Question

graph the function ( y = x ^ { 2 } - 2 x - 8 ) by identifying the domain and any symmetries, finding the derivatives ( y ^ { prime } ) and ( y ^ { prime prime } ), finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any.
d. the curve is neither concave up nor concave down.
find any vertical asymptotes. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one vertical asymptote, (type an equation.)
b. the function has two vertical asymptotes. the leftmost asymptote is and the rightmost asymptote is (type equations.)
c. the function has no vertical asymptotes.

Explanation:

Step1: Recall the definition of vertical asymptote

A vertical asymptote occurs where the function is undefined (for rational functions, where the denominator is zero). But the given function \(y = x^{2}-2x - 8\) is a polynomial.

Step2: Analyze the domain of a polynomial

The domain of a polynomial \(y=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0}\) (\(n\) is a non - negative integer, \(a_{n}
eq0\)) is all real numbers, \(x\in(-\infty,\infty)\). Polynomials are continuous everywhere on their domain.

Answer:

C. The function has no vertical asymptotes.