Build the Cotangent Function from Its Graph Quiz

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| Questions: 20 | Updated: Oct 31, 2025
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1) A cotangent graph has vertical asymptotes at x = 0 and x = π, and an x-intercept at x = π/2. Which equation matches?

Explanation

Step 1: For cot(x), vertical asymptotes occur at x = nπ.

Step 2: The x-intercept at π/2 matches the standard cotangent graph.

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

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About This Quiz
Build The Cotangent Function From Its Graph Quiz - Quiz

From sketch to symbol! You’ll read off asymptotes and intercepts, note whether the graph is decreasing, and then write the cotangent equation that matches. Expect practice picking the right period and phase shift so your function lines up with the graph every time.

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2) The graph of a cotangent function has period π/3. Which is a correct equation?

Explanation

Step 1: Formula for cotangent period = π/|b|.

Step 2: π/b = π/3 ⇒ b = 3.

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

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3) A cotangent graph has vertical asymptotes at x = −π and x = 0, and an x-intercept at x = −π/2. Which equation matches?

Explanation

Step 1: For y = cot(x), asymptotes occur at multiples of π.

Step 2: Between −π and 0, y = cot(x) crosses the x-axis at −π/2.

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

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4) A graph shows a cotangent curve with vertical asymptotes at x = −π/4 and x = 3π/4, an x-intercept at x = π/4, and decreasing from left to right. Which equation fits?

Explanation

Step 1: Asymptotes for y = cot(x − c) occur at x = c + nπ.

Step 2: Choose c = −π/4 ⇒ asymptotes at −π/4 and 3π/4; zero at c + π/2 = π/4.

Step 3: Standard cotangent decreases for a > 0.

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

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5) A cotangent function has amplitude scaled by 3, period π, and no shifts. Which is correct?

Explanation

Step 1: Period π ⇒ b = 1.

Step 2: Vertical stretch by 3 ⇒ multiply by 3.

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

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6) The graph shows vertical asymptotes at x = 1 and x = 3, and an x-intercept at x = 2, decreasing between asymptotes. What is the equation?

Explanation

Step 1: Distance between asymptotes is 2.

Step 2: Period π/b = 2 ⇒ b = π/2.

Step 3: Left asymptote at x = 1 ⇒ shift c = 1; zero at x = c + π/(2b) = 2.

So, the final answer is y = cot((π/2)(x − 1)).

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7) A cotangent graph has period π/2 and is shifted right by π/4. Which equation matches?

Explanation

Step 1: Period π/b = π/2 ⇒ b = 2.

Step 2: Right shift π/4 ⇒ (x − π/4).

So, the final answer is y = cot(2(x − π/4)).

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8) Which function has vertical asymptotes at x = −2 and x = −1 and an x-intercept at x = −1.5, decreasing on that interval?

Explanation

Step 1: Distance between asymptotes is 1 ⇒ period 1 ⇒ b = π.

Step 2: Left asymptote at −2 ⇒ shift (x + 2).

Step 3: Standard cot decreases.

So, the final answer is y = cot(π(x + 2)).

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9) A graph of y = a cot(b(x − c)) + d has asymptotes at x = c and x = c + π/b, and passes through (c + π/(2b), d). Given asymptotes at x = −π/6 and x = π/6, midline y = 2, find an equation with a positive a.

Explanation

Step 1: Distance between asymptotes is π/3 ⇒ π/b = π/3 ⇒ b = 3.

Step 2: Midline y = 2 ⇒ add +2.

Step 3: With c = 0 and a > 0, use y = cot(3x) + 2.

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10) The graph is y = cot(x) reflected over the x-axis and shifted up by 1. Which is correct?

Explanation

Step 1: Reflection across x-axis ⇒ negative sign.

Step 2: Vertical shift up by 1 ⇒ +1 outside.

So, the final answer is y = −cot(x) + 1.

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11) Identify the period of y = cot(4x − π).

Explanation

Step 1: For y = cot(bx − c), period = π/|b|.

