Question

graph the rational function.
$f(x)=\frac{6}{-x + 6}$
start by drawing the vertical and points on each piece of the graph button.

Explanation:

Step1: Find vertical asymptote

Set denominator to 0:
$-x + 6 = 0$
Solve for $x$: $x = 6$

Step2: Find horizontal asymptote

Compare degrees of numerator/denominator:
Numerator degree = 0, denominator degree = 1. For $\frac{\text{constant}}{\text{linear}}$, horizontal asymptote is $y=0$.

Step3: Plot points left of $x=6$

Choose $x=0$:
$f(0)=\frac{6}{-0+6}=1$ → Point $(0,1)$
Choose $x=3$:
$f(3)=\frac{6}{-3+6}=2$ → Point $(3,2)$
Choose $x=-6$:
$f(-6)=\frac{6}{-(-6)+6}=\frac{6}{12}=0.5$ → Point $(-6, 0.5)$

Step4: Plot points right of $x=6$

Choose $x=7$:
$f(7)=\frac{6}{-7+6}=-6$ → Point $(7,-6)$
Choose $x=9$:
$f(9)=\frac{6}{-9+6}=-2$ → Point $(9,-2)$
Choose $x=12$:
$f(12)=\frac{6}{-12+6}=-1$ → Point $(12,-1)$

Answer:

1. Draw vertical asymptote at $x=6$ (dashed vertical line)
2. Draw horizontal asymptote at $y=0$ (dashed horizontal line)
3. Plot points $(-6, 0.5)$, $(0,1)$, $(3,2)$ and draw a curve through them, approaching $x=6$ (from left, $y\to+\infty$) and $y=0$ as $x\to-\infty$
4. Plot points $(7,-6)$, $(9,-2)$, $(12,-1)$ and draw a curve through them, approaching $x=6$ (from right, $y\to-\infty$) and $y=0$ as $x\to+\infty$