Question

a. find the slant asymptote of the graph of the rational function.
b. follow the seven - step strategy and use the slant asymptote to graph the rational function.
f(x)=\frac{x^{2}+2x - 8}{x - 4}
a. select the correct choice below and, if necessary, fill in the answer box to complete the choice.
a. the equation of the slant asymptote is y = x + 6.
(type an equation.)
b. there is no slant asymptote.
b. to graph the function, first determine the symmetry of the graph of f. choose the correct answer below.
origin symmetry
y - axis symmetry
neither y - axis symmetry nor origin symmetry
what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to complete the choice.
a. the y - intercept is 2.
(type an integer or a simplified fraction.)
b. there is no y - intercept.
what is/are the x - intercept(s)? select the correct choice below and, if necessary, fill in the answer box within your choice.
a. the x - intercept(s) is/are - 4,2.
(type an integer or a simplified fraction. use a comma to separate answers if needed.)
b. there are no x - intercepts.
find the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete the choice.
a. the equation of the vertical asymptote is

Explanation:

Step1: Find vertical asymptote

For a rational function \(f(x)=\frac{N(x)}{D(x)}\), vertical asymptotes occur at the values of \(x\) that make \(D(x) = 0\).
Given \(f(x)=\frac{x^{2}+2x - 8}{x - 4}\), set \(D(x)=x - 4=0\).

Step2: Solve for \(x\)

\(x-4 = 0\) gives \(x = 4\).

Answer:

A. The equation of the vertical asymptote is \(x = 4\)