Vertical Asymptotes of Rational Functions - Interactive

An online graphing calculator to graph and explore the vertical asymptotes of rational functions of the form:

\[f(x) = \dfrac{1}{(a x + b)(c x + d)}\]

This calculator also allows you to numerically explore the behavior around the zeros of the denominator by evaluating the function values close to those zeros.

The vertical asymptotes of the rational function are located at the zeros of the denominator, found by solving the equations: \[a x + b = 0 \quad \text{and} \quad c x + d = 0\] This gives the equations of the vertical asymptotes as: \[x = - \dfrac{b}{a} \quad \text{and} \quad x = - \dfrac{d}{c}\]

Example Walkthrough

Let \[f(x) = \dfrac{1}{( x + 2)(2 x - 6)}\]

The denominator \(( x + 2)(2 x - 6)\) is equal to zero when: \[( x + 2)(2 x - 6) = 0\] Which gives \(x = -2\) and \(x = 3\).

If we evaluate \(f(x)\) at exactly \(x = -2\) or \(x = 3\), we get an undefined value (division by zero). However, as \(x\) approaches these values, \(f(x)\) either increases indefinitely (\(+\infty\)) or decreases indefinitely (\(-\infty\)).

Use the Graphing Calculator

Enter values for the constants \(a\), \(b\), \(c\), and \(d\) below and click "Graph". Change the values to investigate how they alter the vertical asymptotes.

Hover your mouse cursor over the graph to trace the coordinates. Use the toolbar at the top right to pan, zoom, or download the graph as a PNG.

Numerical Exploration of Limits Near Asymptotes

Enter an \(x\)-value that causes the denominator to equal zero (e.g., \(x = 3\)) and click "Evaluate Around". This generates data tables showing \(f(x)\) as you approach the asymptote from the left and the right. Notice how \(f(x)\) explodes to infinity or negative infinity.

Interactive Tutorial

  1. Set different values of the parameters \(a\), \(b\), \(c\) and \(d\) in the calculator and verify that the vertical asymptotes are located exactly at \(x = - \dfrac{b}{a}\) and \(x = - \dfrac{d}{c}\).
  2. Use the Numerical Exploration tool to evaluate the function around the asymptotes. Observe how the function value grows indefinitely.
  3. Set parameter \(a = 2\), \(b = 2\), \(c = 1\), and \(d = 1\). Explain why the graph is located entirely above the x-axis. Examine the behavior of the function around the vertical asymptote \(x = -1\).

Practice Exercises

Find the vertical asymptotes of the following functions analytically, then check your answers graphically using the graphing calculator.

  1. \(f(x) = \dfrac{1}{(x-2)(x+3)}\)
  2. \(g(x) = \dfrac{1}{(-3x+6)(-x+3)}\)
  3. \(h(x) = \dfrac{1}{(-x+2)(4x-8)}\)
View Solutions to Exercises
  1. Setting the factors to zero gives \(x - 2 = 0\) and \(x + 3 = 0\).
    Two vertical asymptotes at \(x = 2\) and \(x = -3\).
  2. Setting the factors to zero gives \(-3x + 6 = 0 \Rightarrow x = 2\) and \(-x + 3 = 0 \Rightarrow x = 3\).
    Two vertical asymptotes at \(x = 2\) and \(x = 3\).
  3. Setting the factors to zero gives \(-x + 2 = 0 \Rightarrow x = 2\) and \(4x - 8 = 0 \Rightarrow x = 2\). Since both factors share the same zero:
    One vertical asymptote at \(x = 2\).

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