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.
c. the function has no vertical asymptotes
find any horizontal asymptotes. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one horizontal asymptote,
(type an equation.)
b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is
(type equations.)
c. the function has no horizontal asymptotes

Explanation:

Step1: Recall the definition of horizontal asymptote

For a function \(y = f(x)\), if \(\lim_{x
ightarrow\pm\infty}f(x)=L\) (a finite number), then \(y = L\) is a horizontal asymptote.
For the quadratic function \(y=x^{2}-2x - 8\), we use the formula for the limit of a polynomial function. The general form of a polynomial is \(y=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0}\), and \(\lim_{x
ightarrow\pm\infty}(a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0})=\lim_{x
ightarrow\pm\infty}a_{n}x^{n}\) when \(n>0\).
For \(y=x^{2}-2x - 8\), \(n = 2\), \(a_{n}=1\).

Step2: Calculate \(\lim_{x

ightarrow\pm\infty}(x^{2}-2x - 8)\)
We know that \(\lim_{x
ightarrow\pm\infty}(x^{2}-2x - 8)=\lim_{x
ightarrow\pm\infty}x^{2}(1-\frac{2}{x}-\frac{8}{x^{2}})\).
Using the limit rules \(\lim_{x
ightarrow\pm\infty}\frac{1}{x}=0\) and \(\lim_{x
ightarrow\pm\infty}\frac{1}{x^{2}} = 0\), and \(\lim_{x
ightarrow\pm\infty}x^{2}=\infty\).
Since \(\lim_{x
ightarrow\infty}(x^{2}-2x - 8)=\infty\) and \(\lim_{x
ightarrow-\infty}(x^{2}-2x - 8)=\infty\) (because \(x^{2}\to\infty\) as \(x\to\pm\infty\) and the lower - degree terms \(-2x-8\) are negligible compared to \(x^{2}\) for large \(|x|\)).

Answer:

C. The function has no horizontal asymptotes.