Vertical Shift & Midline Quiz � Graph Sine and Cosine (HSF-TF.B.5)

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

Y = 2 sin(x) − 3

Y = 2 sin(x + π) + 3

Y = −2 cos(x) + 3

Y = 2 cos(x) + 3

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.

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.

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

Y = −4 cos(x) − 1

Y = −4 cos(x) + 1

Y = 4 sin(x) − 1

Y = −4 sin(x) − 1

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.

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.

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

Replacing x with x − c

Multiplying f(x) by k

Adding a constant D to f(x)

Replacing x with Bx

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).

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).

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

Midline y = 2.5; average 2.5 ft

Midline y = 8; average 8 ft

Midline y = 10.5; average 10.5 ft

Midline y = 5.5; average 5.5 ft

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.

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.

5) 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?

Y = 3 sin(x) + 6

Y = −3 sin(x) + 6

Y = 3 cos(x) + 6

Y = −3 cos(x) + 6

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.

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.

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

Y = 0

Y = −2

Y = 4

Y = 2

Given: up 2 ⇒ midline is y = 2.

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

So, the final answer is y = 2.

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.

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

−4

0

4

8

Given: max 9, min −1.

Step 1: Midline D = (9 + (−1))/2 = 4.

So, the final answer is D = 4.

Explanation

Given: max 9, min −1.

Step 1: Midline D = (9 + (−1))/2 = 4.

So, the final answer is D = 4.

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

Y = sin(x − 5)

Y = sin(x) − 5

Y = −5 sin(x)

Y = sin(x) + 5

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.

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.

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

Max 19; Min 5

Max 7; Min −7

Max 5; Min 19

Max 12; Min 0

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.

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.

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

Max 4.5; Min 3.5

Max 0.5; Min −0.5

Max 3.5; Min 4.5

Max 5; Min 3.5

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.

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.

11) For the function y = 3 sin(x) + 2, what is the vertical shift and the midline?

Shift up 3; midline y = 3

Shift up 2; midline y = 2

Shift down 2; midline y = −2

No vertical shift; midline y = 0

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.

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.

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

Midline y = −7; amplitude decreases by 7

Midline y = 0; amplitude unchanged

Midline y = −7; amplitude unchanged

Midline y = 7; amplitude unchanged

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.

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.

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

Shift up 10; midline y = 10

Shift up 4; midline y = 4

Shift down 2; midline y = −2

Shift up 8; midline y = 8

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.

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.

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

Amplitude 5; midline y = 1

Amplitude −5; midline y = 1

Amplitude 5; midline y = 0

Amplitude −5; midline y = 0

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.

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.

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

Y = 2 cos(x) + 6

Y = 2 cos(x) − 6

Y = cos(2x) − 6

Y = −2 cos(x) − 6

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.

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.

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

Y = 4 sin(x) − 3

Y = 3 sin(4x) + 4

Y = 4 sin(x) + 3

Y = sin(4x) + 3

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.

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.

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

0

−1.5

−2

2

Given: midline equals the outside constant k.

Step 1: Midline is −2.

So, the final answer is k = −2.

Explanation

Given: midline equals the outside constant k.

Step 1: Midline is −2.

So, the final answer is k = −2.

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

Y = 3

Y = 4

Y = 6

Y = 2

Given: max 7, min −1.

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

So, the final answer is y = 3.

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.

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

Shift up 4; midline y = 4

Shift down 2; midline y = −2

Shift down 4; midline y = −4

No vertical shift; midline y = 0

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.

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.

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

Y = cos(x) + 5; midline y = 5

Y = cos(x − 5); midline y = 0

Y = 5 cos(x); midline y = 0

Y = cos(x) − 5; midline y = −5

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.

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.

Vertical Shift & Midline Identification Quiz

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

2)

What first name or nickname would you like us to use?

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

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

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

5) 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?

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

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

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

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

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

11) For the function y = 3 sin(x) + 2, what is the vertical shift and the midline?

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

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

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

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

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

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

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

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

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