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Next Back Esc 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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Next Back Esc This is the graph for y = sin x. This is the graph for y = cos x.

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Next Back Esc 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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Next Back Esc 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. 1 1

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Next Back Esc 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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Next Back Esc A sine graph has a phase shift if the key point is shifted to the left or to the right.

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Next Back Esc 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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Next Back Esc A cosine graph has a phase shift if the key point is shifted to the left or to the right.

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Next Back Esc 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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Next Back Esc 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. 3 2 1 -2 -3

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Next Back Esc 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 -2

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Next Back Esc 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. 3 2 1 -2 -3

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Next Back Esc 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. 1

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Next Back Esc 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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Next Back Esc 2 1 -2 The period for this graph is. 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 Use the period to calculate b.

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Next Back Esc 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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Next Back Esc 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. This sine graph moved units to the right. “c”, the phase shift, is. An equation for this graph can be written as 1 y = sin x

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Next Back Esc This cosine graph above moved units to the left. “c”, the phase shift, is. An equation for this graph can be written as 1 y = cos x

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Next Back Esc 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). 4 3 2 1 -2 -3 -4

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Next Back Esc 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 The key point has shifted to, so the phase shift is 4 3 2 1 -2 -3 -4

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Next Back Esc y = a sin b (x – c) a = 4 4 3 2 1 -2 -3 -4 This is an equation for the graph written as a sine function.

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Next Back Esc 4 3 2 1 -2 -3 -4 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 The key point has not shifted, so there is no phase shift. That means that c = 0.

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Next Back Esc a = 4 y = a cos b (x – c) 4 3 2 1 -2 -3 -4 This is an equation for the graph written as a cosine function.

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Next Back Esc 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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Next Back Esc 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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Next Back Esc 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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Next Back Esc 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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Next Back Esc 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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Next Back Esc 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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Next Back Esc 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. 5 -5 5 -5

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Next Back Esc 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. -π + π This is the graph for 0 + ππ + π 5 -5

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Next Back Esc 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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Next Back Esc 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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Next Back Esc To draw, begin with the graph for Draw a new horizontal axis at y = 3. Then shift the graph up 3 units. 5 -5 543202 8 3 The graph now represents

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Next Back Esc This concludes Sine and Cosine Graphs.

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