Write Equations with Vertical Shifts � Sine & Cosine (HSF-TF.B.5)

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?

Shift up 5; amplitude becomes 3

Shift down 5; amplitude remains 2

Shift up 5; amplitude remains 2

Shift down 5; amplitude becomes 7

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.

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.

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)?

Shift up 4

Shift down 8

Shift up 8

Shift down 4

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.

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.

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

Y = 2 sin(3x) − 6

Y = −6 sin(2x) + 2

Y = 2 sin(3x − 6)

Y = sin(2x) − 6

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.

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.

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

9

13

5

−13

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.

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.

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

7

8

11

14

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

Step 1: 2 × 7 = 14.

So, the final answer is 14.

Explanation

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

Step 1: 2 × 7 = 14.

So, the final answer is 14.

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

Y = 3 sin(x + 8)

Y = 3 sin(x) − 8

Y = 3 sin(x − 8)

Y = −3 sin(x) − 8

Given: vertical shift down 8.

Step 1: Subtract 8 outside the sine.

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

Explanation

Given: vertical shift down 8.

Step 1: Subtract 8 outside the sine.

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

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

0

−1

5

1

Given: midline equals the outside constant c.

Step 1: Midline 1 ⇒ c = 1.

So, the final answer is 1.

Explanation

Given: midline equals the outside constant c.

Step 1: Midline 1 ⇒ c = 1.

So, the final answer is 1.

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

Y = −6 cos(x) shifted up 2 units

Y = −6 cos(x) shifted down 2 units

Y = 6 cos(x) shifted up 2 units

Y = 6 cos(x) shifted down 2 units

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.

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.

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

Amplitude 4; midline y = −3

Amplitude 1; midline y = 4

Amplitude 7; midline y = 0

Amplitude 4; midline y = 0

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.

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.

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

Y = −1

Y = −5

Y = −9

Y = 4

Given: max −1, min −9.

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

So, the final answer is y = −5.

Explanation

Given: max −1, min −9.

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

So, the final answer is y = −5.

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

Y = sin(x) − 7

Y = −7 sin(x)

Y = sin(x + 7)

Y = −sin(x) − 7

Given: need midline −7.

Step 1: Subtract 7 outside.

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

Explanation

Given: need midline −7.

Step 1: Subtract 7 outside.

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

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

5

−1

−5

1

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.

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.

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

Vertical shift up 4 units

Vertical stretch by factor 4

Vertical shift down 4 units

Horizontal shift right 4 units

Given: subtracting 4 outside.

Step 1: This moves the graph down 4 units.

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

Explanation

Given: subtracting 4 outside.

Step 1: This moves the graph down 4 units.

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

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?

Midline y = −2; amplitude 2

Midline y = 8; amplitude 2

Midline y = −2; amplitude 7

Midline y = −5; amplitude 2

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.

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.

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

Amplitude increases to 7

Amplitude stays 5; midline y = −2

Amplitude becomes 3; midline y = 0

Amplitude stays 5; midline y = 2

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.

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.

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

Y = cos(x) − 6

Y = cos(x) + 6

Y = 6 cos(x)

Y = x cos(x) + 6

Given: need midline 6.

Step 1: Add +6 outside.

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

Explanation

Given: need midline 6.

Step 1: Add +6 outside.

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

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

Y = 4 sin(x) − 2

Y = −2 sin(x) + 4

Y = 4 sin(x + 2)

Y = −2 + sin(4x)

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.

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.

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

Y = −3

Y = 1.5

Y = 0

Y = 3

Given: original midline 0.

Step 1: Down 3 ⇒ new midline −3.

So, the final answer is y = −3.

Explanation

Given: original midline 0.

Step 1: Down 3 ⇒ new midline −3.

So, the final answer is y = −3.

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

Y = −2 cos(x − 5)

Y = −2 cos(x) + 5

Y = −2 cos(x + 5)

Y = −2 cos(x) − 5

Given: original midline 0.

Step 1: Down 3 ⇒ new midline −3.

So, the final answer is y = −3.

Explanation

Given: original midline 0.

Step 1: Down 3 ⇒ new midline −3.

So, the final answer is y = −3.

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

Y = 4

Y = 3

Y = 0

Y = −4

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

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

So, the final answer is y = 4.

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.

Write Sinusoidal Equations with Vertical Shifts Quiz

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?

2)

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

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)?

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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