Modeling with Phase Shifts: Real-World Trig Translations

  • 11th Grade
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| Questions: 20 | Updated: Dec 11, 2025
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1) Which function represents a cosine graph shifted π/3 units to the right?

Explanation

A right shift by π/3 means replacing x with (x − π/3).

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

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About This Quiz
Modeling With Phase Shifts: Real-world Trig Translations - Quiz

Line up your model with real timing. You’ll choose phase shifts so maxima, minima, or zero crossings land at the right moments. Explain how the shift affects when events happen—without changing how big or how often they occur.

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2) The function y = sin(x + π/2) is equivalent to which basic trig function with no horizontal shift?

Explanation

Identity — sin(x + π/2) = cos(x).

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

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3) Consider y = 2sin(3(x − π/6)) + 1. What is the horizontal (phase) shift?

Explanation

(x − π/6) indicates a right shift by π/6.

So, the final answer is right π/6.

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4) The graph of y = cos(x) is shifted to become one with its maximum at x = −π instead of x = 0. Which equation matches this shift?

Explanation

Moving the max from 0 to −π means shifting left by π → (x + π).

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

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5) Rewrite y = sin(2x − π) in the form y = sin(2(x − c)). What is c?

Explanation

Step 1: 2x − π = 2(x − π/2).

Step 2: Therefore, c = π/2 (right shift).

So, the final answer is c = π/2.

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6) Which of the following is a phase shift of the parent tangent function to the left by π/4?

Explanation

Left shift by a ⇒ (x + a).

So, the final answer is y = tan(x + π/4).

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7) The function y = −cos(x − π/2) has been transformed from y = cos(x) by:

Explanation

Step 1: (x − π/2) → right shift π/2.

Step 2: The negative sign → reflection over the x-axis.

So, the final answer is right shift π/2 and reflection across x-axis.

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8) A sinusoid has midline y = 0, amplitude 1, period 2π, and passes through (x, y) = (π/2, 1) as a maximum. Which equation matches?

Explanation

Step 1: The cosine curve reaches a maximum where x = phase shift.

Step 2: Since max is at π/2, use y = cos(x − π/2).

So, the final answer is y = cos(x − π/2).

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9) A graph described as "the basic sine wave shifted left by π" can be written as:

Explanation

Left shift by a means y = sin(x + a).

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

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10) Consider y = 3cos(4x + π). What is the phase shift, written y = 3cos(4(x − c))?

Explanation

Step 1: 4x + π = 4(x + π/4) = 4(x − (−π/4)).

Step 2: Hence c = −π/4 (left shift).

So, the final answer is c = −π/4.

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11) A cosine graph with amplitude 2, period 2π, and phase shift right π/3 is:

Explanation

Step 1: Right shift π/3 → (x − π/3).

Step 2: Amplitude 2 → multiply by 2.

So, the final answer is y = 2cos(x − π/3).

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12) Which pair of functions are horizontally shifted versions of each other with the same period and amplitude?

Explanation

The second function is a right-shifted version of sin(x).

So, the final answer is y = sin(x) and y = sin(x − π/6).

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13) The function y = sin(x) is shifted to y = sin(x − a). If the maximum originally at x = π/2 moves to x = 5π/6, what is a?

Explanation

Step 1: Shift amount = new − old = 5π/6 − π/2 = π/3.

Step 2: Since it moved right, a = π/3.

So, the final answer is a = π/3.

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14) A tangent graph has vertical asymptotes now at x = −π/2 and x = 3π/2 (instead of −π/2 and π/2). What horizontal shift occurred?

Explanation

Step 1: The interval between asymptotes is π.

Step 2: The new rightmost asymptote moved π units to the right.

So, the final answer is right π.

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15) Rewrite y = −sin(x + 3π/4) as y = −sin(x − c). What is c?

Explanation

Step 1: (x + 3π/4) = (x − (−3π/4)) ⇒ c = −3π/4, meaning shift left 3π/4.

However, written as y = −sin(x − c), c = 3π/4 for the rightward equivalent.

So, the final answer is c = 3π/4.

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16) A sine curve with period 2π has its first maximum at x = π/2 and crosses the midline going upward at x = 0. Which model best fits?

Explanation

The base sine function already crosses at 0 and peaks at π/2.

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

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17) Which statement about y = cos(2(x + π/6)) is true?

Explanation

Step 1: (x + π/6) ⇒ left shift π/6.

Step 2: b = 2 ⇒ period = π.

So, the final answer is shift left π/6; period = π.

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18) To move y = sin(x) so that its zero at x = 0 becomes a zero at x = −π/3 (with identical shape and period), which function works?

Explanation

Step 1: New zero at −π/3 → shift left π/3.

Step 2: Left shift → (x + π/3).

So, the final answer is y = sin(x + π/3).

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19) A cosine function has period 2π, amplitude 1, and now has a minimum at x = 0 (instead of a maximum). Which is correct?

Explanation

To invert maximum to minimum, multiply by −1.

So, the final answer is y = −cos(x).

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20) The function y = 4sin(½x − π/8) can be written y = 4sin(½(x − c)). What is c and the horizontal shift direction?

Explanation

Step 1: ½x − π/8 = ½(x − π/4).

Step 2: Therefore, c = π/4 (right shift).

So, the final answer is c = π/4; shift right.

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Which function represents a cosine graph shifted π/3 units to the...
The function y = sin(x + π/2) is equivalent to which basic trig...
Consider y = 2sin(3(x − π/6)) + 1. What is the horizontal...
The graph of y = cos(x) is shifted to become one with its maximum at x...
Rewrite y = sin(2x − π) in the form y = sin(2(x − c)). What is c?
Which of the following is a phase shift of the parent tangent function...
The function y = −cos(x − π/2) has been transformed from y =...
A sinusoid has midline y = 0, amplitude 1, period 2π, and passes...
A graph described as "the basic sine wave shifted left by π" can be...
Consider y = 3cos(4x + π). What is the phase shift, written y =...
A cosine graph with amplitude 2, period 2π, and phase shift right...
Which pair of functions are horizontally shifted versions of each...
The function y = sin(x) is shifted to y = sin(x − a). If the maximum...
A tangent graph has vertical asymptotes now at x = −π/2 and x =...
Rewrite y = −sin(x + 3π/4) as y = −sin(x − c). What is c?
A sine curve with period 2π has its first maximum at x = π/2 and...
Which statement about y = cos(2(x + π/6)) is true?
To move y = sin(x) so that its zero at x = 0 becomes a zero at x =...
A cosine function has period 2π, amplitude 1, and now has a minimum...
The function y = 4sin(½x − π/8) can be written y =...
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