# Horizontal Asymptotes of Rational Functions - Interactive

An online graphing calculator to graph and explore horizontal asymptotes of
rational functions of the form \[ f(x) = \dfrac{a x + b}{c x + d} \] is presented.

This graphing calculator also allows you to explore the behavior of the function as the variable \( x \) increases or decreases indefinitely.

If we let \( x \) take larger values, the numerator \( a x + b \) takes values closer to \( ax \) and the denominator \( c x + d \) takes values closer to \( c x \) and the value of the function \( f(x) \) takes values closer to:

\[ \dfrac{a x }{c x} = \dfrac{a}{c}\]

and we call the line \( y = \dfrac{a}{c} \) the horizontal asymptote.

Similar behavior, as \( x \) takes smaller values such as \( -10^6\), \( -10^{10} \), ...., is observed.

This graphing calculator also allows you to explore the horizontal asymptote behavior by evaluating the function at very large and very small values of the variable.

Example

Let \( f (x) = \dfrac{a x + b}{c x + d} = \dfrac{-2x + 1}{2 x - 3} \)

\( a = -2 \) is the leading coefficient in the numerator and \( c = 2\) is the leading coefficient in the denominator.

As \( x \) becomes large, \( f(x) \) approaches the value \( \dfrac{a}{c} = \dfrac{-2}{2} = - 1\)

The line \( y = - 1 \) is called the horizontal asymptote.

In what follows, the behaviors of the horizontal asymptotes of rational functions may be explored graphically and numerically.

## Use The Graphing Calculator to Graph Rational Functions

Enter values for the constants \( a \) and \( b \) and press on "Graph". Change the values of \( a \) and \( b \) and investigate the horizontal asymptotes.

## Numerical Exploration of the Function Behavior Close the Horizontal Asymptote

Click on "Evaluate" to obtain ordered pairs of the form \( (x, f(x)) \) for very small values ( \(x\) decreases to small values) and very large values ( \(x\) increases to large values ). Examine the values of the function as \( x \) decreases and as \( x \) increases.

## Interactive Tutorial

Set different values of the parameters \( a \), \( b \), \( c \) and \( d \) and verify that the horizontal asymptote is given by the equation \( y = \dfrac{a}{c} \) and therefore does not depend on \( b \) and \( d \)## Exercises

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

a) \( f(x) = \dfrac{x-2}{x+3} \) b) \( g(x) = \dfrac{-3x+6}{-x+3} \) c) \( h(x) = \dfrac{x+2}{-x-8} \)

## Solutions to Above Exercises

a) Horizontal asymptote at \( y = 1\)

b) Horizontal asymptote at \( y = 3\)

c) Horizontal asymptote at \( y = -1\)

## More References and Links

Rational FunctionsSlant Asymptotes of Rational Functions - Interactive

Vertical Asymptotes of Rational Functions - Interactive

Graphing Calculators .