Learn how to draw graphs of trigonometric graphs, with transformations (stretch, shifts, reflections)

1. Sine and Cosine Graphs
Reading and Drawing
Sine and Cosine Graphs
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2. This is the graph for y = sin x.
− 2π −
3π
2
−π
−
π
2
π
2
0
3π
2
π
2π
This is the graph for y = cos x.
− 2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
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2π
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3. y = sin x
− 2π −
3π
2
−π
−
π
2
One complete period is
highlighted on each of
these graphs.
0
π
2
π
3π
2
2π
y = cos x
− 2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
2π
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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4. y = sin x
1
− 2π −
3π
2
−π
−
π
2
Amplitude deals with the
height of the graphs.
0
π
2
π
3π
2
2π
-1
y = cos x
1
− 2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
2π
-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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5. For y = sin x, there is no phase shift.
− 2π −
3π
2
−π
−
π
2
0
π
2
π
3π
2
2π
The y-intercept is located at the point (0,0).
We will call that point, the key point.
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6. − 2π −
3π
2
−π −
π
2
0
π
2
π
3π
2
2π
A sine graph has a phase shift if the key point
is shifted to the left or to the right.
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7. For y = cos x, there is no phase shift.
1
− 2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
2π
-1
The y-intercept is located at the point (0,1).
We will call that point, the key point.
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8. A cosine graph has a phase shift if the key point
is shifted to the left or to the right.
− 2π
−
3π
2
−π
−
π
2
0
π
2
π
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3π
2
2π
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9. 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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10. 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.
3
2
Consider this sine graph. Since
the height of this graph is 3, then
a = 3.
1
− 2π −
The equation for this graph can be
written as y = 3 sin x.
3π
π
− π − -1 0
2
2
-2
π
2
π
3π
2
2π
-3
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11. Consider this cosine graph. The height of this graph is 2, so a = 2.
2
1
− 2π
−
3π
2
−π
−
π
0
2 -1
π
2
π
3π
2
2π
-2
The equation for this graph can be written as y = 2 cos x.
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12. If a sine graph is “flipped” over the x-axis, the value of a will be negative.
3
2
1
− 2π −
3π
2
−π
−
π -1
0
2
-2
π
2
π
3π
2
2π
-3
For the graph above, a = -3.
An equation for this graph is y = -3 sin x.
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13. If a cosine graph is “flipped” over the x-axis, the value of a will be negative.
1
− 2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
2π
-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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14. 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
2π
period =
.
b
Conversely, when you already know the period of a sine or cosine
graph, b can be determined by
2π
b=
.
period
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15. The period for this graph is
4π
.
3
Use the period to calculate b.
2
1
−
4π
3
−π −
2π
3
−
π
3
0
-1
π
3
2π
3
π
4π
3
b=
( 2π ) = 3
2π
=
period  4π  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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16. − 2π −
3π
2
−π −
π
2
0
π
2
π
3π
2
2π
A sine graph has a phase shift
if its key point has shifted to the
left or to the right.
− 2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
2π
A cosine graph has a phase shift
if its key point has shifted to the
left or to the right.
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17. 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
−
3π
2
−π −
π
2
This sine graph moved
0
π
2
π
3π
2
2π
5π
2
π
2
units to the right. “c”, the phase
π
shift, is
.
2
-1


π
2
An equation for this graph can be written as y = sin  x − .
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18. y = cos x
1
−
5π
3π
π
− 2π −
−π −
2
2
2
0
π
2
π
3π
2
2π
-1
This cosine graph above moved
“c”, the phase shift, is −
π
units to the left.
2
π
.
2
An equation for this graph can be written as

π
 π 

y = cos x −  −   or y = cos x +  .


