Asymptotes and intercepts of trig functions worksheet for Precalculus

Tangent and secant blow up where cosine is zero, at π/2 plus every π; cotangent and cosecant where sine is zero, at every π. Sine crosses the axis at every π and cosine at π/2 plus every π. With bx inside, divide every one of those by b: the first one moves, and so does the spacing.

Also taught in Algebra 2, at its own level and with its own examples.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Trig Asymptotes and Intercepts Date Period Answer each question in terms of π. Locate the asymptotes $y = \sec (x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $y = \csc (2 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $y = \sec (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $y = \cot (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $y = \tan (4 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $y = \csc (4 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $y = \sec (2 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? Locate the $x$ -intercepts $y = 4 \cos (5 x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $y = 2 \sin (x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $y = \cos ⁣ (\frac{3}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $y = \sin ⁣ (\frac{1}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $y = 3 \cos ⁣ (\frac{1}{3} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $y = 4 \sin ⁣ (\frac{3}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $y = 2 \cos ⁣ (\frac{1}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? List the vertical asymptotes on an interval List the vertical asymptotes of $y = \cot ⁣ (x + \frac{π}{4})$ on $[0, 2 π)$ List the vertical asymptotes of $y = \sec ⁣ (x - \frac{2 π}{3})$ on $[0, 2 π)$ List the vertical asymptotes of $y = \tan ⁣ (x + \frac{5 π}{3})$ on $[0, 2 π)$ List the vertical asymptotes of $y = \tan ⁣ (x + \frac{π}{2})$ on $[0, 2 π)$ List the vertical asymptotes of $y = \cot ⁣ (x + \frac{4 π}{3})$ on $[0, 2 π)$ List the vertical asymptotes of $y = \cot ⁣ (x - \frac{5 π}{4})$ on $[0, 2 π)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Trig Asymptotes and Intercepts

Date Period

Answer each question in terms of π.

Locate the asymptotes
$y = \sec (x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes?
$y = \csc (2 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes?
$y = \sec (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes?
$y = \cot (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes?
$y = \tan (4 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes?
$y = \csc (4 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes?
$y = \sec (2 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes?

Locate the $x$ -intercepts
$y = 4 \cos (5 x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts?
$y = 2 \sin (x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts?
$y = \cos ⁣ (\frac{3}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts?
$y = \sin ⁣ (\frac{1}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts?
$y = 3 \cos ⁣ (\frac{1}{3} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts?
$y = 4 \sin ⁣ (\frac{3}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts?
$y = 2 \cos ⁣ (\frac{1}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts?

List the vertical asymptotes on an interval
List the vertical asymptotes of $y = \cot ⁣ (x + \frac{π}{4})$ on $[0, 2 π)$
List the vertical asymptotes of $y = \sec ⁣ (x - \frac{2 π}{3})$ on $[0, 2 π)$
List the vertical asymptotes of $y = \tan ⁣ (x + \frac{5 π}{3})$ on $[0, 2 π)$
List the vertical asymptotes of $y = \tan ⁣ (x + \frac{π}{2})$ on $[0, 2 π)$
List the vertical asymptotes of $y = \cot ⁣ (x + \frac{4 π}{3})$ on $[0, 2 π)$
List the vertical asymptotes of $y = \cot ⁣ (x - \frac{5 π}{4})$ on $[0, 2 π)$



Sigma Prep · sigmaprep.io/worksheets Name Trig Asymptotes and InterceptsAnswer keyVersion 1 Date Period Answer each question in terms of π. Locate the asymptotes $y = \sec (x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{2}$ , every $π$ $y = \csc (2 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{2}$ , every $\frac{π}{2}$ $y = \sec (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{6}$ , every $\frac{π}{3}$ $y = \cot (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{3}$ , every $\frac{π}{3}$ $y = \tan (4 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{8}$ , every $\frac{π}{4}$ $y = \csc (4 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{4}$ , every $\frac{π}{4}$ $y = \sec (2 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{4}$ , every $\frac{π}{2}$ Locate the $x$ -intercepts $y = 4 \cos (5 x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = \frac{π}{10}$ , every $\frac{π}{5}$ $y = 2 \sin (x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = π$ , every $π$ $y = \cos ⁣ (\frac{3}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = \frac{π}{3}$ , every $\frac{2 π}{3}$ $y = \sin ⁣ (\frac{1}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = 2 π$ , every $2 π$ $y = 3 \cos ⁣ (\frac{1}{3} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = \frac{3 π}{2}$ , every $3 π$ $y = 4 \sin ⁣ (\frac{3}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = \frac{2 π}{3}$ , every $\frac{2 π}{3}$ $y = 2 \cos ⁣ (\frac{1}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = π$ , every $2 π$ List the vertical asymptotes on an interval List the vertical asymptotes of $y = \cot ⁣ (x + \frac{π}{4})$ on $[0, 2 π)$ $x = \frac{3 π}{4}, \frac{7 π}{4}$ List the vertical asymptotes of $y = \sec ⁣ (x - \frac{2 π}{3})$ on $[0, 2 π)$ $x = \frac{π}{6}, \frac{7 π}{6}$ List the vertical asymptotes of $y = \tan ⁣ (x + \frac{5 π}{3})$ on $[0, 2 π)$ $x = \frac{5 π}{6}, \frac{11 π}{6}$ List the vertical asymptotes of $y = \tan ⁣ (x + \frac{π}{2})$ on $[0, 2 π)$ $x = 0, π$ List the vertical asymptotes of $y = \cot ⁣ (x + \frac{4 π}{3})$ on $[0, 2 π)$ $x = \frac{2 π}{3}, \frac{5 π}{3}$ List the vertical asymptotes of $y = \cot ⁣ (x - \frac{5 π}{4})$ on $[0, 2 π)$ $x = \frac{π}{4}, \frac{5 π}{4}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Trig Asymptotes and InterceptsAnswer keyVersion 1

Date Period

Answer each question in terms of π.

