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.

This is the graph for y = sin x. This is the graph for y = cos x.

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.)

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.

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.

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

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.

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

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)

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.

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

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.

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.

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

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

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.

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

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

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).

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

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.

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.

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.

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.

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

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.

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

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

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

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.

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

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.

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

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

This concludes Sine and Cosine Graphs.