Vertical Shift Quiz: Vertical Shift Fundamentals

1) In y = sin(x) + 3, what transformation has been applied to y = sin(x)?

Vertical stretch by 3

Shift up by 3 units

Shift right by 3 units

Shift left by 3 units

Adding D = 3 outside the sine function moves every y-value up by 3: y_new = y_old + 3. Shape, amplitude, and period remain unchanged.

Explanation

Adding D = 3 outside the sine function moves every y-value up by 3: y_new = y_old + 3. Shape, amplitude, and period remain unchanged.

2) For y = −2cos(2x) + 4, the maximum value is and the minimum value is .

Base −2cos(2x) has amplitude 2 with range [−2, 2]. Adding 4 shifts to [2, 6]. Thus max = 6, min = 2.

Explanation

Base −2cos(2x) has amplitude 2 with range [−2, 2]. Adding 4 shifts to [2, 6]. Thus max = 6, min = 2.

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3) Select all true statements about y = 2sin(x) − 1 compared to y = 2sin(x).

Amplitude is unchanged

Graph is shifted down by 1

Period is unchanged

Graph is shifted right by 1

Midline is y = 0

Adding D = −1 moves the curve down 1. Amplitude stays |2| and period remains 2π. The midline shifts from y = 0 to y = −1.

Explanation

Adding D = −1 moves the curve down 1. Amplitude stays |2| and period remains 2π. The midline shifts from y = 0 to y = −1.

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4) For y = A·sin(Bx) + D, the maximum value is and the minimum value is (in terms of A and D).

Amplitude is |A|, centered at y = D, so maxima/minima are D ± |A|.

Explanation

Amplitude is |A|, centered at y = D, so maxima/minima are D ± |A|.

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5) The midline of y = 4cos(x) − 5 is y = ____.

For A·cos(x) + D, the midline is y = D. Here D = −5, so the midline is y = −5.

Explanation

For A·cos(x) + D, the midline is y = D. Here D = −5, so the midline is y = −5.

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6) Adding D to f(x) changes the x-intercepts by moving them up or down, which can create or remove intercepts.

True

False

A vertical shift changes where the graph crosses y = 0. Points with f(x) = 0 move to y = D; new zeros occur where f(x) = −D.

Explanation

A vertical shift changes where the graph crosses y = 0. Points with f(x) = 0 move to y = D; new zeros occur where f(x) = −D.

7) Given y = A·sin(Bx + C) + D, which parameter controls the vertical shift?

A

B

C

D

In the standard form, D vertically translates the graph. A affects amplitude, B affects period, and C affects phase (horizontal shift).

Explanation

In the standard form, D vertically translates the graph. A affects amplitude, B affects period, and C affects phase (horizontal shift).

8) Which equation has midline y = −3?

Y = 2cos(x) − 3

Y = 2cos(x + 1) − 1

Y = 2cos(2x) + 3

Y = −3cos(x)

In A·cos(...) + D, the midline is y = D. Choice A has D = −3. Others have D = −1, +3, and 0 respectively.

Explanation

In A·cos(...) + D, the midline is y = D. Choice A has D = −3. Others have D = −1, +3, and 0 respectively.

9) Which functions are vertical shifts of y = sin(x) only (no other changes)?

Y = sin(x) + 4

Y = sin(x) − 1/2

Y = 2sin(x) + 3

Y = sin(x + π) + 2

Y = sin(x)

Pure vertical shift requires same amplitude and same input. A and B add constants only; E is the original. C changes amplitude; D changes phase and adds a shift.

Explanation

Pure vertical shift requires same amplitude and same input. A and B add constants only; E is the original. C changes amplitude; D changes phase and adds a shift.

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10) The midline of y = A·cos(Bx + C) + D is y = D.

True

False

The average of the maximum and minimum values is D, so the horizontal center of oscillation is y = D.

Explanation

The average of the maximum and minimum values is D, so the horizontal center of oscillation is y = D.

11) For y = f(x) + D, D > 0 shifts the graph upward by D units without changing its period or amplitude.

True

False

The vertical shift adds D to all outputs: y_new = f(x) + D. This does not alter input spacing (period) or vertical scaling (amplitude).

Explanation

The vertical shift adds D to all outputs: y_new = f(x) + D. This does not alter input spacing (period) or vertical scaling (amplitude).

12) For y = −3sin(2x) + 6, select all true statements.

Amplitude is 3

Period is π

Midline is y = 6

Maximum value is 3

Minimum value is −9

Amplitude |−3| = 3; period 2π/2 = π; midline y = 6. Range is [6 − 3, 6 + 3] = [3, 9], so max = 9 and min = 3.

Explanation

Amplitude |−3| = 3; period 2π/2 = π; midline y = 6. Range is [6 − 3, 6 + 3] = [3, 9], so max = 9 and min = 3.

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13) What is the range of y = −2sin(x) − 4?

[−6, −2]

[−2, 2] shifted down 4

[−4, 4]

[−2, 2]

Range of −2sin(x) is [−2, 2]. Adding −4 shifts to [−6, −2]. Option B is descriptive but not precise; A is exact.

