Apply and Interpret Vertical Shifts in Trig Graphs Quiz

1) The function y = sin(x) + 4 is a vertical translation of y = sin(x). What is the midline of the new function?

Y = 0

Y = −4

Y = 1

Y = 4

Given: y = sin(x) + 4.

Step 1: +4 outside shifts the midline from 0 to 4.

So, the final answer is y = 4.

Explanation

Given: y = sin(x) + 4.

Step 1: +4 outside shifts the midline from 0 to 4.

So, the final answer is y = 4.

2) The graph of y = cos(x) is shifted down 6 units. Which equation represents the new function?

Y = cos(x) + 6

Y = cos(x) − 6

Y = 6cos(x)

Y = cos(x − 6)

Given: down 6 units.

Step 1: Subtract 6 outside the cosine.

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

Explanation

Given: down 6 units.

Step 1: Subtract 6 outside the cosine.

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

3) Consider f(x) = 3sin(x) − 2. What are the amplitude and midline?

Amplitude 3; midline y = −2

Amplitude 2; midline y = 3

Amplitude 1; midline y = −2

Amplitude 3; midline y = 0

Given: f(x) = 3 sin(x) − 2.

Step 1: Amplitude = |3| = 3.

Step 2: Midline is −2.

So, the final answer is amplitude 3; midline y = −2.

Explanation

Given: f(x) = 3 sin(x) − 2.

Step 1: Amplitude = |3| = 3.

Step 2: Midline is −2.

So, the final answer is amplitude 3; midline y = −2.

4) The function g(x) = −2cos(3x) + 5 has which range?

[3, 7]

[−7, 3]

[−5, 5]

[5, 7]

Given: −2 cos(3x).

Step 1: Range of −2 cos(3x) is [−2, 2].

Step 2: Add 5 ⇒ [3, 7].

So, the final answer is [3, 7].

Explanation

Given: −2 cos(3x).

Step 1: Range of −2 cos(3x) is [−2, 2].

Step 2: Add 5 ⇒ [3, 7].

So, the final answer is [3, 7].

5) Which transformation changes y = sin(x) to a graph with the same amplitude but a midline at y = −7?

Y = sin(x) − 7

Y = −7sin(x)

Y = sin(x + 7)

Y = 7 + sin(7x)

Given: same amplitude, midline −7.

Step 1: Subtract 7 outside.

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

Explanation

Given: same amplitude, midline −7.

Step 1: Subtract 7 outside.

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

6) A Ferris wheel’s height is modeled by h(t) = 12cos(t) + 38. What does 38 represent?

The amplitude of the motion

The period of rotation

The midline (average height)

The maximum height

Given: h(t) = A cos(⋯) + D.

Step 1: D is the vertical shift/midline.

So, the final answer is the midline (average height).

Explanation

Given: h(t) = A cos(⋯) + D.

Step 1: D is the vertical shift/midline.

So, the final answer is the midline (average height).

7) A sound wave is modeled by y = 0.8sin(2x) + 1.2. What is its maximum value?

0.8

1.2

2

2.8

Given: amplitude 0.8 and midline 1.2.

Step 1: Max = 1.2 + 0.8 = 2.0.

So, the final answer is 2.0.

Explanation

Given: amplitude 0.8 and midline 1.2.

Step 1: Max = 1.2 + 0.8 = 2.0.

So, the final answer is 2.0.

8) The function y = cos(x) normally oscillates between −1 and 1. After a vertical shift, the graph oscillates between −4 and −2. Which equation matches this transformation?

Y = 2cos(x) − 3

Y = cos(x) + 3

Y = 2 + cos(x)

Y = cos(x) − 3

Given: target range [−4, −2].

Step 1: Midline is (−4 + −2)/2 = −3; amplitude 1.

Step 2: y = cos(x) − 3 has that range.

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

Explanation

Given: target range [−4, −2].

Step 1: Midline is (−4 + −2)/2 = −3; amplitude 1.

Step 2: y = cos(x) − 3 has that range.

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

9) Which equation represents a sine function with amplitude 5 and midline y = 2?

Y = 5sin(x) + 2

Y = 2sin(x) + 5

Y = 5cos(x) + 2

Y = sin(5x) + 2

Given: amplitude 5, midline 2.

Step 1: Use coefficient 5.

Step 2: Add +2 outside.

So, the final answer is y = 5 sin(x) + 2.

Explanation

Given: amplitude 5, midline 2.

Step 1: Use coefficient 5.

Step 2: Add +2 outside.

So, the final answer is y = 5 sin(x) + 2.

10) The function p(x) = −4sin(x) + c has a minimum value of −1. What is c?

0

3

−3

1

Given: range of −4 sin(x) is [−4, 4].

Step 1: After shift: [c − 4, c + 4].

Step 2: Minimum c − 4 = −1 ⇒ c = 3.

So, the final answer is 3.

Explanation

Given: range of −4 sin(x) is [−4, 4].

Step 1: After shift: [c − 4, c + 4].

Step 2: Minimum c − 4 = −1 ⇒ c = 3.

So, the final answer is 3.

11) A temperature model is given by T(t) = 6cos(ωt) + 72. What is the range of T over one period?

