Q. 1 - Precalculus Enhanced with Graphing Utilities

Question

The graph of $y=\frac{3x-6}{x-4}$ has a vertical asymptote. What is it? 

Step-by-Step Solution

Answer

The vertical asymptote of the graph is  $x=4$.

Step 1. Definition

A rational function $R\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$, in lowest terms, will have a vertical asymptote $x=r$ if $r$ is a real zero of the denominator $Q\left(x\right)$. 

Step 2. Find the vertical asymptote

The function $y=\frac{3x-6}{x-4}$ is in lowest terms. The zero of the denominator is $4$.

$\begin{array}{rcl}x-4& =& 0\\ x& =& 4\end{array}$

So the line $x=4$ is the vertical asymptote of the given rational function.