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Sine and Cosine Graphs

Reading and Drawing

Sine and Cosine Graphs

Some slides in this presentation contain animation. Slides will be

more meaningful if you allow each slide to finish its presentation

before moving to the next one.

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This is the graph for y = sin x.

This is the graph for y = cos x.

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y = sin x

y = cos x

One complete period is

highlighted on each of

these graphs.

For both y = sin x and y = cos x, the period is 2π. (From the beginning of

a cycle to the end of that cycle, the distance along the x-axis is 2π.)

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y = sin x

y = cos x

Amplitude deals with the

height of the graphs.

For both y = sin x and y = cos x, the amplitude is 1. Each of these

graphs extends 1 unit above the x-axis and 1 unit below the x-axis.

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For y = sin x, there is no phase shift.

The y-intercept is located at the point (0,0).

We will call that point, the key point.

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A sine graph has a phase shift if the key point

is shifted to the left or to the right.

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For y = cos x, there is no phase shift.

The y-intercept is located at the point (0,1).

We will call that point, the key point.

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A cosine graph has a phase shift if the key point

is shifted to the left or to the right.

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y = a sin b (x - c)

For a sine graph which has no vertical shift, the equation for the

graph can be written as

For a cosine graph which has no vertical shift, the equation for the

graph can be written as

y = a cos b (x - c)

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y = a sin b (x - c) y = a cos b (x – c)

|a| is the amplitude of the sine or cosine graph.

The amplitude describes the height of the graph.

Consider this sine graph. Since

the height of this graph is 3, then

a = 3.

The equation for this graph can

be written as y = 3 sin x.

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Consider this cosine graph. The height of this graph is 2, so a = 2.

The equation for this graph can be written as y = 2 cos x.

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If a sine graph is “flipped” over the x-axis, the value of a will be negative.

For the graph above, a = -3.

An equation for this graph is y = -3 sin x.

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If a cosine graph is “flipped” over the x-axis, the value of a will be negative.

For the graph above, a = -1.

An equation for this graph is y = -1 cos x or just y = - cos x.

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y = a sin b (x - c) y = a cos b (x - c)

“b” affects the period of the sine or cosine graph.

For sine and cosine graphs, the period can be determined by

.b

2period

Conversely, when you already know the period of a sine or cosine

graph, b can be determined by

.period

2b

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The period for this graph is . 3

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period

2b

Notice that a =2 on this graph since the graph extends 2 units above

the x-axis.

.x2

3sin2y

2

3b Since and a = 2, the sine equation for this graph is

Use the period to calculate b.

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A sine graph has a phase shift

if its key point has shifted to the

left or to the right.

A cosine graph has a phase shift

if its key point has shifted to the

left or to the right.

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y = a sin b (x - c) y = a sin b (x - c)

“c” indicates the phase shift of the sine graph or of the

cosine graph. The x-coordinate of the key point is c.

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This sine graph moved

units to the right. “c”, the phase

shift, is .

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An equation for this graph can be written as .2

xsiny

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y = sin x

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This cosine graph above moved units to the left.

“c”, the phase shift, is .

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An equation for this graph can be written as

.2

xcosyor2

xcosy

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y = cos x

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Graphs whose equations can be written as a sine function can also be

written as a cosine function.

Given the graph above, it is possible to write an equation for the

graph. We will look at how to write both a sine equation that describes

this graph and a cosine equation that describes the graph.

The sine function will be written as y = a sin b (x – c).

The cosine function will be written as y = a cos b (x – c).

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y = a sin b (x – c)

For the sine function, the values for a, b, and c must be determined.

The height of the graph is 4, so a = 4.

The period of the graph is .2

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periodb .

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The key point has shifted to , so the phase shift is 3

.

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c

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y = a sin b (x – c)

a = 4 2

3b

3c

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3sin4

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3sin4 xyorxy

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This is an equation for the graph written as a sine function.

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y = a cos b (x – c)

To write the equation as cosine function, the values for a, b, and c

must be determined. Interestingly, a and b are the same for cosine as

they were for sine. Only c is different. The height of the graph is 4, so a = 4.

The period of the graph is .2

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periodb

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The key point has not shifted, so there is no phase shift. That means

that c = 0.

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a = 4 2

3b 0c

x2

3cos4yor0x

2

3cos4y

y = a cos b (x – c)

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This is an equation for the graph written as a cosine function.

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It is important to be able to draw a sine graph when you are given the

corresponding equation. Consider the equation

Begin by looking at a, b, and c.

.xsiny

822

.xsiny

822

822

cba

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The amplitude is 2. Maximums will be at 2.

Minimums will be at -2.

The negative sign means that the graph has “flipped” about the x-axis.

82sin2 xy

2a2a

2

-2

2

-2

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The phase shift is

That means that the key point

shifts from the origin to

82sin2 xy 8

c

.8

.8

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2b

Use b = 2 to calculate the period of the graph.

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2

b

2period

One complete period is highlighted here.

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In order to correctly label the x-intercepts, maximums, and minimums on

the graph, you will need to divide the period into 4 equal parts or

increments.

An increment, ¼ of the period, is the distance between an x-intercept and

a maximum or minimum.

One increment

The increment is ¼ of the period. Since the period for

is π, the increment is .4

or4

1

82sin2 xy

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To label the graph, begin at the phase shift. Add one increment at a time

to label x-intercepts, maximums, and minimums.

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What does the graph for the equation look like? x2

1cos5y

cba2

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Maximums will be at 5.

Minimums will be at -5.

This means that the

amplitude of the graph is 5.

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The phase shift is

That means that the key point

shifts from the origin to

c

.

.

Use to calculate the period of the graph.

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bperiod

One complete period is highlighted here.

.2

1cos5 xy

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1b

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Remember that the increment (¼ of the period) is the distance between

an x-intercept and a maximum or minimum.

Since the period for is 4π, the increment is π.

Don’t forget that x-intercepts, maximums, and minimums can be labeled

by beginning at the phase shift and adding one increment at a time.

xy2

1cos5

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-π + π

This is the graph for

.2

1cos5 xy

0 + π π + π

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Sometimes a sine or cosine graph may be shifted up or down. This is

called a vertical shift.

y = a sin b (x - c) +d.

The equation for a sine graph with a vertical shift can be written as

The equation for a cosine graph with a vertical shift can be written as

y = a cos b (x - c) +d.

In both of these equations, d represents the vertical shift.

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A good strategy for graphing a sine or cosine function that has a

vertical shift:

•Graph the function without the vertical shift

• Shift the graph up or down d units.

Consider the graph for

The equation is in the form y = a cos b (x - c) +d.

“d” equals 3, so the vertical shift is 3.

The graph of

was drawn in the previous example.

.32

1cos5 xy

xy2

1cos5

xy2

1cos5

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To draw , begin with the graph for 32

1cos5 xy .x

2

1cos5y

xy2

1cos5

Draw a new horizontal axis at

y = 3.

Then shift the graph up 3 units.

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The graph now represents .3x2

1cos5y

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This concludes

Sine and Cosine Graphs.