4. Graphing Trigonometric Functions

Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.

y = -2 sin x

In Exercises 37–40, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, graph one period of the equation. Then find the following: a. the maximum displacement b. the frequency c. the time required for one cycle d. the phase shift of the motion. d = 3 cos(πt + π/2)

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = 2 sin (x + π)

An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.

𝒮(t) = 5 cos 2t

What is the frequency?

In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 1/3 sin 2t

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.

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Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.

y = -2 cos 3x

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = -½ cos 3x

In Exercises 75–78, graph one period of each function. y = |2 cos x/2|

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −2 sin(2x + π/2)

Graph each function over a one-period interval.

y = -4 sin(2x - π)

Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.

y = ⅔ sin x

Fill in the blank(s) to correctly complete each sentence.

The graph of y = 6 + 3 sin x is obtained by shifting the graph of y = 3 sin x ________ unit(s) __________ (up/down).

Fill in the blank(s) to correctly complete each sentence.

The graph of y = 3 + 5 cos (x + π/5) is obtained by shifting the graph of y = cos x horizontally ________ unit(s) to the __________, (right/left) stretching it vertically by a factor of ________, and then shifting it vertically ________ unit(s) __________. (up/down)

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.

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