Cotangent Graphs: Period, Asymptotes, Shifts & Midlines Quiz

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

Explanation

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

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3) 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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4) 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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5) 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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6) 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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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) On a graph, which represents y = cot(2x)?

Explanation

b=2 halves the period ⇒ horizontal compression.

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

Explanation

Factor 2 multiplies outputs ⇒ vertical stretch.

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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) 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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12) 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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13) 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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14) 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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15) Which of the following is the reciprocal identity defining cotangent?

Explanation

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

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16) 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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17) 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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18) 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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19) For y = −cot(x), which is true?

Explanation

The leading minus reflects across the x-axis.

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