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4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

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Graph of Sine and Cosine Function

Graph of Sine and Cosine Function

Amplitude and Reflection of Sine and Cosine

Amplitude and Reflection of Sine and Cosine

Period of Sine and Cosine Functions

Period of Sine and Cosine Functions

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What are the critical points of a sine and cosine graph

What are the critical points of a sine and cosine graph

Graphing Sin and Cos

Graphing Sin and Cos

Mario's Math Tutoring

The Graph of the Function f(x) =sin(x)

The Graph of the Function f(x) =sin(x)

The Math Sorcerer

Graphing the sin(x)

Graphing the sin(x)

MooMooMath and Science

How do you graph y=sinx and y=cosx?

How do you graph y=sinx and y=cosx?

How do you find the amplitude of the sine and cosine graph

How do you find the amplitude of the sine and cosine graph

Amplitude and Period of Sine and Cosine

Amplitude and Period of Sine and Cosine

Graphing Sine and Cosine Functions

Graphing Sine and Cosine Functions

Graphing a Sine Function by Finding the Amplitude and Period

Graphing a Sine Function by Finding the Amplitude and Period

How do you graph y=sinx and y=cosx?

How do you graph y=sinx and y=cosx?

Graphing Sin and Cos

Graphing Sin and Cos

Mario's Math Tutoring

Graph of y=cosx

Graph of y=cosx

Carolee Pederson

How do you find the amplitude of the sine and cosine graph

How do you find the amplitude of the sine and cosine graph

Amplitude and Period of Sine and Cosine

Amplitude and Period of Sine and Cosine

Graphing a Trig Function with Cosine

Graphing a Trig Function with Cosine

Graphing Sine and Cosine Functions

Graphing Sine and Cosine Functions

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Practice this topic

Multiple Choice

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

Multiple Choice

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

Multiple Choice

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

Multiple Choice

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

Multiple Choice

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

Multiple Choice

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

Textbook Question

In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 4 sin x

Textbook Question

In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 1/3 sin x

Textbook Question

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?

Textbook Question

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).

Textbook Question

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

y = 3 - ¼ cos ⅔ x

Textbook Question

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

y = cos (x - π/4)

A. <IMAGE> B. <IMAGE> C. <IMAGE>

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

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

Textbook Question

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

y = 3 cos (x + π/2)

Textbook Question

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

y = -sin (x - 3π/4)

Textbook Question

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

y = 2 cos x

Textbook Question

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

y = ⅔ sin x

Textbook Question

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

y = -1 + cos x

A. <IMAGE> B. <IMAGE> C. <IMAGE>

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

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

Textbook Question

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

y = -2 sin x

Textbook Question

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

Textbook Question

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).

Textbook Question

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?

Textbook Question

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

Textbook Question

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

y = sin ⅔ x

Textbook Question

Graph each function over a one-period interval.

y = -2 cos x

Textbook Question

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

y = sin 3x

Textbook Question

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.

<IMAGE>

Textbook Question

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.

<IMAGE>

Textbook Question

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

y = 2 sin ¼ x

Textbook Question

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.

<IMAGE>

Textbook Question

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?

Textbook Question

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 ________.

Textbook Question

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

y = -2 cos 3x

Textbook Question

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

y = 2 sin (x + π)

Textbook Question

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

y = -¼ cos (½ x + π/2)

Textbook Question

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.

Textbook Question

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

y = -2 sin 2 πx

Textbook Question

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

y = 3 cos [π/2 (x - ½)]

Textbook Question

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

y = ½ cos π x

2

Textbook Question

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

y = 2 - sin(3x - π/5)

Textbook Question

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

y = π sin πx

Textbook Question

Graph each function over a two-period interval.

y = cos (x - π/2 )

Textbook Question

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.

Textbook Question

Graph each function over a one-period interval.

y = -½ cos (πx - π)

Textbook Question

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.

<IMAGE>

Textbook Question

Graph each function over a two-period interval.

y = sin (x + π/4)

Textbook Question

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.

<IMAGE>

Textbook Question

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.

<IMAGE>

Textbook Question

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

<IMAGE>

Textbook Question

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.

<IMAGE>

Textbook Question

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

<IMAGE>

Textbook Question

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.

<IMAGE>

Textbook Question

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

y = (3/2) sin [2(x + π/4)]

Textbook Question

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.

<IMAGE>

Textbook Question

Graph each function over a one-period interval.

y = -4 sin(2x - π)

Textbook Question

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

y = 2 sin 2x

Textbook Question

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).

Textbook Question

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

y = -3 + 2 sin x

Textbook Question

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

y = -1 - 2 cos 5x

Textbook Question

Graph each function over a two-period interval.

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

Textbook Question

Graph each function over a two-period interval.

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

Textbook Question

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).

