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**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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**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 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 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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**1 -1 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) y = a cos b (x - c)**

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. 3 2 1 -1 -2 -3 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. 2 1 -1 -2

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**An equation for this graph is y = -3 sin x.**

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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**An equation for this graph is y = -1 cos x or just y = - cos x.**

If a cosine graph is “flipped” over the x-axis, the value of a will be negative. 1 -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 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 An equation for this graph can be written as

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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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**The sine function will be written as y = a sin b (x – c). **

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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**For the sine function, the values for a, b, and c must be determined.**

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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**The height of the graph is 4, so a = 4.**

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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**This is an equation for the graph written as a cosine function.**

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. 2 -2 2 -2

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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 π, the increment is

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**To label the graph, begin at the phase shift**

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. 5 -5

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**That means that the key point shifts from the origin to**

The phase shift is That means that the key point shifts from the origin to 5 -5 Use to calculate the period of the graph. 5 -5 One complete period is highlighted here.

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**Since the period for is 4π, the increment is π.**

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. 5 -5 This is the graph for -π + π 0 + π π + π

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**y = a sin b (x - c) +d. y = a cos b (x - c) +d.**

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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**Graph the function without the vertical shift **

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. 5 -5

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**p - 5 4 3 2 To draw , begin with the graph for**

8 Draw a new horizontal axis at y = 3. Then shift the graph up 3 units. 5 -5 3 p - 5 4 3 2 The graph now represents

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**This concludes Sine and Cosine Graphs.**

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