[66, 72]

[66, 78]

[72, 78]

[60, 72]

Given: amplitude 6, midline 72.

Step 1: Range is [72 − 6, 72 + 6] = [66, 78].

So, the final answer is [66, 78].

Explanation

Given: amplitude 6, midline 72.

Step 1: Range is [72 − 6, 72 + 6] = [66, 78].

So, the final answer is [66, 78].

12) The midline of r(x) = 2sin(5x) − 9 is:

Y = 2

Y = −9

Y = −7

Y = 0

Given: r(x) = A sin(⋯) + D.

Step 1: D = −9 is the midline.

So, the final answer is y = −9.

Explanation

Given: r(x) = A sin(⋯) + D.

Step 1: D = −9 is the midline.

So, the final answer is y = −9.

13) If y = sin(x) is shifted vertically to become y = sin(x) + k so that its minimum value becomes 2, what is k?

2

1

3

−2

Given: original min is −1.

Step 1: −1 + k = 2 ⇒ k = 3.

So, the final answer is 3.

Explanation

Given: original min is −1.

Step 1: −1 + k = 2 ⇒ k = 3.

So, the final answer is 3.

14) A cosine wave reaches a maximum at y = 11 and a minimum at y = 1. Which equation could represent the graph?

Y = 5cos(x) + 6

Y = 10cos(x) + 1

Y = 6cos(x) + 5

Y = 5cos(2x) + 11

Given: max 11, min 1.

Step 1: Amplitude = (11 − 1)/2 = 5.

Step 2: Midline = (11 + 1)/2 = 6.

So, the final answer is y = 5 cos(x) + 6.

Explanation

Given: max 11, min 1.

Step 1: Amplitude = (11 − 1)/2 = 5.

Step 2: Midline = (11 + 1)/2 = 6.

So, the final answer is y = 5 cos(x) + 6.

15) A sine graph has a midline at y = −3 and amplitude 2. Which equation represents it?

Y = 2sin(x) − 3

Y = −3sin(x) + 2

Y = 2sin(2x) + 3

Y = sin(x) − 2

Given: midline −3, amplitude 2.

Step 1: Use coefficient 2.

Step 2: Subtract 3 outside.

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

Explanation

Given: midline −3, amplitude 2.

Step 1: Use coefficient 2.

Step 2: Subtract 3 outside.

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

16) The function y = 7cos(x) + k oscillates between −1 and 13. What is the value of k?

−7

6

7

8

Given: 7 cos(x) ranges [−7, 7].

Step 1: After shift: [k − 7, k + 7] matches [−1, 13].

Step 2: Solve k − 7 = −1 ⇒ k = 6.

So, the final answer is 6.

Explanation

Given: 7 cos(x) ranges [−7, 7].

Step 1: After shift: [k − 7, k + 7] matches [−1, 13].

Step 2: Solve k − 7 = −1 ⇒ k = 6.

So, the final answer is 6.

17) A spring’s displacement is modeled by d(t) = −1.5sin(πt) + 0.5. What is the minimum value of d(t)?

−2.0

−1.0

−0.5

0.5

Given: −1.5 sin(πt) ranges [−1.5, 1.5].

Step 1: Add 0.5 ⇒ range [−1.0, 2.0].

Step 2: Minimum is −1.0.

So, the final answer is −1.0.

Explanation

Given: −1.5 sin(πt) ranges [−1.5, 1.5].

Step 1: Add 0.5 ⇒ range [−1.0, 2.0].

Step 2: Minimum is −1.0.

So, the final answer is −1.0.

18) The graph of y = cos(x) is shifted up 4 units and reflected across the x-axis. Which equation represents the result?

Y = −cos(x) − 4

Y = cos(x) − 4

Y = −cos(x) + 4

Y = cos(−x) + 4

Given: up 4 ⇒ cos(x) + 4.

Step 1: Reflect across x-axis ⇒ −(cos(x) + 4) = −cos(x) − 4.

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

Explanation

Given: up 4 ⇒ cos(x) + 4.

Step 1: Reflect across x-axis ⇒ −(cos(x) + 4) = −cos(x) − 4.

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

19) Which statement correctly describes g(x) = 3sin(2x) − 10?

The amplitude is −10; midline y = 3

The amplitude is 3; midline y = −10

The amplitude is 2; midline y = −3

The amplitude is 3; midline y = 0

Given: g(x) = A sin(⋯) + D.

Step 1: Amplitude is |A| = 3.

Step 2: Midline is D = −10.

So, the final answer is amplitude 3; midline y = −10.

Explanation

Given: g(x) = A sin(⋯) + D.

Step 1: Amplitude is |A| = 3.

Step 2: Midline is D = −10.

So, the final answer is amplitude 3; midline y = −10.

20) A vertical shift is applied to y = −2cos(x) so that the maximum value becomes 9. What shift was applied?

Shift down 7 units

Shift up 7 units

Shift up 9 units

Shift down 9 units

Given: −2 cos(x) has max 2.

Step 1: To make the max 9, add 7 outside.

So, the final answer is shift up 7 units.

Explanation

Given: −2 cos(x) has max 2.

Step 1: To make the max 9, add 7 outside.

So, the final answer is shift up 7 units.