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Graphs of Sine and Cosine Functions Objective: To recognize a sine, cosine, or tangent graph and to graph one yourself. Sin Curve • Lets us our knowledge of the unit circle to try and come up with a graph for y = sinx. Sin Curve • Lets us our knowledge of the unit circle to try and come up with a graph for y = sinx. • The amplitude is the distance from the middle to the highest or lowest point. You can find the amplitude by sight or by taking ½ (Max – Min). For a standard sine curve, the amplitude is 1. Sin Curve • Lets us our knowledge of the unit circle to try and come up with a graph for y = sinx. • The sine function is periodic, meaning it repeats itself over and over again over a certain amount of time, or period. The amount of time it takes to repeat is the period. To find the period, find the distance from a peak to a peak. The period of y = sinx is 2p. Cosine Curve • Again, lets use our knowledge of the unit circle to graph the cosine curve. Cosine Curve • Again, lets use our knowledge of the unit circle to graph the cosine curve. • The period and amplitude are the same as the sine curve. The period is 2p and the amplitude is 1. Cosine Curve • Again, lets use our knowledge of the unit circle to graph the cosine curve. • The period and amplitude are the same as the sine curve. This curve is actually the same exact graph as the sine curve only shifted over p/2 units. Negative Sine Curve • Again, lets use our knowledge of the unit circle to graph negative curves. y sin x y sin x • A negative sine or cosine curve is the exact same, only a reflection on the y-axis. In other words, upside down. It will have the same period and amplitude. Negative Cosine Curve • Again, lets use our knowledge of the unit circle to graph negative curves. • A negative cosine curve is the exact same, only a reflection on the y-axis. In other words, upside down. It will have the same period and amplitude. Sine/Cosine Curves • Find the period and amplitude for the following graph. 2p p 2 p 3p 2 • Sometimes the x-axis scale has integers and sometimes it has multiples of pi. Sine/Cosine Curves • Find the period and amplitude for the following graph. 2p p 2 p 3p 2 y 2 sin 2 x amplitude 2 period p • Sometimes the x-axis scale has integers and sometimes it has multiples of pi. Sine/Cosine Curves • Find the period and amplitude for the following graph. 1 2 3 4 • Sometimes the x-axis scale has integers and sometimes it has multiples of pi. Sine/Cosine Curves • Find the period and amplitude for the following graph. y 3 cos px 1 2 3 4 amplitude 3 period 2 • Sometimes the x-axis scale has integers and sometimes it has multiples of pi. General Form • We can have curves with a different period or amplitude. We need to add something to our equations. The general form of the equations of the sine and cosine curves now is: y = A sin bx • The amplitude is the |A|. • The period is 2p/b. y = A cos bx General Form • When the amplitude is increased (|A| > 1) it is called a vertical stretch. When it is decreased (-1 < A <1) it is called a vertical shrink. y = A sin bx • The amplitude is the |A|. • The period is 2p/b. y = A cos bx Vertical Shrink/Stretch • Look at the graph to see the vertical shrink/stretch. Notice that the period is unchanged. Vertical Shrink/Stretch • Look at the graph to see the vertical shrink/stretch. Notice that the period is unchanged. General Form • When the period is increased it is called a horizontal stretch. When it is decreased it is called a horizontal shrink. y = A sin bx • The amplitude is the |A|. • The period is 2p/b. y = A cos bx Horizontal Stretch/Shrink • Look at this equation and graph. B in this equation is ½, so the period is 2p/.5 or 2p(2) = 4p. The amplitude is unchanged since A = 1. Period/Amplitude • Given the equation y = 3 sin 2x, find the amplitude and period. Period/Amplitude • Given the equation y = 3 sin 2x, find the amplitude and period. • The amplitude is 3. • The period is 2p/2 or p. Period/Amplitude • Given the equation y = -2 cospx, find the amplitude and period. Period/Amplitude • Given the equation y = -2 cospx, find the amplitude and period. • The amplitude is 2. • The period is 2p/p = 2. General Form • Now, lets look at how to graph these two equations. y = 2 sin 2x y = -3 cos px General Form • Now, lets look at how to graph these two equations. y = -3 cos px y = 2 sin 2x p 2 p 3p 2 2p 1 2 3 4 General Form • Now, you try to graph these two equations. y = -2 sin 4x y = 3 cos (p/2)x General Form • Now, you try to graph these two equations. y = 3 cos (p/2)x y = -2 sin 4x p 4 p 2 3p 4 p 1 2 3 4 Homework • Pages 490-491 • 1-13 odd, 23, 24, 35-42 all