Vertical Shift Effect Quiz: Effect Of Vertical Shift On Maxima, Minima, And Midline

1) Which changes occur when D increases from 0 to 3 for y = sin(x) + D?

Midline moves from y = 0 to y = 3

Maximum moves from 1 to 4

Minimum moves from −1 to 2

Amplitude increases from 1 to 4

Period changes from 2π to 6π

All y-values increase by 3, so midline, max, and min each rise by 3. Amplitude and period are unaffected.

Explanation

All y-values increase by 3, so midline, max, and min each rise by 3. Amplitude and period are unaffected.

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2) If the original maximum is M and minimum is m, then after adding D they become M + D and m + D.

True

False

Vertical shift translates the entire range by D, preserving amplitude and period.

Explanation

Vertical shift translates the entire range by D, preserving amplitude and period.

3) The new maximum of y = 3cos(x) + D is ____.

Original max of 3cos(x) is 3. After vertical shift by D, every y-value increases by D, so max becomes D + 3.

Explanation

Original max of 3cos(x) is 3. After vertical shift by D, every y-value increases by D, so max becomes D + 3.

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4) Compare y = sin(x) and y = sin(x) + 2. Select all correct statements.

The midline changes from y = 0 to y = 2

The maximum changes from 1 to 3

The minimum changes from −1 to 1

The period changes from 2π to 4π

The amplitude remains 1

Adding 2 shifts the whole graph up by 2; amplitude stays 1 and period stays 2π.

Explanation

Adding 2 shifts the whole graph up by 2; amplitude stays 1 and period stays 2π.

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5) Adding D to a trig function increases both the maximum and minimum values by exactly D.

True

False

If the original range is [y_min, y_max], then f(x)+D has range [y_min + D, y_max + D]. The amplitude and period are unchanged.

Explanation

If the original range is [y_min, y_max], then f(x)+D has range [y_min + D, y_max + D]. The amplitude and period are unchanged.

6) If y = 3sin(x) + D has minimum −1, what is D?

1/2

12/28

1/1

12/26

Min = D − |A| = D − 3. Given −1 = D − 3 ⇒ D = 2.

Explanation

Min = D − |A| = D − 3. Given −1 = D − 3 ⇒ D = 2.

7) Vertical shift changes the spacing between consecutive maxima along the x-axis.

True

False

Spacing between maxima is determined by the period, which depends on B, not on D.

Explanation

Spacing between maxima is determined by the period, which depends on B, not on D.

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9) A vertical shift can change whether the graph crosses the x-axis within a given interval.

True

False

Shifting up/down moves intercepts; solutions to f(x)=0 become solutions to f(x)=−D after the shift, which may add or remove zeros in an interval.

Explanation

Shifting up/down moves intercepts; solutions to f(x)=0 become solutions to f(x)=−D after the shift, which may add or remove zeros in an interval.

10) Select all true statements about y = −2sin(x) + 5.

Midline is y = 5

Amplitude is 2

Maximum value is 7

Minimum value is 3

Period is 2π

Amplitude is |−2| = 2; period is 2π. Range of −2sin(x) is [−2,2], so adding 5 gives [3,7]; midline is y=5.

Explanation

Amplitude is |−2| = 2; period is 2π. Range of −2sin(x) is [−2,2], so adding 5 gives [3,7]; midline is y=5.

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11) For y = A·cos(Bx + C) + D, the minimum value is ____.

Max/min are D ± |A| because the base cosine oscillates between −1 and 1.

Explanation

Max/min are D ± |A| because the base cosine oscillates between −1 and 1.

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12) The range of y = A·sin(Bx) + D is [____, ____].

Because sin ranges from −1 to 1, multiplying by A scales to [−|A|, |A|], then adding D shifts to [D−|A|, D+|A|].

Explanation

Because sin ranges from −1 to 1, multiplying by A scales to [−|A|, |A|], then adding D shifts to [D−|A|, D+|A|].

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13) For y = A·sin(Bx) + D, what is the midline?

Y = 0

Y = A

Y = D

Y = B

A vertical shift by D translates every point up/down by D, so the horizontal center of oscillation (midline) moves from y=0 to y=D.

Explanation

A vertical shift by D translates every point up/down by D, so the horizontal center of oscillation (midline) moves from y=0 to y=D.

14) Given y = 5cos(2x) + D has maximum 12, what is D?

7

12

1/5

1/2

Max = D + |A| = D + 5. If max is 12, D + 5 = 12 ⇒ D = 7.

Explanation

Max = D + |A| = D + 5. If max is 12, D + 5 = 12 ⇒ D = 7.

15) The graph of y = 4sin(x) has range [−4, 4]. What is the range of y = 4sin(x) − 6?

[−10, −2]

[−4, 4]

[−6, 6]

[−2, 10]

Vertical shift by −6 moves every value down by 6: [−4−6, 4−6] = [−10, −2].

Explanation

Vertical shift by −6 moves every value down by 6: [−4−6, 4−6] = [−10, −2].

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17) Changing D affects the amplitude of the function.

True

False

Amplitude depends on |A|, not on D. D only shifts the graph up or down.

Explanation

Amplitude depends on |A|, not on D. D only shifts the graph up or down.

18) For y = 2cos(x) + D, which statements are always true regardless of D?

Amplitude is 2

Maximum is D + 2

Minimum is D − 2

Period is 2π

Midline is y = D

All follow from the standard form: amplitude |A|, max/min D±|A|, period 2π, and midline y=D.

Explanation

All follow from the standard form: amplitude |A|, max/min D±|A|, period 2π, and midline y=D.

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19) Select all functions whose midline is y = −3.

Y = 4sin(x) − 3

Y = −2cos(5x) − 3

Y = sin(x − 1) − 3

Y = 3 + 2sin(x)

Y = −3cos(x)

Midline equals D. Choices A, B, C have D = −3. D has D = 3; E has D = 0.

Explanation

Midline equals D. Choices A, B, C have D = −3. D has D = 3; E has D = 0.

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20) For y = −4cos(x) + 1, what are (maximum, minimum)?

(1 + 4, 1 − 4) = (5, −3)

(1 − 4, 1 + 4) = (−3, 5)

(4, −4)

(−4, 4)

Max = D + |A| = 1 + 4 = 5; Min = D − |A| = 1 − 4 = −3. Order is (min,max) depends on wording; pair (−3,5) corresponds to (minimum, maximum).

Explanation

Max = D + |A| = 1 + 4 = 5; Min = D − |A| = 1 − 4 = −3. Order is (min,max) depends on wording; pair (−3,5) corresponds to (minimum, maximum).

Vertical Shift Effect Quiz: Effect of Vertical Shift on Maxima, Minima, and Midline