QUATER-1-PE-HEALTH-LC2- this is just a sample of unpacked lesson
Trigonometric Functions and their Graphs
1. 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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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
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1
( x + π)
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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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