4.5 Graphing Sine and Cosine

Transcription

1 .5 Graphing Sine and Cosine Imagine taking the circumference of the unit circle and peeling it off the circle and straightening it out so that the radian measures from 0 to π lie on the x axis. This is the framework we re going to use to introduce the sine and cosine graphs. ex) Complete the table of values listed below. Below the table is a stylized piece of graph paper suited for plotting the points from the table. π π π x π 3π π 3π π y= sin( x) This is the graph of y= sin( x) : Domain: Range:

2 ex) Complete the table of values listed below. π π π x π 3π π 3π π y= cos( x) This is the graph of y= cos( x) : Domain: Range: The trigonometric functions of sine and cosine are periodic functions. This means they repeat the same output values over a specific interval on the x axis. For sine and cosine the period is The sinusoidal (i.e. wavy ) shape of the sine and cosine graphs can be used to model any phenomenon with a periodic patter to it: motion of a pendulum, ocean tide cycles, even some predator prey relationships. The transformations we ll perform on these graphs will affect the: 1. Amplitude (stretch or shrink vertically). Period (stretch or shrink horizontally) 3. Phase Shift (shift the graphs left or right)

3 1. Amplitude Changes Amplitude is the distance a wave travels above and below its equilibrium point. For graphs of sine and cosine the amplitude can change by using this transformation: y = Asin( x) and y= Acos( x)... where A is a vertical stretch or shrink factor. ex) Graph two full periods (or cycles) of y= 3cos( x) For graphs of sine and cosine you ll need 5 critical points for each period. You ll need to label the critical points on the x axis as well as label the amplitude values on the y axis to scale your graph. 1 ex) Graph two full periods of y= sin( x) 5

4 . Period Changes The period can be changed by causing a horizontal stretch or shrink. The transformations which will change amplitude and period are: y = Asin( Bx) and y= Acos( Bx)... where B will cause the horizontal stretch or shrink. To complete one full period the argument Bx needs to go from 0 to π. Set up the inequality 0 Bx π π... solve for x x This is the period interval. 0 B The period distance can always be calculated by π B. When the graph doesn t involve a horizontal phase shift (coming up next) the period interval can be divided into parts to label each critical point. The increment between each critical point is always ¼ of the period. 1 ex) Graph two full periods of y= sin( x). Amplitude: Period distance: Increment:

5 ex) Graph two full periods of y= 8cos( π x) Amplitude: Period distance: Increment: 3. Phase Shift (Horizontal shifting) The full transformations which will cause amplitude, period and phase shift changes are: y= Asin( Bx C) and y = Acos( Bx C)... where the period interval s endpoints can be found by using the same inequality. Set up the inequality 0 Bx C π... solve for x. The initial period interval from [0,π ] will be changed to a new period interval. After solving the inequality for x, the left value of the inequality will be the new 0 starting point. The period distance is still B π and you ll need to remember to use addition to locate the other critical points with the increment = ¼ of the period to label the x axis critical points.

6 ex) Graph two full periods of y= 1.5cos( x π ) When a phase shift is involved, it s best to use the x axis shown below which has the y axis omitted. Label the critical points first with the shifted 0 point in the center and go one period right and left of this shifted center point. Amplitude: Period distance: Phase Shift Center point: Increment: On phase shift graphs, make sure the phase shift center point is here and graph one period left and one period right of this point. Make sure to label your critical points! AND make sure to determine the correct positioning of the y axis. You need it to indicate amplitude.

7 1 π ex) Graph two full periods of y= sin( x+ ). 3 6 Amplitude: Period distance: Phase Shift Center point: Increment: Don t forget the shifted 0 point goes in the center! Don t forget to use the increment to label all critical points! Don t forget to locate the position of the y axis and label it for amplitude!