+3)] Amplitude = 2, graph is flipped over the x-axis (a is negative), period = 6, phase shift is 3 units to the left, no vertical shift.

3

1

6© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 2 Graphing y = a sin x

Graph y = 3 sin x, and y = sin x

on the same coordinate system over theinterval [−2π, 2π].

Solution

Begin with the graph of y = sin x and multiply the y-coordinate of each point (including the key points) on this graph by 3 to get the graph of y = 3 sin x.

7© 2011 Pearson Education, Inc. All rights reserved

This stretches the graph vertically by a factor of 3 without changing the x-intercepts.

Then, multiply the y-coordinate of each

point on the graph of y = sin x by to get

the graph of This compresses the graph.

EXAMPLE 2 Graphing y = a sin x

Solution continued

8© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 2 Graphing y = a sin x

Solution continued

x

3x

2 223

2

2

3

3

9© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 4 Graphing y = sin bx

Sketch one cycle of the graphs of y = sin 3x,

and y = sin x on the same

coordinate system.

Solution

Begin with the graph of y = sin x and divide

its period, 2π, by 3 to get the period of

y = sin 3x. One cycle of y = sin 3x will be

compressed into the interval

Period: 2 b

*Divide each key x-value by 3 and keep the y-values the same.*

11© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 4 Graphing y = sin bxSolution continued

Divide this interval into four equal parts to find the x-coordinates for the key points:

. (y values do not change.)

To find the period of divide 2π by

to get 6π. This yields an interval of [0, 6π]

with these x-coordinates for the key points:

.3

2 and ,

2 ,

3 ,

6 ,0

12© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 4 Graphing y = sin bx

Solution continued

13© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 5 Graphing y = a cos bx

Graph over a one-period interval.

13cos

2y x

Solution

Amplitude is 3. Period is .

Divide the period, 4π, into four quarters:0 to ππ to 2π2π to 3π3π to 4π

The five endpoints give the highest and lowest points and the x-intercepts of the graph.

14© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 5

Solution continued

Graphing y = a cos bx

15© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 6 Graphing y = a sin (x – c)

Graph over a one-period

interval.

Solution

The graph of f (x – c) is the graph of f (x) shifted right c units if c > 0 and left |c| units if c < 0.

16© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 6 Graphing y = a sin (x – c)

Solution continued

; so the graph of is

the graph of y = sin x shifted right units.

17© 2011 Pearson Education, Inc. All rights reserved

PROCEDURE FOR GRAPHING y = a sin [b(x – c)] & y = a cos [b(x – c)]

Step 1 Find the amplitude, period, and phase shift.

amplitude = |a| phase shift = c

period =

If c > 0, shift to the right. If c < 0, shift to the left.

Step 2 The cycle begins at x = c. One complete

cycle occurs over the interval .

bcc

2,

18© 2011 Pearson Education, Inc. All rights reserved

Step 3 Divide the interval into four

equal parts, each of length

. This gives the

x-coordinates for the five key points:

PROCEDURE FOR GRAPHING y = a sin [b(x – c)] AND y = a cos

[b(x – c)]

bcc

2,

b

2

4

1period

4

1

.2

and ,2

4

3 ,

2

2

1 ,

2

4

1 ,

bc

bc

bc

bcc

19© 2011 Pearson Education, Inc. All rights reserved

PROCEDURE FOR GRAPHING y = a sin [b(x – c)] AND y = a cos

[b(x – c)]

Step 4 If a > 0, for y = a sin [b(x – c)], sketch one cycle of the sine curve through the key

, ,2

c ab

points (c,0), ,0 ,cb

20© 2011 Pearson Education, Inc. All rights reserved

PROCEDURE FOR GRAPHING y = a sin [b(x – c)] AND y = a cos

[b(x – c)]Step 4 continued

For y = a cos [b(x – c)], sketch one cycle of the cosine curve through the key points

,0 ,2

cb

(c,a),

21© 2011 Pearson Education, Inc. All rights reserved

PROCEDURE FOR GRAPHING y = a sin b(x – c) AND y = a cos b(x – c) Step 4 continued

For a < 0, reflect the graph ofy = |a| sin b(x – c) or y = |a| cos b(x – c), in the x-axis.

22© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 10 Graphing y = a sin b(x – k)

Graph over a one-period interval.

3sin 22

y x

Solution

Rewrite in standard form as

Amplitude = 3

PeriodPhase shift

Starting point

One cycle5

, ,4 4 4 4

23© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 10 Graphing y = a sin b(x – k)

Solution continued

1 1period

4 4 4

starting point 4

12nd pt

4 4 2

1 33rd pt

4 2 4

34th pt

4 4

5

end pt 4 4

24© 2011 Pearson Education, Inc. All rights reserved

EXAMPLE 10 Graphing y = a sin b(x – k)

Solution continued