Vertical Shift & Midline Identification Quiz

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| By Thames
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Thames
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Quizzes Created: 7116 | Total Attempts: 9,522,086
| Questions: 20 | Updated: Oct 31, 2025
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1) For the function y = 3 sin(x) + 2, what is the vertical shift and the midline?

Explanation

Given: y = 3 sin(x) + 2.

Step 1: +2 shifts the graph up by 2.

Step 2: Midline moves from y = 0 to y = 2.

So, the final answer is shift up 2; midline y = 2.

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About This Quiz
Vertical Shift & Midline Identification Quiz - Quiz

See the baseline at a glance! You’ll find the midline, read how far the graph moved up or down, and connect amplitude to the max and min values. Expect quick “what’s the midline?” and “what’s the new range?” wins throughout.

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2) The graph of y = cos(x) is shifted upward by 5 units. Which equation represents the new function and what is its midline?

Explanation

Given: shift up 5 units.

Step 1: Add +5 outside the cosine.

Step 2: The midline becomes y = 5.

So, the final answer is y = cos(x) + 5; midline y = 5.

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3) The function y = −2 sin(2x) − 4 has which vertical shift and midline?

Explanation

Given: y = −2 sin(2x) − 4.

Step 1: −4 outside shifts the graph down by 4.

Step 2: The midline is y = −4.

So, the final answer is shift down 4; midline y = −4.

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4) A sinusoidal graph has maximum 7 and minimum −1. What is the midline?

Explanation

Given: max 7, min −1.

Step 1: Midline = (max + min)/2 = (7 + (−1))/2 = 6/2 = 3.

So, the final answer is y = 3.

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5) The function y = 1.5 cos(πx) + k has midline y = −2. What is k?

Explanation

Given: midline equals the outside constant k.

Step 1: Midline is −2.

So, the final answer is k = −2.

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6) A sine function has amplitude 4, vertical shift up 3, and period 2π. Which equation matches?

Explanation

Given: amplitude 4, period 2π, shift up 3.

Step 1: Amplitude ⇒ coefficient 4.

Step 2: Period 2π ⇒ inside is just x.

Step 3: Up 3 ⇒ +3 outside.

So, the final answer is y = 4 sin(x) + 3.

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7) A cosine graph is shifted downward 6 units, amplitude 2, period 2π. Which is correct?

Explanation

Given: amplitude 2, down 6.

Step 1: Use 2 cos(x).

Step 2: Down 6 ⇒ −6 outside.

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

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8) Consider y = −5 sin(x) + 1. Which statement is true?

Explanation

Given: y = −5 sin(x) + 1.

Step 1: Amplitude is |−5| = 5.

Step 2: +1 outside ⇒ midline y = 1.

So, the final answer is amplitude 5; midline y = 1.

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9) A sinusoid oscillates between 10 and −2. What is the vertical shift and midline?

Explanation

Given: range from −2 to 10.

Step 1: Midline = (10 + (−2))/2 = 4.

Step 2: Midline moved from 0 to 4 ⇒ shift up 4.

So, the final answer is shift up 4; midline y = 4.

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10) If y = sin(x) is transformed to y = sin(x) − 7, what happens to the midline and amplitude?

Explanation

Given: y = sin(x) − 7.

Step 1: −7 outside ⇒ midline y = −7.

Step 2: Vertical shifts do not change amplitude (remains 1).

So, the final answer is midline y = −7; amplitude unchanged.

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11) A graph shows a sinusoid with midline y = −3 and amplitude 2. Which equation could represent it?

Explanation

Given: amplitude 2, midline −3.

Step 1: Use coefficient 2.

Step 2: Subtract 3 to set midline.

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

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12) The function y = 0.5 cos(3x) + 4 has what maximum and minimum?

Explanation

Given: amplitude 0.5, midline 4.

Step 1: Max = 4 + 0.5 = 4.5.

