Graphing Sine and Cosine

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Peter Aldous McDonald
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2 The problem with average monthly temperatures on the preview worksheet is an example of a periodic function. Periodic functions are defined on p.254 Periodic functions repeat themselves each period. The period is the amount of time it takes the function to complete one cycle or oscillation.

3 The sine function has these properties: The period is 2π. Sine is an odd function. - For all x, sin (-x) = -sin (x). - The graph is symmetric with respect to the origin. The domain is the set of real numbers. The range is [-1,1]. The zeros occur at multiples of π.

4 The cosine function has these properties: The period is 2π. Cosine is an even function. - For all x, cos (-x) = cos (x). - The graph is symmetric with respect to the y-axis. The domain is the set of real numbers. The range is [-1,1]. π The zeros occur at odd multiples of. 2

5 Graph f(x) = x 2 and g(x) = (x 1) without your calculator. Scale a graph by a fourth of the period and graph y = sin x. π Then graph y = sin(x ) π 2

6 Graph f(x) = x 2 and g(x) = x 2 without your calculator. Scale a graph by a fourth of the period and graph y = sin x. π 2 Graph y = sin(x)

7 Graph f(x) = x 2 and g(x) = 2x 2 without your calculator. Scale a graph by a fourth of the period and graph y = sin x. π 2 Graph y = 2 sin(x)

8 Graph y = -sin x Put the period (2π) on the 4 th grid mark to the right of the y-axis. i.e.: scale by onefourth of the period. A reflection of y = sin x over the x-axis.

9 Graph y = 2 cos x Put the period (2π) on the 4 th grid mark to the right of the y-axis. i.e.: scale by one-fourth of the period. y = cos x dilated by 2.

10 Graph y = cos x + 2 Put the period (2π) on the 4 th grid mark to the right of the y-axis. i.e.: scale by one-fourth of the period. y = cos x translated up 2.

11 Graph y = sin (x - π / 4 ) Put the period (2π) on the 4 th grid mark to the right of the y-axis. i.e.: scale by one-fourth of the period. y = sin x translated π / 4 right (one-half unit right if scaled by π / 2 ).

12 A good way to check a sinusoidal graph you ve drawn is to graph it on your calculator in a window you think one cycle should exactly fill. The graph of y = sin (x - π / 4 ) is shown below. Find such a window and graph the equation in that window.

13 AMPLITUDE If a periodic function has maximum value M and minimum value m, amplitude is given by: A 1 2 M m If a function doesn t have both a minimum and maximum value, it has no amplitude. Find the amplitude of y = sin x.

14 PERIOD, AMPLITUDE, PHASE SHIFT, VERTICAL SHIFT The period of f(x) = a sin b(x c) + d is The amplitude is a c causes f(x) = a sin b(x c) + d to be shifted c units horizontally (phase shift). d causes f(x) = a sin b(x c) + d to be shifted d units vertically (vertical shift). The same is true of f(x) = a cos b(x c) + d.

15 Graph y = sin 2x The period is The amplitude There is no phase shift or vertical shift. Notice that at 0 radians, y = sin 2x yields the same y value as y = sin x. π 1 Scale by, one-fourth of the period. The graph goes through one cycle in π units.

16 Graph y = sin 2x

17 Graph y = 2.5 cos 3x The period is The amplitude 2.5 There is no vertical shift or phase shift. Notice that at 0 radians, y = 2.5 cos 3x yields the same y value as y = 2.5 cos x. What scale should be chosen? Scale by, one-fourth of the period.

18 Graph y = 2.5 cos 3x

19 Graph y = 3 sin ½(x + π) + 1 The period is The amplitude 3 Phase shift -π, or π left Vertical shift 4π 1, or 1 up Notice that at 0 radians, y = 3 sin ½x yields the same y value as y = 3 sin x. What scale should be chosen? Scale by π, one-fourth of the period.

20 Graph y = 3 sin ½(x + π) + 1 Graph y = 3 sin ½x, then use translations to graph y = 3 sin ½(x + π) + 1. y = 3 sin ½x y = 3 sin ½(x + π) + 1

21 Find the sine function with amplitude ½, period π, phase shift, and a vertical shift of -1. Graph.

Section 8.4: The Equations of Sinusoidal Functions

Section 8.4: The Equations of Sinusoidal Functions In this section, we will examine transformations of the sine and cosine function and learn how to read various properties from the equation. Transformed