Step 2: b = 4 ⇒ period = π/4.

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12) A cotangent graph shows asymptotes at x = 2 and x = 2 + π/2, midline y = 0. Which equation matches?

Explanation

Step 1: Distance between asymptotes is π/2 ⇒ π/b = π/2 ⇒ b = 2.

Step 2: Shift right 2 ⇒ (x − 2).

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

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13) The point (π/3, 0) lies on a cotangent graph whose nearest asymptotes are x = 0 and x = 2π/3. Which equation fits?

Explanation

Step 1: Distance between asymptotes is 2π/3 ⇒ π/b = 2π/3 ⇒ b = 3/2.

Step 2: Then zeros occur at x = (2n+1)π/(2b) = (2n+1)π/3; for n=0 ⇒ π/3.

So, the final answer is y = cot(1.5x).

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14) The graph of y = cot(x − π/3) is shifted:

Explanation

(x − π/3) shifts the graph right by π/3.

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15) The graph shows y = cot(bx) with x-intercepts at x = ±π/6, ±π/2, …. Which value of b fits?

Explanation

Step 1: Distance between consecutive zeros is π/b.

Step 2: From −π/6 to π/6 the spacing is π/3 ⇒ π/b = π/3 ⇒ b = 3.

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16) Find vertical asymptotes of y = cot(2x + π/2).

Explanation

Step 1: Undefined where sin(2x + π/2) = 0.

Step 2: 2x + π/2 = nπ ⇒ x = −π/4 + nπ/2.

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17) A cotangent graph has asymptotes at x = −π and x = 0, midline y = −1, and decreasing. Which equation matches?

Explanation

Step 1: Decreasing behavior corresponds to standard cotangent (a > 0).

Step 2: Midline y = −1 ⇒ subtract 1.

So, the final answer is y = cot(x) − 1.

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18) The graph shows y = a cot(x) with steeper slope at x = π/4 than y = cot(x). Which is true?

Explanation

Step 1: Larger |a| yields steeper vertical scaling.

Step 2: Steeper and same orientation ⇒ a > 1.

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19) Which function has period π and asymptotes at x = π/6 + kπ?

Explanation

Step 1: Standard period is π.

Step 2: Shifting right by π/6 gives asymptotes at x = π/6 + kπ.

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

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20) A graph shows y = cot(3(x − π/9)) + 2. Which feature is correct?

Explanation

Step 1: (x − π/9) ⇒ shift right π/9.

Step 2: +2 ⇒ vertical shift up 2.

Step 3: b = 3 ⇒ period π/3.

So, the final answer is phase shift right π/9.

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A cotangent graph has vertical asymptotes at x = 0 and x = π, and...
The graph of a cotangent function has period π/3. Which is a...
A cotangent graph has vertical asymptotes at x = −π and x = 0, and...
A graph shows a cotangent curve with vertical asymptotes at x =...
A cotangent function has amplitude scaled by 3, period π, and no...
The graph shows vertical asymptotes at x = 1 and x = 3, and an...
A cotangent graph has period π/2 and is shifted right by π/4. Which...
Which function has vertical asymptotes at x = −2 and x = −1 and an...
A graph of y = a cot(b(x − c)) + d has asymptotes at x = c and x = c...
The graph is y = cot(x) reflected over the x-axis and shifted up by 1....
Identify the period of y = cot(4x − π).
A cotangent graph shows asymptotes at x = 2 and x = 2 + π/2, midline...
The point (π/3, 0) lies on a cotangent graph whose nearest asymptotes...
The graph of y = cot(x − π/3) is shifted:
The graph shows y = cot(bx) with x-intercepts at x = ±π/6, ±π/2,...
Find vertical asymptotes of y = cot(2x + π/2).
A cotangent graph has asymptotes at x = −π and x = 0, midline y =...
The graph shows y = a cot(x) with steeper slope at x = π/4 than y =...
Which function has period π and asymptotes at x = π/6 + kπ?
A graph shows y = cot(3(x − π/9)) + 2. Which feature is correct?
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