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Sine and Cosine Graphs
Reading and Drawing
Sine and Cosine Graphs
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This is the graph for y = sin x.
 2

3
2



2

2
0
3
2

2
This is the graph for y = cos x.
 2

3
2



2
0

2

3
2
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2
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y = sin x
 2

3
2



2
One complete period is
highlighted on each of
these graphs.
0

2

3
2
2
y = cos x
 2

3
2



2
0

2

3
2
2
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
1
 2

3
2



2
Amplitude deals with the
height of the graphs.
0

2

3
2
2
-1
y = cos x
1
 2

3
2



2
0

2

3
2
2
-1
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.
 2

3
2



2
0

2

3
2
2
The y-intercept is located at the point (0,0).
We will call that point, the key point.
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 2 
3

 
2
2
0

2

3
2
2
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.
1
 2

3
2



2
0

2

3
2
2
-1
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.
 2

3
2



2
0

2

Esc
3
2
2
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For a sine graph which has no vertical shift, the equation for the
graph can be written as
y = a sin b (x - c)
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 cos b (x – c)
y = a sin b (x - c)
|a| is the amplitude of the sine or cosine graph.
The amplitude describes the height of the graph.
3
2
Consider this sine graph. Since
the height of this graph is 3, then
a = 3.
1
 2 
The equation for this graph can
be written as y = 3 sin x.
3

   -1 0
2
2

2

3
2
2
-2
-3
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Consider this cosine graph. The height of this graph is 2, so a = 2.
2
1
 2

3
2



0
2 -1

2

3
2
2
-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.
3
2
1
 2 
3
2


 -1
0
2
-2

2

3
2
2
-3
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.
1
 2

3
2



2
0

2

3
2
2
-1
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
2
period 
.
b
Conversely, when you already know the period of a sine or cosine
graph, b can be determined by
2
b
.
period
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4
The period for this graph is
.
3
Use the period to calculate b.
2
1

4
3


2
3


3
0
-1

3
2
3

4
3
b
2  3
2

period  4  2
 
 3 
-2
Notice that a =2 on this graph since the graph extends 2 units above
the x-axis.
Since b 
3
and a = 2, the sine equation for this graph is
2
3
y  2 sin x.
2
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 2 
3

 
2
2
0

2

3
2
2
A sine graph has a phase shift
if its key point has shifted to the
left or to the right.
 2

3
2



2
0

2

3
2
2
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.
y = sin x
1

3

 
2
2
This sine graph moved
0

2

3
2
2
5
2

2
units to the right. “c”, the phase
shift, is  .
2
-1



2
An equation for this graph can be written as y  sin  x  .
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y = cos x
1

5
3

 2 
 
2
2
2
0

2

3
2
2
-1
This cosine graph above moved
“c”, the phase shift, is 

2
units to the left.

.
2
An equation for this graph can be written as


  

y  cos x      or y  cos x   .
2
 2 


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Graphs whose equations can be written as a sine function can also be
written as a cosine function.
4
3
2
1

4
3


2
3


3
-1
-2

3
2
3

4
3
-3
-4
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)
4
3
2
1

4
3


2
3

 -1
3 -2

3
2
3

4
3
-3
-4
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
4
.
3
The key point has shifted to 
b
2
2 3

 .
4 2
period
3
b
3
.
2




.
c .
, so the phase shift is
3
3
3
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y = a sin b (x – c)
4
3
2
1

4
3


2
3


3

3
-1
-2
2
3

4
3
-3
-4
a=4
b
3
2

c
3
3
  
y  4 sin  x     
2
 3 
or
3

y  4 sin  x  
2
3
This is an equation for the graph written as a sine function.
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y = a cos b (x – c)
4
3
2
1

4
3


2
3


3

3
-1
-2
2
3

4
3
-3
-4
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
4
.
3
b
2
2 3

 .
4 2
period
3
b
3
2
The key point has not shifted, so there is no phase shift. That means
that c = 0.
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y = a cos b (x – c)
4
3
2
1

4
3


2
3


3
-1
-2

3
2
3

4
3
-3
-4
a=4
b
3
2
c0
3
y  4 cos x  0 
2
or
3
y  4 cos x
2
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


y  2 sin 2  x   .
8

Begin by looking at a, b, and c.


y  2 sin 2  x   .
8

a  2
b2
c

8
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

y  2 sin 2 x  
8

The amplitude is 2.
a  2
a 2
Maximums will be at 2.
2
-2
Minimums will be at -2.
The negative sign means that the graph has “flipped” about the x-axis.
2
-2
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

y  2 sin 2 x  
8

The phase shift is
c

8

.
8

8
That means that the key point
shifts from the origin to  .
8
b2
Use b = 2 to calculate the period of the graph.
period 
2 2


b
2

8
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

8



The increment is ¼ of the period. Since the period for y  2 sin 2 x  
8

1

  or .
is π, the increment is
4
4
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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.
2


 0
8
8
-2

8

3
8

4
5
8
3 

8
4
7
8
9
8
11
8
13 15 17
8
8
8
5 

8
4


y  2 sin 2  x  
8

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What does the graph for the equation y  5 cos
a5
a5
a 5
This means that the
amplitude of the graph is 5.
b
1
x    look like?
2
1
2
c  
Maximums will be at 5.
5
-5
Minimums will be at -5.
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y  5 cos
1
 x   .
2
c  
The phase shift is  .
That means that the key point
shifts from the origin to  .
5

-5
Use b 
period 
1
to calculate the period of the graph.
2
2 2

 4
1
b
2
5

-5
One complete period is highlighted here.
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Remember that the increment (¼ of the period) is the distance between
an x-intercept and a maximum or minimum.
1
Since the period for y  5 cos  x    is 4π, the increment is π.
2
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.
5
 2  0

2
0+π
This is the graph for
1
y  5 cos  x    .
2
-5
-π + π
3 4  5 
π+π
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Sometimes a sine or cosine graph may be shifted up or down. This is
called a vertical shift.
The equation for a sine graph with a vertical shift can be written as
y = a sin b (x - c) +d.
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.
1
Consider the graph for y  5 cos  x     3 .
2
The equation is in the form y = a cos b (x - c) +d.
“d” equals 3, so the vertical shift is 3.
5
1
The graph of y  5 cos  x   
2
was drawn in the previous example.
 2  
 2 3 4  5 
0
-5
y  5 cos
Esc
1
x  
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To draw y  5 cos
1
 x     3 , begin with the graph for y  5 cos 1 x   .
2
2
8
Draw a new horizontal axis at
y = 3.
5
3
2 
0
 2 3 4 5
Then shift the graph up 3 units.
-5
1
y  5 cos  x   
2
3
The graph now represents y  5 cos
1
x     3 .
2
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This concludes
Sine and Cosine Graphs.
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