4. Graphing Trigonometric Functions

Sketch the function $y=\cos\left(x\right)-1$ on the graph below.

Determine the value of $y=\sin\left(-\frac{\pi}{2}\right)+50$ without using a calculator or the unit circle.

Determine the value of $y=-2\cdot\sin\left(-\frac{3\pi}{2}\right)+10$ without using a calculator or the unit circle.

Graph the function $y=-3\cdot\cos\left(x\right)$.

Given below is the graph of the function $y=\sin\left(bx\right)$. Determine the correct value for b.

The Period for the function $y=\cos\left(bx\right)$ is $T=20\pi$. Determine the correct value of b.

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 amplitude of this motion?

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

The graph of y = sin (x + π/4) is obtained by shifting the graph of y = sin x ______ unit(s) to the ________ (right/left).

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

y = 3 - ¼ cos ⅔ x

Match each function with its graph in choices A–I. (One choice will not be used.)

y = cos (x - π/4)

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D. <IMAGE> E. <IMAGE> F. <IMAGE>

G. <IMAGE> H. <IMAGE> I. <IMAGE>

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

y = 3 cos (x + π/2)

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

y = -sin (x - 3π/4)

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

y = 2 cos x

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

y = ⅔ sin x

Match each function with its graph in choices A–I. (One choice will not be used.)

y = -1 + cos x

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D. <IMAGE> E. <IMAGE> F. <IMAGE>

G. <IMAGE> H. <IMAGE> I. <IMAGE>

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?

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

I

y = -4 sin(3x - 2)

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

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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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?

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 ________.

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 = 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.

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

y = -2 sin 2 πx

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

y = 3 cos [π/2 (x - ½)]

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

y = ½ cos π x

2

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

y = 2 - sin(3x - π/5)

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

y = π sin πx

Graph each function over a two-period interval.

y = cos (x - π/2 )

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

The graph of y = -3 sin x is obtained by stretching the graph of y = sin x by a factor of ________ and reflecting across the ________-axis.

Graph each function over a one-period interval.

y = -½ cos (πx - π)

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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Graph each function over a two-period interval.

y = sin (x + π/4)

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

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

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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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Graph each function over a one-period interval. See Example 3.

y = (3/2) sin [2(x + π/4)]

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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Graph each function over a one-period interval.

y = -4 sin(2x - π)

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

y = 2 sin 2x

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).

Graph each function over a two-period interval. See Example 4.

y = -3 + 2 sin x

Graph each function over a two-period interval. See Example 4.

y = -1 - 2 cos 5x

Graph each function over a two-period interval.

y = 1 - 2 cos ((1/2)x)

Graph each function over a two-period interval.

y = -2 + (1/2) sin 3x

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

The graph of y = -5 + 2 cos x is obtained by shifting the graph of y = 2 cos x ________ unit(s) __________ (up/down).

Graph each function over a two-period interval.

y = sin [2(x + π/4) ] + 1/2

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

y = -½ cos 3x

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)

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

y = 2 sin 5x

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

The graph of y = -2 + 3 cos (x - π/6) 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)

Match each function with its graph in choices A–I. (One choice will not be used.)

y = sin (x - π/4)

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D. <IMAGE> E. <IMAGE> F. <IMAGE>

G. <IMAGE> H. <IMAGE> I. <IMAGE>

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

y = 1 + 2 sin ¼ x