Reading and Drawing Sine and Cosine Graphs
This presentation will focus on how to read and draw sine and cosine graphs. Some slides in the presentation contain animation which will help in understanding the concepts. It is recommended to allow
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About Reading and Drawing Sine and Cosine Graphs
PowerPoint presentation about 'Reading and Drawing Sine and Cosine Graphs'. This presentation describes the topic on This presentation will focus on how to read and draw sine and cosine graphs. Some slides in the presentation contain animation which will help in understanding the concepts. It is recommended to allow. The key topics included in this slideshow are . Download this presentation absolutely free.
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Slide1Next 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.
Slide2Next Back Esc This is the graph for y = sin x . This is the graph for y = cos x .
Slide3Next 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 2 π . (From the beginning of a cycle to the end of that cycle, the distance along the x-axis is 2 π .)
Slide4Next 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 1 -1
Slide5Next 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 .
Slide6Next Back Esc A sine graph has a phase shift if the key point is shifted to the left or to the right.
Slide7Next Back Esc 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 .
Slide8Next Back Esc A cosine graph has a phase shift if the key point is shifted to the left or to the right.
Slide9Next 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 )
Slide10Next 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 -1 -2 -3
Slide11Next 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 -1 -2
Slide12Next 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 -1 -2 -3
Slide13Next 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 -1
Slide14Next 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
Slide15Next Back Esc 2 1 -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 .
Slide16Next 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.
Slide17Next 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 -1 y = sin x
Slide18Next 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 -1 y = cos x
Slide19Next 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 -1 -2 -3 -4
Slide20Next 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 -1 -2 -3 -4
Slide21Next Back Esc y = a sin b (x – c ) a = 4 4 3 2 1 -1 -2 -3 -4 This is an equation for the graph written as a sine function.
Slide22Next Back Esc 4 3 2 1 -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.
Slide23Next Back Esc a = 4 y = a cos b (x – c ) 4 3 2 1 -1 -2 -3 -4 This is an equation for the graph written as a cosine function.
Slide24Next 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 .
Slide25Next 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
Slide26Next 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.
Slide27Next 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
Slide28Next 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
Slide29Next 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
Slide30Next 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
Slide31Next 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 4 π , 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 + π π + π 5 -5
Slide32Next 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.
Slide33Next 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
Slide34Next 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 5 4 3 2 0 2 8 3 The graph now represents
Slide35Next Back Esc This concludes Sine and Cosine Graphs.