4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

4. Graphing Trigonometric Functions

# Graphs of the Sine and Cosine Functions - Video Tutorials & Practice Problems

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concept

## Graph of Sine and Cosine Function

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Problem

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

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B

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D

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Problem

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

A

$y=50$

B

$y=51$

C

$y=49$

D

$y=50+\frac{\sqrt3}{2}$

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example

## Example 1

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concept

## Amplitude and Reflection of Sine and Cosine

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Problem

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

A

$y=8$

B

$y=10$

C

$y=-2$

D

$y=12$

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Problem

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

A

B

C

D

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example

## Example 1

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concept

## Period of Sine and Cosine Functions

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Problem

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

A

$b=\pi$

B

$b=2$

C

$b=\frac12$

D

$b=4$

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Problem

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

A

$b=\frac{1}{10}$

B

$b=10$

C

$b=20$

D

$b=\frac{1}{20}$

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PRACTICE PROBLEMS AND ACTIVITIES (114)

- In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same ...
- In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same ...
- An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follo...
- Fill in the blank(s) to correctly complete each sentence.The graph of y = sin (x + π/4) is obtained by shiftin...
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = 3 - ¼ ...
- Match each function with its graph in choices A–I. (One choice will not be used.)y = cos (x - π/4)A. <IMAGE...
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = 3 cos ...
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = -sin (...
- 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 xA. <...
- 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 ...
- Fill in the blank(s) to correctly complete each sentence.The graph of y = cos (x - π/6) is obtained by shiftin...
- An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as foll...
- Match each function in Column I with the appropriate description in Column II.Iy = -4 sin(3x - 2)IIA. amplitud...
- 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 ...
- 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 ...
- 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 ...
- An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follo...
- Fill in the blank(s) to correctly complete each sentence.The graph of y = 4 sin x is obtained by stretching th...
- 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 ...
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = -¼ cos...
- Decide whether each statement is true or false. If false, explain why.The graph of y = sec x in Figure 37 sugg...
- Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.y = -2 sin 2 π...
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = 3 cos ...
- Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.y = ½ cos π ...
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = 2 - si...
- 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 t...
- 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 Examp...
- 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 Examp...
- Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Examp...
- Graph each function over a two-period interval.y = 2 cos (x - π/3)
- Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.<IM...
- Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.<IM...
- Graph each function over a one-period interval. See Example 3.y = (3/2) sin [2(x + π/4)]
- Graph each function over a one-period interval.y = -4 sin(2x - π)
- Graph each function over a one-period interval.y = (1/2) cos ((1/2)x - π/4)
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = 2 sin ...
- Fill in the blank(s) to correctly complete each sentence.The graph of y = 6 + 3 sin x is obtained by shifting ...
- 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
- Graph each function over a two-period interval.y = -3 + 2 sin (x + π/2)
- Fill in the blank(s) to correctly complete each sentence.The graph of y = -5 + 2 cos x is obtained by shifting...
- 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...
- Fill in the blank(s) to correctly complete each sentence.The graph of y = 3 + 5 cos (x + π/5) is obtained by s...
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = 2 sin ...
- Fill in the blank(s) to correctly complete each sentence.The graph of y = -2 + 3 cos (x - π/6) is obtained by...
- Match each function with its graph in choices A–I. (One choice will not be used.)y = sin (x - π/4)A. <IMAGE...
- For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = 1 + 2 ...
- In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same ...
- Graph y = 1/2 sin x + cos x, 0 ≤ x ≤ 2π.
- In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function....
- In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function....
- In Exercises 12–13, use a vertical shift to graph one period of the function. y = 2 cos 1/3 x − 2
- In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function....
- In Exercises 14–15, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = sin x +...
- Graph each function. See Examples 1 and 2. ƒ(x) = 3|x|
- In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period o...
- Graph each function. See Examples 1 and 2. ƒ(x) = ⅔ |x|
- Graph each function. See Examples 1 and 2. g(x) = 2x²
- In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period o...
- Graph each function. See Examples 1 and 2. g(x) = ½ x²
- In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period o...
- Graph each function. See Examples 1 and 2. ƒ(x) = -½ x²
- Graph each function. See Examples 1 and 2. ƒ(x) = -3|x|
- Graph each function. See Examples 1 and 2. h(x) = |-½ x|
- Graph each function. See Examples 1 and 2. h(x) = √4x
- Graph each function. See Examples 1 and 2. ƒ(x) = -√-x
- Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with ...
- Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with ...
- Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with ...
- Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = -x³ + 2x
- Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = 0.5x⁴ - 2x² + 6
- In Exercises 53–60, use a vertical shift to graph one period of the function. y = sin x + 2
- Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = x³ - x + 9
- Determine whether each function is even, odd, or neither. See Example 5. 1 ƒ(x) = x + —— x⁵
- In Exercises 53–60, use a vertical shift to graph one period of the function. y = cos x + 3
- Graph each function. See Examples 6 – 8. ƒ(x) = x² - 1
- In Exercises 53–60, use a vertical shift to graph one period of the function. y = −3 sin 2πx + 2
- Graph each function. See Examples 6 – 8. g(x) = (x - 4)²
- In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = 3 cos x...
- In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x +...
- Graph each function. See Examples 6 – 8. g(x) = |x| - 1
- In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x +...
- Graph each function. See Examples 6 – 8. h(x) = -(x + 1)³
- In Exercises 67–68, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 4. y = cos πx +...
- Graph each function. See Examples 6 – 8. h(x) = 2x² - 1
- Graph each function. See Examples 6 – 8. ƒ(x) = 2(x - 2)² - 4
- Graph each function. See Examples 6 – 8. ƒ(x) = √x + 2
- Graph each function. See Examples 6 – 8. ƒ(x) = √-x
- Graph each function. See Examples 6 – 8. _ ƒ(x) = 2√x + 1
- Graph each function. See Examples 6 – 8. g(x) = ½ x³ - 4
- In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the gra...
- In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the gra...