Graphing Rational Functions

To graph the given rational function, we will start by identifying the values of $a,$ $b,$ $c,$ and $d.$ $$\begin{array}{c}h(x)=\frac{8x+3}{2x-6}\iff h(x)=\frac{8x+3}{2x+(\text{-}6)}\end{array}$$ We see above that $a$ $=$ $8,$ $b$ $=$ $3,$ $c$ $=$ $2,$ and $d$ $=$ $\text{-}6.$ With this information, we will find the asymptotes and graph the function.

Asymptotes

Let's start by calculating the vertical asymptote. To do so, we need to solve the equation $cx+d=0.$ Since we already know the values of $c$ and $d,$ we can substitute them in this equation and solve for $x.$ The vertical asymptote is the line $x=3.$ The horizontal asymptote is given by the equation $y=\frac{a}{c}.$ Since we already know the values of $a$ and $c,$ we will substitute them in this equation. $$\begin{array}{c}y=\frac{a}{c}\stackrel{\text{substitute}}{\to }y=\frac{8}{2}\\ y=4\end{array}$$ The horizontal asymptote is the line $y=4.$

Graph

Let's make a table of values to find points on the curve. Make sure to include values of $x$ to the left and to the right of the vertical asymptote.

$x$	$\frac{8x+3}{2x-6}$	$h(x)$
$\text{-}3$	$\frac{8(\text{-}3)+3}{2(\text{-}3)-6}$	$1.75$
$\text{-}1$	$\frac{8(\text{-}1)+3}{2(\text{-}1)-6}$	$0.625$
$1$	$\frac{8(1)+3}{2(1)-6}$	$\text{-}2.75$
$5$	$\frac{8(5)+3}{2(5)-6}$	$10.75$
$7$	$\frac{8(7)+3}{2(7)-6}$	$7.375$
$9$	$\frac{8(9)+3}{2(9)-6}$	$6.25$

Let's plot and connect the obtained points. Keep in mind that rational functions have two branches. Do not forget to graph the asymptotes!