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Pre-Calc Exam Notes 116

Pre-Calc Exam Notes 116 - angle goes from 0 to 2 π Here we...

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116 Chapter 5 Graphing and Inverse Functions §5.2 Generalizing Example 5.9, an expression of the form a sin ω x + b cos ω x is equivalent to radicallow a 2 + b 2 sin ( x + θ ), where θ is an angle such that cos θ = a radicallow a 2 + b 2 and sin θ = b radicallow a 2 + b 2 . So y = a sin ω x + b cos ω x will have amplitude radicallow a 2 + b 2 . Note that this method only works when the angle ω x is the same in both the sine and cosine terms. We have seen how adding a constant to a function shifts the entire graph vertically. We will now see how to shift the entire graph of a periodic curve horizontally. x y 0 A A π ω 2 π ω period = 2 π ω Figure 5.2.10 y = A sin ω x Consider a function of the form y = A sin ω x , where A and ω are nonzero constants. For simplicity we will assume that A > 0 and ω > 0 (in general either one could be nega- tive). Then the amplitude is A and the period is 2 π ω . The graph is shown in Figure 5.2.10. Now consider the function y = A sin ( ω x φ ), where φ is some constant. The amplitude is still A , and the period is still 2 π ω , since ω x φ is a linear function of x . Also, we know that the sine function goes through an entire cycle when its
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Unformatted text preview: angle goes from 0 to 2 π . Here, we are taking the sine of the angle ω x − φ . So as ω x − φ goes from 0 to 2 π , an entire cycle of the function y = A sin ( ω x − φ ) will be traced out. That cycle starts when ω x − φ = ⇒ x = φ ω and ends when ω x − φ = 2 π ⇒ x = 2 π ω + φ ω . Thus, the graph of y = A sin ( ω x − φ ) is just the graph of y = A sin ω x shifted horizontally by φ ω , as in Figure 5.2.11. The graph is shifted to the right when φ > 0, and to the left when φ < 0. The amount φ ω of the shift is called the phase shift of the graph. x y A − A 2 π ω + φ ω φ ω period = 2 π ω phase shift (a) φ > 0: right shift x y A − A 2 π ω + φ ω φ ω period = 2 π ω phase shift (b) φ < 0: left shift Figure 5.2.11 Phase shift for y = A sin ( ω x − φ )...
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