Explanation

Range of −2sin(x) is [−2, 2]. Adding −4 shifts to [−6, −2]. Option B is descriptive but not precise; A is exact.

14) What is the range of y = 3sin(x) + 2?

[2 − 3, 2 + 3] = [−1, 5]

[−3, 3]

[−1, 1] shifted up 2

[0, 6]

Range of 3sin(x) is [−3, 3]. Adding 2 shifts the range up: [−3+2, 3+2] = [−1, 5]. Choice C is descriptive but not interval notation.

Explanation

Range of 3sin(x) is [−3, 3]. Adding 2 shifts the range up: [−3+2, 3+2] = [−1, 5]. Choice C is descriptive but not interval notation.

15) If y = sin(x) has zeros at x = kπ, then y = sin(x) + D has zeros where sin(x) = ____.

Setting sin(x) + D = 0 gives sin(x) = −D. The vertical shift moves the x-intercepts to solutions of this equation.

Explanation

Setting sin(x) + D = 0 gives sin(x) = −D. The vertical shift moves the x-intercepts to solutions of this equation.

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16) Identify all graphs that represent a vertical shift of y = cos(x) by +2 (and nothing else).

Y = cos(x) + 2

Y = 2 + cos(x)

Y = cos(x + 2)

Y = 2cos(x) + 2

Y = cos(x) − (−2)

Adding +2 outside produces a pure vertical shift. Writing −(−2) also equals +2. Options with phase or amplitude changes are not pure vertical shifts.

Explanation

Adding +2 outside produces a pure vertical shift. Writing −(−2) also equals +2. Options with phase or amplitude changes are not pure vertical shifts.

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17) Vertical shifts affect the spacing of successive peaks along the x-axis.

True

False

Peak spacing depends on the period (controlled by B), not on D. Vertical shift moves peaks up or down but does not change x-spacing.

Explanation

Peak spacing depends on the period (controlled by B), not on D. Vertical shift moves peaks up or down but does not change x-spacing.

18) A vertical shift can change the number of solutions to sin(x) = k within a given interval.

True

False

Shifting sin(x) by D changes intersections with horizontal lines y = k. This can create or remove intersections in a fixed interval.

Explanation

Shifting sin(x) by D changes intersections with horizontal lines y = k. This can create or remove intersections in a fixed interval.

19) Translate y = cos(x) downward by 7 units: y = ____.

Vertical shift by D = −7 gives y = cos(x) + (−7) = cos(x) − 7. Shape and period are preserved.

Explanation

Vertical shift by D = −7 gives y = cos(x) + (−7) = cos(x) − 7. Shape and period are preserved.

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20) Select all correct statements for y = 5sin(x) + D.

Amplitude equals 5 for any D

Period equals 2π for any D

Midline equals y = D

Range length equals 10 regardless of D

Nodes (max/min) shift up by D

D translates the graph. Amplitude |5| and period 2π are unchanged. Range is [−5, 5] shifted by D so length 10 remains. Extrema shift to y = D ± 5.

Explanation

D translates the graph. Amplitude |5| and period 2π are unchanged. Range is [−5, 5] shifted by D so length 10 remains. Extrema shift to y = D ± 5.

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Vertical Shift Quiz: Vertical Shift Fundamentals

1) In y = sin(x) + 3, what transformation has been applied to y = sin(x)?

2)

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

2) For y = −2cos(2x) + 4, the maximum value is ____ and the minimum value is ____.

3) Select all true statements about y = 2sin(x) − 1 compared to y = 2sin(x).

4) For y = A·sin(Bx) + D, the maximum value is ____ and the minimum value is ____ (in terms of A and D).

5) The midline of y = 4cos(x) − 5 is y = ____.

6) Adding D to f(x) changes the x-intercepts by moving them up or down, which can create or remove intercepts.

7) Given y = A·sin(Bx + C) + D, which parameter controls the vertical shift?

8) Which equation has midline y = −3?

9) Which functions are vertical shifts of y = sin(x) only (no other changes)?

10) The midline of y = A·cos(Bx + C) + D is y = D.

11) For y = f(x) + D, D > 0 shifts the graph upward by D units without changing its period or amplitude.

12) For y = −3sin(2x) + 6, select all true statements.

13) What is the range of y = −2sin(x) − 4?

14) What is the range of y = 3sin(x) + 2?

15) If y = sin(x) has zeros at x = kπ, then y = sin(x) + D has zeros where sin(x) = ____.

16) Identify all graphs that represent a vertical shift of y = cos(x) by +2 (and nothing else).

17) Vertical shifts affect the spacing of successive peaks along the x-axis.

18) A vertical shift can change the number of solutions to sin(x) = k within a given interval.

19) Translate y = cos(x) downward by 7 units: y = ____.

20) Select all correct statements for y = 5sin(x) + D.

2) For y = −2cos(2x) + 4, the maximum value is and the minimum value is .

4) For y = A·sin(Bx) + D, the maximum value is and the minimum value is (in terms of A and D).