Question

graph the rational function.
$f(x) = \frac{-2x + 3}{-x + 3}$
start by drawing the vertical and horizontal 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: $-x + 3 = 0$
Solve for $x$: $x = 3$

Step2: Find horizontal asymptote

Compare degrees of numerator/denominator (both degree 1). Horizontal asymptote is ratio of leading coefficients: $y = \frac{-2}{-1} = 2$

Step3: Simplify the function

$f(x)=\frac{-2x+3}{-x+3}=\frac{2x-3}{x-3}=\frac{2(x-3)+3}{x-3}=2+\frac{3}{x-3}$

Step4: Find points left of $x=3$

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

Step5: Find points right of $x=3$

For $x=4$: $f(4)=\frac{-2(4)+3}{-4+3}=\frac{-5}{-1}=5$ → Point $(4,5)$
For $x=6$: $f(6)=\frac{-2(6)+3}{-6+3}=\frac{-9}{-3}=3$ → Point $(6,3)$

Answer:

• Vertical Asymptote: $x=3$ (dashed vertical line)
• Horizontal Asymptote: $y=2$ (dashed horizontal line)
• Points to plot: $(0,1)$, $(2,-1)$, $(4,5)$, $(6,3)$
• The graph is a hyperbola shifted such that it approaches $x=3$ and $y=2$, passing through the plotted points.