\( \) \( \) \( \) \( \)

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.

\( a = \)            \( b = \)

\( c = \)
           \( d = \)


Hover the mousse cursor over the graph to trace the coordinates.
Hover the mousse cursor on the top right of the graph to have the option of downloading the graph as a png file, zooming in and out, shifting the graphs, ....


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 Functions
Slant Asymptotes of Rational Functions - Interactive
Vertical Asymptotes of Rational Functions - Interactive
Graphing Calculators .