Locate the asymptotes
$y = \sec (x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{2}$ , every $π$
$y = \csc (2 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{2}$ , every $\frac{π}{2}$
$y = \sec (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{6}$ , every $\frac{π}{3}$
$y = \cot (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{3}$ , every $\frac{π}{3}$
$y = \tan (4 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{8}$ , every $\frac{π}{4}$
$y = \csc (4 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{4}$ , every $\frac{π}{4}$
$y = \sec (2 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? $x = \frac{π}{4}$ , every $\frac{π}{2}$

Locate the $x$ -intercepts
$y = 4 \cos (5 x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = \frac{π}{10}$ , every $\frac{π}{5}$
$y = 2 \sin (x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = π$ , every $π$
$y = \cos ⁣ (\frac{3}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = \frac{π}{3}$ , every $\frac{2 π}{3}$
$y = \sin ⁣ (\frac{1}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = 2 π$ , every $2 π$
$y = 3 \cos ⁣ (\frac{1}{3} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = \frac{3 π}{2}$ , every $3 π$
$y = 4 \sin ⁣ (\frac{3}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = \frac{2 π}{3}$ , every $\frac{2 π}{3}$
$y = 2 \cos ⁣ (\frac{1}{2} x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? $x = π$ , every $2 π$

List the vertical asymptotes on an interval
List the vertical asymptotes of $y = \cot ⁣ (x + \frac{π}{4})$ on $[0, 2 π)$ $x = \frac{3 π}{4}, \frac{7 π}{4}$
List the vertical asymptotes of $y = \sec ⁣ (x - \frac{2 π}{3})$ on $[0, 2 π)$ $x = \frac{π}{6}, \frac{7 π}{6}$
List the vertical asymptotes of $y = \tan ⁣ (x + \frac{5 π}{3})$ on $[0, 2 π)$ $x = \frac{5 π}{6}, \frac{11 π}{6}$
List the vertical asymptotes of $y = \tan ⁣ (x + \frac{π}{2})$ on $[0, 2 π)$ $x = 0, π$
List the vertical asymptotes of $y = \cot ⁣ (x + \frac{4 π}{3})$ on $[0, 2 π)$ $x = \frac{2 π}{3}, \frac{5 π}{3}$
List the vertical asymptotes of $y = \cot ⁣ (x - \frac{5 π}{4})$ on $[0, 2 π)$ $x = \frac{π}{4}, \frac{5 π}{4}$

How to do these

$y = \tan (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes?

tan is undefined where 3x = π/2, so x = π/6.
Asymptotes of tan are π apart, so here they are π/3 apart.

The answer is $x = \frac{π}{6}$ , every $\frac{π}{3}$ .

Where students go wrong

Forgetting to divide by b. The asymptotes of tan(3x) are three times as close together as those of tan x.

Also called vertical asymptotes of tangent, zeros of sine and cosine, period and asymptotes or trig graph features.

What you can put on this worksheet

Where the asymptotes are: $y = \sec (3 x)$ : where is the first asymptote to the right of the $y$ -axis, and how far apart are the asymptotes? → $x = \frac{π}{6}$ , every $\frac{π}{3}$
Where the x-intercepts are: $y = \cos (3 x)$ : where is the first $x$ -intercept to the right of the $y$ -axis, and how far apart are the intercepts? → $x = \frac{π}{6}$ , every $\frac{π}{3}$
Shifted graphs, listed on an interval: List the vertical asymptotes of $y = \sec ⁣ (x - \frac{4 π}{3})$ on $[0, 2 π)$ → $x = \frac{5 π}{6}, \frac{11 π}{6}$

Questions about these worksheets

Yes. Every asymptotes and intercepts of trig functions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the asymptotes and intercepts of trig functions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: where the asymptotes are, where the x-intercepts are and shifted graphs, listed on an interval. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting to divide by b. The asymptotes of tan(3x) are three times as close together as those of tan x.

Asymptotes and intercepts of trig functions worksheet for Precalculus

Locate the asymptotes

Locate the xxx-intercepts

List the vertical asymptotes on an interval

Locate the asymptotes

Locate the xxx-intercepts

List the vertical asymptotes on an interval

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these asymptotes and intercepts of trig functions worksheets free?+

Do the asymptotes and intercepts of trig functions worksheets come with an answer key?+

Are these asymptotes and intercepts of trig functions worksheets printable?+

What grade are these asymptotes and intercepts of trig functions worksheets for?+

Can I get a different version of the same asymptotes and intercepts of trig functions worksheet?+

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What kinds of question are on a asymptotes and intercepts of trig functions worksheet?+

What do students get wrong with asymptotes and intercepts of trig functions?+

Locate the $x$ -intercepts

Locate the $x$ -intercepts