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.

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 vertical asymptotes.

Numerical Exploration the Function behavior Close the Vertical Asymptotes

Interactive Tutorial

1. Set different values of the parameters \( a \), \( b \), \( c \) and \( d \) and verify that the vertical asymptotes are located at the zeros of the denominator and are given by \( x = - \dfrac{b}{a} \) and \( x = - \dfrac{d}{c} \).
   
2. Evaluate the function around the vertical asymptotes and examine the behavior of the function whose value increases or decreases indefinitely.
   
3. Set parameter \( a = 2 \), \( b = 2\), \( c = 1\) and \( d = 1 \) and explain why the graph is located above the x axis. Examine the behavior (values) of the function around the vertical asymptote(s) \( x = - 1 \).
   

Exercises

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) \( h(x) = \dfrac{1}{(-x+2)(4x-8)} \)

Solutions to Above Exercises

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 \( x = 2 \)

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