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.
Enter values for the constants \( a \) and \( b \) and press on "Graph". Change the values of \( a \) and \( b \) and investigate the horizontal asymptotes.
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) \( h(x) = \dfrac{x+2}{-x-8} \)
a) Horizontal asymptote at \( y = 1\)
b) Horizontal asymptote at \( y = 3\)
c) Horizontal asymptote at \( y = -1\)