4. Graphing Trigonometric Functions

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

y = -2 sin x

Match each function in Column I with the appropriate description in Column II.

I

y = 3 sin(2x - 4)

II

A. amplitude = 2, period = π/2, phase shift = ¾

B. amplitude = 3, period = π, phase shift = 2

C. amplitude = 4, period = 2π/3, phase shift = ⅔

D. amplitude = 2, period = 2π/3, phase shift = 4⁄3

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

The graph of y = cos (x - π/6) is obtained by shifting the graph of y = cos x ______ unit(s) to the ________ (right/left).

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 period of this motion?

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

y = sin ⅔ x

Graph each function over a one-period interval.

y = -2 cos x

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

y = sin 3x

Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.

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Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.

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

y = 2 sin ¼ x

Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.

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

The graph of y = 4 sin x is obtained by stretching the graph of y = sin x vertically by a factor of ________.

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

y = 2 sin (x + π)

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

y = -¼ cos (½ x + π/2)

Decide whether each statement is true or false. If false, explain why.

The graph of y = sec x in Figure 37 suggests that sec(-x) = sec x for all x in the domain of sec x.