Questions: Graph the rational function. f(x) = (-2x^2 + 15x - 15) / (3x - 9) Start by drawing the asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph-a-function button

Graph the rational function.
f(x) = (-2x^2 + 15x - 15) / (3x - 9)

Start by drawing the asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph-a-function button

Transcript text: Graph the rational function. \[ f(x)=\frac{-2 x^{2}+15 x-15}{3 x-9} \] Start by drawing the asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph-a-function button

Solution

Solution Steps

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero. Set the denominator equal to zero and solve for \( x \): \[ 3x - 9 = 0 \] \[ x = 3 \]

Step 2: Identify Horizontal Asymptotes

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Both the numerator and the denominator are of degree 2. The horizontal asymptote is given by the ratio of the leading coefficients: \[ y = \frac{-2}{3} \]

Step 3: Plot Points on the Graph

Choose values of \( x \) to find corresponding \( y \)-values to plot points on the graph. For example:

For \( x = 0 \): \[ f(0) = \frac{-2(0)^2 + 15(0) - 15}{3(0) - 9} = \frac{-15}{-9} = \frac{5}{3} \]
For \( x = 1 \): \[ f(1) = \frac{-2(1)^2 + 15(1) - 15}{3(1) - 9} = \frac{-2 + 15 - 15}{3 - 9} = \frac{-2}{-6} = \frac{1}{3} \]