Question

graph the rational function.

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

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:

Find the vertical asymptote

Using the Asymptotes knowledge point

$$
LATEXBLOCK0
$$

Find the horizontal asymptote

Using the Asymptotes knowledge point

$$
LATEXBLOCK1
$$

Plot points to the left of the vertical asymptote

We choose \(x\)-values less than \(2\), such as \(x = 0\) and \(x = 1\):

• For \(x = 1\):

$$ f(1) = \frac{3(1) - 3}{-(1) + 2} = \frac{0}{1} = 0 \implies (1, 0) $$

• For \(x = 0\):

$$ f(0) = \frac{3(0) - 3}{-(0) + 2} = \frac{-3}{2} = -1.5 \implies (0, -1.5) $$

Plot points to the right of the vertical asymptote

We choose \(x\)-values greater than \(2\), such as \(x = 3\) and \(x = 4\):

• For \(x = 3\):

$$ f(3) = \frac{3(3) - 3}{-(3) + 2} = \frac{6}{-1} = -6 \implies (3, -6) $$

• For \(x = 4\):

$$ f(4) = \frac{3(4) - 3}{-(4) + 2} = \frac{9}{-2} = -4.5 \implies (4, -4.5) $$

Summarize the graphing steps

We have identified the asymptotes and two points on each branch of the hyperbola to construct the graph.

Answer:

To graph the rational function \(f(x) = \frac{3x - 3}{-x + 2}\):

1. Asymptotes:

• Vertical Asymptote: \(x = 2\)
• Horizontal Asymptote: \(y = -3\)

1. Points on the left branch (\(x < 2\)):

• \((1, 0)\)
• \((0, -1.5)\)

1. Points on the right branch (\(x > 2\)):

• \((3, -6)\)
• \((4, -4.5)\)