4. Graphing Trigonometric Functions

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 2 πx

Graph one period of each function. y = 3 sin 2x

Determine the amplitude and period of each function. Then graph one period of the function. y = 3 sin (1/2) x

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)

In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = 2 cos x, g(x) = cos 2x, h(x) = (f + g)(x)

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)