Sine and Cosine Graphs

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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.
This is the graph for y cos x.
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y sin x
One complete period is highlighted on each of
these graphs.
y cos x
For both y sin x and y cos x, the period is
2p. (From the beginning of a cycle to the end of
that cycle, the distance along the x-axis is 2p.)
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y sin x
Amplitude deals with the height of the graphs.
1 -1
y cos x
1 -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.
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.
1 -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.
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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 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.
3 2 1 -1 -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.
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
Conversely, when you already know the period of a
sine or cosine graph, b can be determined by
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The period for this graph is .
Use the period to calculate b.
2 1 -1 -2
Notice that a 2 on this graph since the graph
extends 2 units above the x-axis.
Since and a 2, the sine equation for
this graph is
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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.
y sin x
1 -1
This sine graph moved units to the right. c,
the phase shift, is .
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y cos x
1 -1
This cosine graph above moved units to
the left. c, the phase shift, is .
An equation for this graph can be written as
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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 -1 -2 -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 -1 -2 -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
The key point has shifted to , so the
phase shift is
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y a sin b (x c)
4 3 2 1 -1 -2 -3 -4
a 4
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 -1 -2 -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
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 -1 -2 -3 -4
a 4
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.
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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.
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The phase shift is That means that the key
point shifts from the origin to
Use b 2 to calculate the period of the graph.
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 p, the increment is
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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 -2
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What does the graph for the equation
look like?
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
Use to calculate the period of the
graph.
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. Since the period for is
4p, the increment is p. Dont forget that
x-intercepts, maximums, and minimums can be
labeled by beginning at the phase shift and
adding one increment at a time.
This is the graph for
-p p
0 p
p p
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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.
• 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.

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To draw , begin with the graph for
Draw a new horizontal axis at y 3. Then
shift the graph up 3 units.
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5 -5
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The graph now represents
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This concludesSine and Cosine Graphs.