Questions: Graph the rational function. f(x) = (-x^2 + 10x) / (x^2 - 10x + 21) Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote. Finally, click on the graph-a-function button.

Graph the rational function.
f(x) = (-x^2 + 10x) / (x^2 - 10x + 21)

Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote. Finally, click on the graph-a-function button.

Transcript text: Graph the rational function. \[ f(x)=\frac{-x^{2}+10 x}{x^{2}-10 x+21} \] Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote. Finally, click on the graph-a-function button.

Solution

Solution Steps

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. Set the denominator equal to zero and solve for \( x \): \[ x^2 - 10x + 21 = 0 \] \[ (x - 3)(x - 7) = 0 \] So, the vertical asymptotes are at \( x = 3 \) and \( x = 7 \).

Step 2: Identify Horizontal Asymptote

For 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: \[ \frac{-1}{1} = -1 \] So, the horizontal asymptote is \( y = -1 \).

Step 3: Find Intercepts

To find the y-intercept, set \( x = 0 \): \[ f(0) = \frac{-0^2 + 10(0)}{0^2 - 10(0) + 21} = \frac{0}{21} = 0 \] So, the y-intercept is at \( (0, 0) \).

To find the x-intercepts, set the numerator equal to zero and solve for \( x \): \[ -x^2 + 10x = 0 \] \[ x(-x + 10) = 0 \] So, the x-intercepts are at \( x = 0 \) and \( x = 10 \).