Asymptotes: Videos & Practice Problems

Which of the following best describes an asymptote?

If a graph approaches the line y = 3 as x goes to infinity, what is the equation of the horizontal asymptote?

A function f(x) has a vertical asymptote at x = 2. Which of the following could be the equation of f(x)?

What typically happens to the values of a function as the graph approaches a vertical asymptote from both directions?

When sketching the function f(x) = 1/x, how should the asymptotes be drawn on the graph?

Given the function f(x) = (2x^2 + 3)/(x^2 + 1), identify the horizontal asymptote.

For the function g(x) = (x^2 - 4)/(x^2 - 1), where are the vertical asymptotes located?

Without knowing the exact function, how can you sketch a graph that has a horizontal asymptote at y = -3 and a vertical asymptote at x = 4?

If a function has a vertical asymptote at x = 2 and is symmetric about the y-axis, what is another possible asymptote?

What is the significance of a horizontal asymptote in the behavior of a graph?

For the function h(x) = (3x + 2)/(x + 1), what is the horizontal asymptote?

What are the vertical asymptotes of the function j(x) = (x^2 - 1)/(x^2 - 4)?

Describe the behavior of f(x) = 5/(x+1) as x approaches positive and negative infinity.

For the function k(x) = 1/(x-3), what happens as x approaches 3 from the left and right?

How should you represent the asymptotes of the function p(x) = 1/(x+2) on a graph?

The function q(x) = (2x^3 + 3x^2)/(x^3 + 1) approaches which horizontal asymptote as x approaches infinity?

Which x-values are vertical asymptotes for the function r(x) = (x+1)/(x^2 - 9)?

How would you sketch a graph with a horizontal asymptote at y = 1 and a vertical asymptote at x = -1, without knowing the function?

A function is symmetric about the origin and has a vertical asymptote at x = 4. What other asymptote might the graph have?

A graph has a vertical asymptote at $x=-3$ and a horizontal asymptote at $y=4$. Find the value of $a+k$ where the asymptotes are written as $x=a$ and $y=k$.

For the function $f\left(x\right)=3+\frac{5}{x^2+1}$, determine the y-value of its horizontal asymptote.

Find the x-value of the vertical asymptote of $y=\frac{7}{x-4}+2$.

A graph is symmetric about the y-axis. As $x\to \infty$ and $x\to -\infty$, the graph approaches $y=-1$. The graph becomes unbounded near $x=-5$ and near its symmetric partner. Find the sum of all asymptote constants.

Let $g\left(x\right)=-\frac{2}{(x+1{)}^2}-3$. Find the product of the constants in the equations of its vertical and horizontal asymptotes.

For $f\left(x\right)=\frac{x^2+1}{x^2-9}$, find the positive x-value of a vertical asymptote.

The function $h\left(x\right)=\frac{x}{x-2}$ has a horizontal asymptote. Find the x-coordinate where the graph crosses that horizontal asymptote. Answer as '$0$' if the graph never crosses it.

A function has the form $y=\frac{a}{x-h}+k$. If its asymptotes are $x=6$ and $y=-4$, and the graph passes through $(8,-3)$, find $a$.

Consider $f\left(x\right)=\frac{x+1}{(x-2)(x+2)}$. Find the absolute difference between the two vertical asymptote constants.

For the function $y=5-\frac{12}{(x-1{)}^2}$, determine the y-value of the horizontal asymptote.