Phase Shift Basics: Identify Horizontal Shifts in Trig Functions

  • 10th Grade
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| Questions: 20 | Updated: Dec 11, 2025
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1) What is the phase shift of y = sin(x − π/3)?

Explanation

Inside (x − π/3) means shift right by π/3.

So, the final answer is π/3 to the right.

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About This Quiz
Phase Shift Basics: Identify Horizontal Shifts In Trig Functions - Quiz

How far did the graph slide? This quiz is all about spotting horizontal shifts quickly. You’ll rewrite equations to a clean “shifted” form and decide whether a graph moved left or right—and by how much—across sine, cosine, and tangent.

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2) The function y = cos(2x + π) can be written in the form y = cos[2(x + c)]. What is c?

Explanation

2x + π = 2(x + π/2), so c = π/2 (left shift).

So, the final answer is c = π/2 to the left.

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3) Which function is obtained by shifting y = sin x to the left by π/4?

Explanation

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

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

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4) The function y = −cos(x − π/2) is the result of which transformation applied to y = cos x?

Explanation

(x − π/2) shifts right π/2; the negative sign reflects across x-axis.

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

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5) Identify the horizontal shift for y = 3sin(½x + π/4).

Explanation

For bx + φ, shift = left by φ/b. Here b = 1/2, φ = π/4 ⇒ shift = (π/4)/(1/2) = π/2 left.

So, the final answer is left π/2.

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6) Which function is equivalent to shifting y = tan x right by π/6?

Explanation

Right shift by a means replace x with (x − a).

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

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7) Consider y = sin(3x − 3π/2). What is the phase shift?

Explanation

Shift = left by φ/b with φ = −3π/2, b = 3 ⇒ left by (−3π/2)/3 = −π/2 ⇒ right π/2.

So, the final answer is right π/2.

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8) If y = cos x is shifted to become y = cos(x + a), and the resulting graph starts at a maximum at x = −π/3 instead of x = 0, what is a?

Explanation

To move the max from 0 to −π/3, shift left π/3 ⇒ use (x + π/3).

So, the final answer is a = π/3.

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9) Which of the following has no horizontal shift?

Explanation

Multiplying x by 2 changes period, not phase; others have ± constants.

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

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10) The graph of y = cos x is shifted left by π/5 and reflected across the x-axis. Which equation represents the result?

Explanation

Left shift: cos(x + π/5); reflection: add a negative outside.

So, the final answer is y = −cos(x + π/5).

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11) What horizontal shift transforms y = sin x into y = sin(x + 7π/6)?

Explanation

y = sin(x + a) shifts left by a.

So, the final answer is left 7π/6.

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12) Write y = sin(4x + π) in the form y = sin[4(x − h)]. What is h?

Explanation

4x + π = 4(x + π/4) = 4[x − (−π/4)] ⇒ h = −π/4.

So, the final answer is h = −π/4 (left).

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13) Which function is y = cos(x − π/3) equivalent to in terms of horizontal translation of y = cos x?

Explanation

(x − a) shifts right by a.

So, the final answer is shift right π/3.

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14) The function y = −2sin(3(x + π/9)) has which horizontal shift relative to y = sin x?

Explanation

(x + π/9) is a left shift by π/9; (−2) only reflects/scales vertically.

So, the final answer is left π/9.

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15) If y = tan(2x − π/2), what is the phase shift when written as y = tan[2(x − h)]?

Explanation

2x − π/2 = 2(x − π/4) ⇒ h = π/4 (right).

So, the final answer is h = π/4 (right).

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16) A sine wave y = sin x is shifted so that its midline crossing going upward occurs at x = π instead of x = 0. What is the new equation?

Explanation

The “upward midline crossing” of sin occurs at its zeros; move 0 → π with right shift.

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

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17) Which expression represents a left shift of π/8 for y = cos(5x)?

Explanation

Left shift by a: cos[5(x + a)] = cos(5x + 5a); a = π/8 ⇒ +5π/8.

So, the final answer is y = cos(5x + 5π/8).

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18) Two functions are given: f(x) = sin x and g(x) = sin(x − 2π/3). Which describes g relative to f?

Explanation

(x − a) shifts right by a.

So, the final answer is “shifted right 2π/3”.

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19) Consider y = cos(½x + π/6). What is the phase shift?

Explanation

Shift = left by φ/b with φ = π/6, b = 1/2 ⇒ (π/6)/(1/2) = π/3 left.

So, the final answer is left π/3.

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20) Which function is obtained by shifting y = −sin x to the right by π/10?

Explanation

Right shift by a: replace x with (x − a); keep the leading minus.

So, the final answer is y = −sin(x − π/10).

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What is the phase shift of y = sin(x − π/3)?
The function y = cos(2x + π) can be written in the form y = cos[2(x +...
Which function is obtained by shifting y = sin x to the left by π/4?
The function y = −cos(x − π/2) is the result of which...
Identify the horizontal shift for y = 3sin(½x + π/4).
Which function is equivalent to shifting y = tan x right by π/6?
Consider y = sin(3x − 3π/2). What is the phase shift?
If y = cos x is shifted to become y = cos(x + a), and the resulting...
Which of the following has no horizontal shift?
The graph of y = cos x is shifted left by π/5 and reflected across...
What horizontal shift transforms y = sin x into y = sin(x + 7π/6)?
Write y = sin(4x + π) in the form y = sin[4(x − h)]. What is h?
Which function is y = cos(x − π/3) equivalent to in terms of...
The function y = −2sin(3(x + π/9)) has which horizontal shift...
If y = tan(2x − π/2), what is the phase shift when written as y =...
A sine wave y = sin x is shifted so that its midline crossing going...
Which expression represents a left shift of π/8 for y = cos(5x)?
Two functions are given: f(x) = sin x and g(x) = sin(x − 2π/3)....
Consider y = cos(½x + π/6). What is the phase shift?
Which function is obtained by shifting y = −sin x to the right by...
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