An online graphing calculator to graph and explore the vertical asymptotes of
rational functions of the form \[ f(x) = \dfrac{1}{(a x + b)(c x + d)} \] is presented.
This graphing calculator also allows you to explore the vertical asymptotes behavior around the zeros of the denominator by evaluating the function around these zeros.
The vertical asymptotes of the above rational function are at the zeros of the denominator found by solving the equations:
\( a x + b = 0 \) and \( c x + d = 0 \)
which gives the equations of the vertical asymptotes as
\( x = - \dfrac{b}{a} \) and \( x = - \dfrac{d}{c} \)
Example
Let \( f (x) = \dfrac{1}{( x + 2)(2 x - 6)} \)
The denominator \( ( x + 2)(2 x - 6) \) of \( f(x) \) is equal to zero for
\( ( x + 2)(2 x - 6) = 0 \)
\( x = - 2 \) and \( x = 3 \)
If we try to evaluate \( f(x) \) at the zeros of the denominator \( x = - 2 \) and \( x = 3 \), we will have undefined values. However as \( x \) approaches and becomes very close to these values, \( f(x) \) either increases or decreases indefinitely.
In what follows, the behaviors of the vertical asymptotes 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 vertical asymptotes.
Find the vertical asymptotes of the following functions analytically and check your answers graphically using the graphing calculator.
a) \( f(x) = \dfrac{1}{(x-2)(x+3)} \) b) \( g(x) = \dfrac{1}{(-3x+6)(-x+3)} \) c) \( h(x) = \dfrac{1}{(-x+2)(4x-8)} \)
a) Two vertical asymptotes at \( x = 2 \) and \( x - 3\)
b) Two vertical asymptotes at \( x = 2 \) and \( x = 3 \)
c) One vertical asymptote at \( x = 2 \)