Write Sinusoidal Equations with Vertical Shifts Quiz

  • 10th Grade
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Quizzes Created: 7387 | Total Attempts: 9,541,629
| Questions: 20 | Updated: Dec 11, 2025
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1) The function y = −2 sin(4x) − 5 is formed by shifting y = −2 sin(4x). What is the shift, and how does it affect amplitude?

Explanation

Given: y = −2 sin(4x) − 5.

Step 1: −5 outside ⇒ shift down 5.

Step 2: Amplitude remains |−2| = 2.

So, the final answer is shift down 5; amplitude remains 2.

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About This Quiz
Write Sinusoidal Equations With Vertical Shifts Quiz - Quiz

Build equations that sit at the right height. You’ll translate specs like “midline at y = −3” or “max at 7, min at −1” into clean sine or cosine formulas with the correct vertical shift—no guessing, just method.

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2) A cosine function has maximum 12 and minimum 4. What vertical shift produces an equivalent equation of the form y = A cos(Bx) + d relative to y = A cos(Bx)?

Explanation

Given: max 12, min 4.

Step 1: Midline = (12 + 4)/2 = 8.

Step 2: Shift the midline from 0 to 8 by adding 8.

So, the final answer is shift up 8.

Submit
3) A function y = a sin(bx) + d has amplitude 2 and midline y = −6. Which equation matches?

Explanation

Given: amplitude 2 and midline −6.

Step 1: Use coefficient 2 for amplitude.

Step 2: Subtract 6 outside for the midline.

So, the final answer is y = 2 sin(3x) − 6.

Submit
4) The graph of y = −4 cos(x) + 9 has what maximum value?

Explanation

Given: −4 cos(x) ranges in [−4, 4].

Step 1: Add 9 ⇒ range [5, 13].

Step 2: Maximum is 13.

So, the final answer is 13.

Submit
5) A sinusoid has amplitude 7 and midline y = 4. What is the distance between its maximum and minimum values?

Explanation

Given: distance (max − min) = 2 × amplitude.

Step 1: 2 × 7 = 14.

So, the final answer is 14.

Submit
6) The graph of y = 3 sin(x) is shifted down 8 units. Which equation represents the new graph?

Explanation

Given: vertical shift down 8.

Step 1: Subtract 8 outside the sine.

So, the final answer is y = 3 sin(x) − 8.

Submit
7) A sinusoid has equation y = 5 cos(2x) + c. If its midline is y = 1, what is c?

Explanation

Given: midline equals the outside constant c.

Step 1: Midline 1 ⇒ c = 1.

So, the final answer is 1.

Submit
8) The function y = 2 − 6 cos(x) can be viewed as a vertical translation of a base cosine function. Which description is correct?

Explanation

Given: y = 2 − 6 cos(x) = −6 cos(x) + 2.

Step 1: This is −6 cos(x) shifted up 2 units.

So, the final answer is “−6 cos(x)” shifted up 2 units.

Submit
9) Consider y = 4 sin(x) − 3. Which statement is correct?

Explanation

Given: y = 4 sin(x) − 3.

Step 1: Amplitude = |4| = 4.

Step 2: Midline = −3.

So, the final answer is amplitude 4; midline y = −3.

Submit
10) A cosine function has minimum value −9 and maximum value −1. What is its midline?

Explanation

Given: max −1, min −9.

Step 1: Midline = (−1 + (−9))/2 = −10/2 = −5.

So, the final answer is y = −5.

Submit
11) The graph of y = sin(x) is shifted to produce a midline at y = −7. Which equation represents the result?

Explanation

Given: need midline −7.

Step 1: Subtract 7 outside.

So, the final answer is y = sin(x) − 7.

Submit
12) A sinusoid is given by y = −3 sin(x) + k and has a maximum value of 2. What is k?

Explanation

Given: y = −3 sin(x) + k.

Step 1: Range of −3 sin(x) is [−3, 3]. After shift: [k − 3, k + 3].

Step 2: Max = k + 3 = 2 ⇒ k = −1.

So, the final answer is −1.

Submit
13) Which transformation converts y = 2 cos(3x) into y = 2 cos(3x) − 4?

Explanation

Given: subtracting 4 outside.

Step 1: This moves the graph down 4 units.

So, the final answer is vertical shift down 4 units.

Submit
14) A sine graph has midline y = 3 and amplitude 2. After a downward shift of 5 units, what are the new midline and amplitude?

Explanation

Given: original midline 3, amplitude 2.

Step 1: Down 5 ⇒ new midline 3 − 5 = −2.

Step 2: Amplitude stays 2.

So, the final answer is midline y = −2; amplitude 2.

Submit
15) The function y = −5 sin(x) is shifted up 2 units. What is true about the new graph?

Explanation

Given: up 2 units.

Step 1: Add 2 outside ⇒ midline 2.

Step 2: Amplitude remains |−5| = 5.

So, the final answer is amplitude 5; midline y = 2.

Submit
16) Starting from y = cos(x), a vertical shift produces a midline at y = 6. Which equation matches?

Explanation

Given: need midline 6.

Step 1: Add +6 outside.

So, the final answer is y = cos(x) + 6.

Submit
17) A sinusoidal function has amplitude 4 and midline y = −2. Which equation could represent it?

Explanation

Given: amplitude 4, midline −2.

Step 1: Use coefficient 4.

Step 2: Subtract 2 outside.

So, the final answer is y = 4 sin(x) − 2.

Submit
18) The function y = 1.5 sin(2x) is shifted down 3 units. What is the new midline?

Explanation

Given: original midline 0.

Step 1: Down 3 ⇒ new midline −3.

So, the final answer is y = −3.

Submit
19) The graph of y = −2 cos(x) is shifted up 5 units. Which equation represents the new function?

Explanation

Given: original midline 0.

Step 1: Down 3 ⇒ new midline −3.

So, the final answer is y = −3.

Submit
20) A sinusoid is given by y = 3 sin(x) + 4. What is its midline?

Explanation

Given: y = 3 sin(x) + 4.

Step 1: The constant +4 is the vertical shift and midline.

So, the final answer is y = 4.

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The function y = −2 sin(4x) − 5 is formed by shifting y =...
A cosine function has maximum 12 and minimum 4. What vertical shift...
A function y = a sin(bx) + d has amplitude 2 and midline y = −6....
The graph of y = −4 cos(x) + 9 has what maximum value?
A sinusoid has amplitude 7 and midline y = 4. What is the distance...
The graph of y = 3 sin(x) is shifted down 8 units. Which equation...
A sinusoid has equation y = 5 cos(2x) + c. If its midline is y = 1,...
The function y = 2 − 6 cos(x) can be viewed as a vertical...
Consider y = 4 sin(x) − 3. Which statement is correct?
A cosine function has minimum value −9 and maximum value...
The graph of y = sin(x) is shifted to produce a midline at y =...
A sinusoid is given by y = −3 sin(x) + k and has a maximum value...
Which transformation converts y = 2 cos(3x) into y = 2 cos(3x) −...
A sine graph has midline y = 3 and amplitude 2. After a downward shift...
The function y = −5 sin(x) is shifted up 2 units. What is true...
Starting from y = cos(x), a vertical shift produces a midline at y =...
A sinusoidal function has amplitude 4 and midline y = −2. Which...
The function y = 1.5 sin(2x) is shifted down 3 units. What is the new...
The graph of y = −2 cos(x) is shifted up 5 units. Which equation...
A sinusoid is given by y = 3 sin(x) + 4. What is its midline?
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