graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y…

graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y and y, 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. 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. 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.)

graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y and y, 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. 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. 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.)

Answer

Explanation:

Step1: Recall the form of the function

The given function is $y = x^{2}-2x - 8$, which is a polynomial function of degree 2. The general form of a polynomial function is $y=a_nx^n+\cdots+a_1x + a_0$, where $n$ is a non - negative integer and $a_n\neq0$. For a polynomial function, the domain is $(-\infty,\infty)$.

Step2: Determine vertical asymptotes

Vertical asymptotes occur for rational functions when the denominator is zero. Since $y = x^{2}-2x - 8$ is a polynomial function (not a rational function), there are no values of $x$ for which the function is undefined in the real - number system. So, the function has no vertical asymptotes.

Step3: Determine horizontal asymptotes

For a polynomial function $y=a_nx^n+\cdots+a_1x + a_0$ with $n\gt0$, as $x\to\pm\infty$, if $n\gt0$, there are no horizontal asymptotes. When $n = 2$ (as in our function $y=x^{2}-2x - 8$), as $x\to\pm\infty$, $y\to\infty$. So, the function has no horizontal asymptotes.

Answer:

For vertical asymptotes: C. The function has no vertical asymptotes. For horizontal asymptotes: C. The function has no horizontal asymptotes.