Step 2: Min = 4 − 0.5 = 3.5.

So, the final answer is Max 4.5; Min 3.5.

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13) A sinusoidal model has midline y = 12 and amplitude 7. Which pair is correct?

Explanation

Given: midline 12, amplitude 7.

Step 1: Max = 12 + 7 = 19.

Step 2: Min = 12 − 7 = 5.

So, the final answer is Max 19; Min 5.

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14) Which transformation converts y = sin(x) to a function with midline y = −5 and the same amplitude?

Explanation

Given: need midline −5.

Step 1: Subtract 5 outside to shift the graph down.

Step 2: Amplitude unchanged.

So, the final answer is y = sin(x) − 5.

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15) Suppose y = A cos(Bx) + D has max 9 and min −1. What is D?

Explanation

Given: max 9, min −1.

Step 1: Midline D = (9 + (−1))/2 = 4.

So, the final answer is D = 4.

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16) The graph of a cosine wave is shifted up 2 units, then reflected across its midline. What is the resulting midline?

Explanation

Given: up 2 ⇒ midline is y = 2.

Step 1: Reflection across the midline leaves the midline unchanged.

So, the final answer is y = 2.

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17) A sinusoidal function has a midline at y = 6 and passes through this midline at x = 0 with a decreasing slope. Which equation could represent the function?

Explanation

Given: crossing midline at x = 0 with decreasing slope.

Step 1: −sin starts at 0 and decreases.

Step 2: Midline +6 ⇒ add 6 outside.

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

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18) A tide model h(t) = 2.5 sin(πt / 6) + 8. What are the midline and average water level?

Explanation

Given: h(t) = A sin(⋯) + D.

Step 1: D = 8 is the vertical shift/midline.

Step 2: The average equals the midline.

So, the final answer is midline 8; average 8 ft.

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19) Which change affects only the vertical position (midline) and not amplitude?

Explanation

Given: vertical position is controlled by an outside constant.

Step 1: Adding D shifts the graph vertically.

Step 2: Amplitude remains the same.

So, the final answer is adding a constant D to f(x).

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20) A sinusoid has midline y = −1 and amplitude 4. If its minimum occurs at x = 0, which represents it?

Explanation

Given: minimum at x = 0 and amplitude 4, midline −1.

Step 1: cos(0) = 1, so −4 cos(0) − 1 = −5 (minimum).

Step 2: This matches the condition.

So, the final answer is y = −4 cos(x) − 1.

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For the function y = 3 sin(x) + 2, what is the vertical shift and the...
The graph of y = cos(x) is shifted upward by 5 units. Which equation...
The function y = −2 sin(2x) − 4 has which vertical shift and...
A sinusoidal graph has maximum 7 and minimum −1. What is the...
The function y = 1.5 cos(πx) + k has midline y = −2. What is k?
A sine function has amplitude 4, vertical shift up 3, and period 2π....
A cosine graph is shifted downward 6 units, amplitude 2, period 2π....
Consider y = −5 sin(x) + 1. Which statement is true?
A sinusoid oscillates between 10 and −2. What is the vertical shift...
If y = sin(x) is transformed to y = sin(x) − 7, what happens to the...
A graph shows a sinusoid with midline y = −3 and amplitude 2. Which...
The function y = 0.5 cos(3x) + 4 has what maximum and minimum?
A sinusoidal model has midline y = 12 and amplitude 7. Which pair is...
Which transformation converts y = sin(x) to a function with midline y...
Suppose y = A cos(Bx) + D has max 9 and min −1. What is D?
The graph of a cosine wave is shifted up 2 units, then reflected...
A sinusoidal function has a midline at y = 6 and passes through this...
A tide model h(t) = 2.5 sin(πt / 6) + 8. What are the midline and...
Which change affects only the vertical position (midline) and not...
A sinusoid has midline y = −1 and amplitude 4. If its minimum occurs...
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