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M1410405Part2

# M1410405Part2 - 4.41 PART E PERIOD We now consider the...

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4.41 PART E: PERIOD We now consider the forms f x ( ) = a sin bx ( ) and f x ( ) = a cos bx ( ) . In Sections 4.5 and 4.6 , we assume that b is a positive real number. “Accordion Effects” Recall from Section 1.6: Notes 1.62 that, if x replaced by bx , then the corresponding graph is: horizontally squeezed if b > 1 horizontally stretched if 0 < b < 1 Warning: This is the “opposite” of how a affects vertical stretching and squeezing. The period of y = a sin x or y = a cos x is 2 π . Observe that a has no effect on the period. More generally, the period of y = a sin bx ( ) or y = a cos bx ( ) is 2 b . We assume b > 0 ; otherwise, the period is 2 b . Technical Note: Observe that we obtain exactly one cycle on the x -interval x 0 bx 2 { } , or, equivalently, x 0 x 2 b . Interpretations of b Again, we assume b > 0 . We may think of b as an “aging factor.” (Thanks to Peter Doyle for this idea.) The higher b is, the more rapidly the graph changes (or “ages”). Also, b is the number of cycles one finds on an x -interval of length 2 .

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4.42 What if b < 0? We apply the Even/Odd Properties of the basic trig functions and write a new equation in which b > 0 . See the Example below. The sign of a may change, also. If we fail to apply the Even/Odd Properties, we may be in danger of having to draw our cycle shapes backwards (i.e., from right-to-left). This is something of a “time reversal” idea. Example Graph y = 7sin 3 x ( ) . Solution Because b = 3 < 0 at present, we use the fact that sine is an odd function. y = 7sin 3 x ( ) y = 7 sin 3 x ( ) y = 7sin 3 x ( ) We will focus on our new equivalent equation, which has b = 3 > 0 . Amplitude = a = 7 = 7 Period = 2 π b = 2 3 Observe that we find b = 3 complete cycles on an x -interval of length 2 . “3” is the aging factor.
4.43 Examples (for comparative purposes) Graph of [one cycle of] y = 7sin 3 x ( ) , as in the previous Example: Period = 2 π 3 Graph of y = 7sin x : Period = 2 Graph of y = 7sin 1 3 x , or y = 7sin x 3 : Period = 2 1/ 3 = 2 3 = 6

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4.44 PART F: THE “FRAME METHOD” We can use the “Frame Method” to graph one or more cycles of the graph of y = a sin bx ( ) or
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M1410405Part2 - 4.41 PART E PERIOD We now consider the...

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