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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
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.
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3sin2y
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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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y = a sin b (x – c)
a = 42
3b
3c
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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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The key point has not shifted, so there is no phase shift. That means that c = 0.
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a = 42
3b 0c
x2
3cos4yor0x
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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.
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.xsiny
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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.
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The phase shift is
That means that the key point shifts from the origin to
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2b
Use b = 2 to calculate the period of the graph.
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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
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2sin2 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
.
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Use to calculate the period of the graph.
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bperiod
One complete period is highlighted here.
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1cos5 xy
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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.
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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.
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To draw , begin with the graph for 32
1cos5 xy .x
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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 concludesSine and Cosine Graphs.