Textbook Question

Graph each function over a two-period interval.

y = sin [2(x + π/4) ] + 1/2

Textbook Question

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

y = -½ cos 3x

Textbook Question

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)

Textbook Question

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

y = 2 sin 5x

Textbook Question

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)

Textbook Question

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

y = sin (x - π/4)

A. <IMAGE> B. <IMAGE> C. <IMAGE>

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

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

Textbook Question

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

y = 1 + 2 sin ¼ x

Textbook Question

In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = -3 sin x

Textbook Question

Graph y = 1/2 sin x + cos x, 0 ≤ x ≤ 2π.

Textbook Question

In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function. y = 3 sin 1/2 x

Textbook Question

In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function. y = -3 sin 2πx

Textbook Question

In Exercises 12–13, use a vertical shift to graph one period of the function. y = 2 cos 1/3 x − 2

Textbook Question

In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function. y = -sin 2/3 x

Textbook Question

In Exercises 14–15, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = sin x + cos 1/2 x

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x) = 3|x|

Textbook Question

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

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x) = ⅔ |x|

Textbook Question

Graph each function. See Examples 1 and 2. g(x) = 2x²

Textbook Question

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 3 sin(2x − π)

Textbook Question

Graph each function. See Examples 1 and 2. g(x) = ½ x²

Textbook Question

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 1/2 sin(x + π/2)

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x) = -½ x²

Textbook Question

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)

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x) = -3|x|

Textbook Question

Graph each function. See Examples 1 and 2. h(x) = |-½ x|

Textbook Question

Graph each function. See Examples 1 and 2. h(x) = √4x

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x) = -√-x

Textbook Question

Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (a) x-axis (5, -3)

Textbook Question

Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (b) y-axis (5, -3)

Textbook Question

Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (c) origin. (5, -3)

Textbook Question

Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (a) x-axis (-4, -2)

Textbook Question

Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (b) y-axis. (-4, -2)

Textbook Question

Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (c) origin. (-4, -2)

Textbook Question

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = -x³ + 2x

Textbook Question

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = 0.5x⁴ - 2x² + 6

Textbook Question

In Exercises 53–60, use a vertical shift to graph one period of the function. y = sin x + 2

Textbook Question

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = x³ - x + 9

Textbook Question

Determine whether each function is even, odd, or neither. See Example 5. 1 ƒ(x) = x + —— x⁵

Textbook Question

In Exercises 53–60, use a vertical shift to graph one period of the function. y = cos x + 3

Textbook Question

Graph each function. See Examples 6 – 8. ƒ(x) = x² - 1

Textbook Question

In Exercises 53–60, use a vertical shift to graph one period of the function. y = −3 sin 2πx + 2

Textbook Question

Graph each function. See Examples 6 – 8. g(x) = (x - 4)²

Textbook Question

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = 3 cos x + sin x

Textbook Question

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x + cos 2x

Textbook Question

Graph each function. See Examples 6 – 8. g(x) = |x| - 1

Textbook Question

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x + sin 2x

Textbook Question

Graph each function. See Examples 6 – 8. h(x) = -(x + 1)³

Textbook Question

In Exercises 67–68, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 4. y = cos πx + sin π/2 x

Textbook Question

Graph each function. See Examples 6 – 8. h(x) = 2x² - 1

Textbook Question

Graph each function. See Examples 6 – 8. ƒ(x) = 2(x - 2)² - 4

Textbook Question

Graph each function. See Examples 6 – 8. ƒ(x) = √x + 2

Textbook Question

Graph each function. See Examples 6 – 8. ƒ(x) = √-x

Textbook Question

Graph each function. See Examples 6 – 8. _ ƒ(x) = 2√x + 1

Textbook Question

Graph each function. See Examples 6 – 8. g(x) = ½ x³ - 4

Textbook Question

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)

Textbook Question

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) = cos x, g(x) = sin 2x, h(x) = (f − g)(x)

Textbook Question

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −2 sin(2πx + 4π)

Textbook Question

In Exercises 31–34, determine the amplitude of each function. Then graph the function and y = cos x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 2 cos x

Textbook Question

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = cos 2x

Textbook Question

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = 4 cos 2πx

Textbook Question

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = -4 cos 1/2 x

Textbook Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = cos(x − π/2)

Textbook Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 3 cos(2x − π)

Textbook Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 1/2 cos (3x + π/2)

Textbook Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −3 cos (2x − π/2)

Textbook Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 2 cos (2πx + 8π)

Textbook Question

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

Textbook Question

In Exercises 75–78, graph one period of each function. y = −|3 sin πx|

Textbook Question

In Exercises 1–6, determine the amplitude and period of each function. Then graph one period of the function. y = 3 sin 4x

Textbook Question

In Exercises 1–6, determine the amplitude and period of each function. Then graph one period of the function. y = 1/2 sin π/3 x

Textbook Question

In Exercises 7–11, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −3 cos (x + π)

Textbook Question

In Exercises 7–11, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −3 sin(π/3 x − 3π)

Textbook Question

In Exercises 1–4, graph one period of each function. y = 3 sin 2x

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