Question

graph the following rational function.
$f(x) = \frac{ -x + 1 }{ x^2 - 64 }$
step 2 of 3 : plot the horizontal asymptotes, if any, on the graph.
answer
once the number of asymptotes is selected, that number of lines are available to plot. select a location on the grid to plot a line. lines can be moved by dragging or using the arrow keys.
number of horizontal asymptotes:
\\(\circ\\) none \\(\circ\\) one \\(\circ\\) two
graph with x-axis from -10 to 10 and y-axis from -10 to 10, grid lines, and axes arrows

Explanation:

Step1: Recall Horizontal Asymptote Rules

For a rational function \( f(x) = \frac{N(x)}{D(x)} \), where \( \text{deg}(N(x)) \) is the degree of the numerator and \( \text{deg}(D(x)) \) is the degree of the denominator:

• If \( \text{deg}(N(x)) < \text{deg}(D(x)) \), horizontal asymptote is \( y = 0 \).
• If \( \text{deg}(N(x)) = \text{deg}(D(x)) \), horizontal asymptote is \( y = \frac{\text{leading coefficient of } N(x)}{\text{leading coefficient of } D(x)} \).
• If \( \text{deg}(N(x)) > \text{deg}(D(x)) \), no horizontal asymptote (may have oblique/slant asymptote).

Step2: Determine Degrees of Numerator and Denominator

For \( f(x) = \frac{-x + 1}{x^2 - 64} \):

• Numerator \( N(x) = -x + 1 \), degree \( \text{deg}(N(x)) = 1 \).
• Denominator \( D(x) = x^2 - 64 \), degree \( \text{deg}(D(x)) = 2 \).

Since \( 1 < 2 \) ( \( \text{deg}(N(x)) < \text{deg}(D(x)) \) ), the horizontal asymptote is \( y = 0 \). So there is one horizontal asymptote.

Answer:

One (and the horizontal asymptote is \( y = 0 \))