Cotangent Graphs: Period, Asymptotes, Shifts & Midlines Quiz

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

Explanation

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

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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) 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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3) 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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4) 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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5) On a graph, which represents y = cot(x − π/4)?

Explanation

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

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

Explanation

Factor 2 multiplies outputs ⇒ vertical stretch.

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7) 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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8) 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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9) 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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10) 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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11) 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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12) 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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13) 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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14) 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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15) 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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16) 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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17) 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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18) On a graph, which represents y = cot(2x)?

Explanation

b=2 halves the period ⇒ horizontal compression.

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19) For y = −cot(x), which is true?

Explanation

The leading minus reflects across the x-axis.

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20) 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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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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Which of the following is the reciprocal identity defining cotangent?
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)?
On a graph, which represents y = cot(x − π/4)?
On a graph, which represents y = 2·cot(x)?
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?
Solve on 0 < x < π: cot(x) = 0.
A model y = 1 + 0.5·cot(2x − π/3) has period and phase shift:
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:
The function y = a·cot(bx − c) + d has period:
If y = cot(2x), what is the period?
The function y = cot(x) is:
On a graph, which represents y = cot(2x)?
For y = −cot(x), which is true?
The smallest positive solution to cot(3x) = 0 is:
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