To graph the given rational function, we will start by identifying the values of $a,$ $b,$ $c,$ and $d.$ $h(x)=2x−68x+3 ⇔h(x)=2x+(-6)8x+3 $ We see above that $a$ $=$ $8,$ $b$ $=$ $3,$ $c$ $=$ $2,$ and $d$ $=$ $-6.$ With this information, we will find the asymptotes and graph the function.
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.
$x$ | $2x−68x+3 $ | $h(x)$ |
---|---|---|
$-3$ | $2(-3)−68(-3)+3 $ | $1.75$ |
$-1$ | $2(-1)−68(-1)+3 $ | $0.625$ |
$1$ | $2(1)−68(1)+3 $ | $-2.75$ |
$5$ | $2(5)−68(5)+3 $ | $10.75$ |
$7$ | $2(7)−68(7)+3 $ | $7.375$ |
$9$ | $2(9)−68(9)+3 $ | $6.25$ |
Let's plot and connect the obtained points. Keep in mind that rational functions have two branches. Do not forget to graph the asymptotes!