2
 2 


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19. Graphs whose equations can be written as a sine function can also be
written as a cosine function.
4
3
2
1
−
4π
−π
3
−
2π
3
−
π
3
-1
-2
π
3
2π
3
π
4π
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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20. y = a sin b (x – c)
4
3
2
1
−
4π
−π
3
−
2π
3
−
π -1
3 -2
π
3
2π
3
π
4π
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
4π
.
3
The key point has shifted to −
b=
2π
2π 3
=
= .
period 4 π 2
3
b=
3
.
2
π
π
π
− . c=− .
, so the phase shift is
3
3
3
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21. y = a sin b (x – c)
4
3
2
1
−
4π
−π
3
−
2π
3
−
π
3
π
3
-1
-2
2π
3
π
4π
3
-3
-4
a=4
b=
3
2
π
c=−
3
3
 π 
y = 4 sin  x −  −  

2
 3 

or
3
π
y = 4 sin  x + 
2
3
This is an equation for the graph written as a sine function.
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22. y = a cos b (x – c)
4
3
2
1
−
4π
−π
3
−
2π
3
−
π
3
π
3
-1
-2
2π
3
π
4π
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
4π
.
3
b=
2π
2π 3
=
= .
period 4 π 2
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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23. y = a cos b (x – c)
4
3
2
1
−
4π
−π
3
−
2π
3
−
π
3
-1
-2
π
3
2π
3
π
4π
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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24. It is important to be able to draw a sine graph when you are given the
corresponding equation. Consider the equation
π

y = −2 sin 2  x −  .
8

Begin by looking at a, b, and c.
π

y = −2 sin 2  x −  .
8

a = −2
b=2
c=
π
8
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25. π

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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26. π

y = −2 sin 2 x − 
8

The phase shift is
c=
π
8
π
.
8
π
8
That means that the key point
π
shifts from the origin to .
8
b=2
Use b = 2 to calculate the period of the graph.
period =
2π 2π
=
=π
b
2
π
8
One complete period is highlighted here.
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27. 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
π
8
π


The increment is ¼ of the period. Since the period for y = −2 sin 2 x − 
8

1
π
( π) or .
is π, the increment is
4
4
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28. To label the graph, begin at the phase shift. Add one increment at a time
to label x-intercepts, maximums, and minimums.
2
π
π
− 0
8
8
-2
π π
+
8 4
3π
8
5π
8
3π π
+
8
4
7π
8
9π
8
11π
8
13π
8
15 π 17π
8
8
5π π
+
8
4
π

y = − 2 sin 2  x − 
8

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29. 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 + π ) look like?
2
1
2
c = −π
Maximums will be at 5.
5
-5
Minimums will be at -5.
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30. y = 5 cos
1
( x + π) .
2
c = −π
The phase shift is − π.
That means that the key point
shifts from the origin to − π.
5
−π
-5
Use b =
period =
1
to calculate the period of the graph.
2
2π 2π
=
= 4π
1
b
2
One complete period is highlighted here.
5
−π
-5
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31. 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 + π ) 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
− 2π −π π 0
−
π
2π
-5
-π + π
0+π
3π
4π
5π
This is the graph for
1
y = 5 cos ( x + π ) .
2
π+π
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32. 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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33. 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 2 ( x + π ) + 3 .
The equation is in the form y = a cos b (x - c) +d.
“d” equals 3, so the vertical shift is 3.
5
1
y = 5 cos ( x + π )
The graph of
2
was drawn in the previous example.
− 2π − π
π
0
2π
3π 4 π 5 π
-5
y = 5 cos
Esc
1
( x + π)
2
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34. 1
( x + π ) + 3 , begin with the graph for y = 5 cos 1 ( x + π ) .
To draw y = 5 cos
2
2
8
Draw a new horizontal axis at
y = 3.
5
3
− 2π −π
0
π 2π 3π 4π 5π
Then shift the graph up 3 units.
-5
1
y = 5 cos ( x + π )
2
The graph now represents
+3
y = 5 cos
1
( x + π) + 3 .
2
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35. This concludes
Sine and Cosine Graphs.
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