Question

graph the rational function:
$f(x) = \frac{x^2 - 5x + 3}{x - 3}$
start by drawing the asymptotes. then plot two points on each piece of the graph. finally, click on the graph - a - function button.

Explanation:

Step1: Find vertical asymptote

Set denominator to 0: $x-3=0 \implies x=3$

Step2: Find slant asymptote

Divide numerator by denominator:
$$\polylongdiv{x^2-5x+3}{x-3}$$
Quotient: $x-2$, so slant asymptote is $y=x-2$

Step3: Plot points left of $x=3$

For $x=0$: $f(0)=\frac{0^2-5(0)+3}{0-3}=\frac{3}{-3}=-1$, point $(0,-1)$
For $x=2$: $f(2)=\frac{2^2-5(2)+3}{2-3}=\frac{4-10+3}{-1}=\frac{-3}{-1}=3$, point $(2,3)$

Step4: Plot points right of $x=3$

For $x=4$: $f(4)=\frac{4^2-5(4)+3}{4-3}=\frac{16-20+3}{1}=-1$, point $(4,-1)$
For $x=6$: $f(6)=\frac{6^2-5(6)+3}{6-3}=\frac{36-30+3}{3}=\frac{9}{3}=3$, point $(6,3)$

Answer:

1. Vertical asymptote: $x=3$ (dashed vertical line)
2. Slant asymptote: $y=x-2$ (dashed line)
3. Points to plot:

• Left of $x=3$: $(0,-1)$, $(2,3)$
• Right of $x=3$: $(4,-1)$, $(6,3)$

Connect the points on each side of $x=3$, approaching the asymptotes to form the graph.