Cotangent Graphs: Period, Asymptotes, Shifts & Midlines Quiz

  • 11th Grade
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| Questions: 20 | Updated: Dec 11, 2025
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1) What is the period of y = cot(x)?

Explanation

Step 1: Cotangent shares period π with tangent.

Step 2: Therefore, the period is π.

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About This Quiz
Cotangent Graphs: Period, Asymptotes, Shifts & Midlines Quiz - Quiz

Meet cotangent’s signature look: decreasing curves between evenly spaced asymptotes. In this quiz, you’ll find the period, zeros, and asymptotes, and you’ll see how vertical stretches, reflections, and horizontal shifts change the graph. Get comfortable telling exactly what each parameter does—just by looking.

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2) The vertical asymptotes of y = cot(x) occur at:

Explanation

Step 1: cot(x) = cos(x)/sin(x) is undefined when sin(x)=0.

Step 2: sin(x)=0 at x=nπ.

So, asymptotes are x = nπ.

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3) The x-intercepts of y = cot(x) occur at:

Explanation

Step 1: x-intercepts occur where cot(x) = 0.

Step 2: cot(x) = 0 when cos(x) = 0.

Step 3: cos(x) = 0 at x = (2n + 1)π/2.

So, the final answer is x = (2n + 1)π/2.

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4) Which of the following is the reciprocal identity defining cotangent?

Explanation

By definition, cot(x)=cos(x)/sin(x).

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5) The function y = a·cot(bx − c) + d has period:

Explanation

Step 1: Base period for cot is π.

Step 2: Scaling x by b gives period π/|b|.

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6) Consider y = cot(x − π/4). Its graph is:

Explanation

(x − π/4) indicates a shift to the right by π/4.

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7) For y = 2·cot(x), which statement is true?

Explanation

Step 1: Multiplying by 2 scales output only.

Step 2: That is a vertical stretch by factor 2.

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8) What is the range of y = cot(x)?

Explanation

Step 1: cot(x) is unbounded above and below.

So, the range is (−∞, ∞).

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9) If y = cot(2x), what is the period?

Explanation

Step 1: Period = π/|b|.

Step 2: b=2 ⇒ period = π/2.

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10) The function y = cot(x) is:

Explanation

Step 1: cot(−x)=cos(−x)/sin(−x)=cos(x)/(−sin(x))=−cot(x).

Step 2: Hence, cot is odd.

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11) On a graph, which represents y = cot(x − π/4)?

Explanation

Inside (x − π/4) ⇒ shift right by π/4.

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12) On a graph, which represents y = 2·cot(x)?

Explanation

Factor 2 multiplies outputs ⇒ vertical stretch.

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13) On a graph, which represents y = cot(2x)?

Explanation

b=2 halves the period ⇒ horizontal compression.

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14) Which graph shows the basic y = cot(x)?

Explanation

Basic cot has asymptotes at x=nπ and period π; zeros at x=π/2+nπ.

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15) Which transformation maps y = cot(x) to y = cot(x) − 3?

Explanation

Subtracting 3 outside the function moves the graph down by 3 units.

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16) Which of the following has vertical asymptotes at x = π/6 + nπ?

Explanation

Asymptotes where sin(·)=0 ⇒ (x − π/6) = nπ.

So, asymptotes at x = π/6 + nπ.

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17) For y = −cot(x), which is true?

Explanation

The leading minus reflects across the x-axis.

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18) Solve on 0 < x < π: cot(x) = 0.

Explanation

Step 1: cot(x)=0 ⇔ cos(x)=0.

Step 2: In (0,π), cos(x)=0 at x=π/2.

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19) The smallest positive solution to cot(3x) = 0 is:

Explanation

cot(3x)=0 ⇒ 3x=π/2 ⇒ x=π/6 (smallest positive).

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20) A model y = 1 + 0.5·cot(2x − π/3) has period and phase shift:

Explanation

Step 1: Period for cot is π/b with b=2 ⇒ π/2.

Step 2: Phase shift = c/b where c=π/3 ⇒ (π/3)/2 = π/6 to the right.

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What is the period of y = cot(x)?
The vertical asymptotes of y = cot(x) occur at:
The x-intercepts of y = cot(x) occur at:
Which of the following is the reciprocal identity defining cotangent?
The function y = a·cot(bx − c) + d has period:
Consider y = cot(x − π/4). Its graph is:
For y = 2·cot(x), which statement is true?
What is the range of y = cot(x)?
If y = cot(2x), what is the period?
The function y = cot(x) is:
On a graph, which represents y = cot(x − π/4)?
On a graph, which represents y = 2·cot(x)?
On a graph, which represents y = cot(2x)?
Which graph shows the basic y = cot(x)?
Which transformation maps y = cot(x) to y = cot(x) − 3?
Which of the following has vertical asymptotes at x = π/6 + nπ?
For y = −cot(x), which is true?
Solve on 0 < x < π: cot(x) = 0.
The smallest positive solution to cot(3x) = 0 is:
A model y = 1 + 0.5·cot(2x − π/3) has period and phase shift:
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