Horizontal Asymptotes of Rational Functions - Interactive
An online graphing calculator to graph and explore the horizontal asymptotes of rational functions of the form \[f(x) = \dfrac{a x + b}{c x + d}\] is presented below.
This calculator allows you to explore the behavior of the function as the variable \(x\) increases or decreases indefinitely. As \(x\) grows arbitrarily large (approaches \(\infty\) or \(-\infty\)), the constant terms \(b\) and \(d\) become insignificant compared to the terms \(ax\) and \(cx\).
Thus, the value of the function \(f(x)\) behaves like: \[f(x) \approx \dfrac{a x }{c x} = \dfrac{a}{c}\] Because of this, we call the horizontal line \(y = \dfrac{a}{c}\) the horizontal asymptote. Similar behavior is observed as \(x\) takes smaller negative values (such as \(-10^6\), \(-10^{10}\), etc.).
You can also numerically explore this horizontal asymptote behavior by generating data tables of the function evaluated at extreme values of \(x\).
Example
Let \[f(x) = \dfrac{a x + b}{c x + d} = \dfrac{-2x + 1}{2 x - 3}\]
Here, \(a = -2\) is the leading coefficient in the numerator, and \(c = 2\) is the leading coefficient in the denominator. As \(x\) becomes very large, \(f(x)\) approaches the ratio of these coefficients: \[\dfrac{a}{c} = \dfrac{-2}{2} = -1\]
The line \(y = -1\) is the horizontal asymptote. Explore this graphically below by inputting these exact coefficients.
Use the Graphing Calculator
Enter values for the constants \(a\), \(b\), \(c\), and \(d\) and click "Graph". Change the values to investigate how they affect the horizontal 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
Click "Evaluate" below to obtain ordered pairs \((x, f(x))\) as \(x\) approaches \(-\infty\) (decreases to small values) and \(\infty\) (increases to large values). Notice how the \(y\)-values get closer and closer to \(a/c\).
Interactive Tutorial & Exercises
Tutorial Step: Set different values of the parameters \(a\), \(b\), \(c\), and \(d\) in the calculator above. Verify that the horizontal asymptote is always given by the equation \(y = \dfrac{a}{c}\) and therefore does not depend on \(b\) and \(d\).
Practice: Find the horizontal asymptotes of the following functions analytically, and then check your answers graphically using the calculator above.
- \(f(x) = \dfrac{x-2}{x+3}\)
- \(g(x) = \dfrac{-3x+6}{-x+3}\)
- \(h(x) = \dfrac{x+2}{-x-8}\)
View Solutions
- a) The leading coefficients are \(a = 1\) and \(c = 1\).
Horizontal asymptote at \(y = 1\). - b) The leading coefficients are \(a = -3\) and \(c = -1\).
Horizontal asymptote at \(y = 3\). - c) The leading coefficients are \(a = 1\) and \(c = -1\).
Horizontal asymptote at \(y = -1\).