graph the function ( y=\frac{2x}{x^{2}-16} ) by identifying the domain and any symmetries, finding the…

graph the function ( y=\frac{2x}{x^{2}-16} ) 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.

graph the function ( y=\frac{2x}{x^{2}-16} ) 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.

Answer

Explanation:

Step1: Find the domain

The function is (y = \frac{2x}{x^{2}-16}). The denominator (x^{2}-16=(x + 4)(x - 4)). The domain is all real numbers except (x=-4) and (x = 4) since the denominator cannot be zero.

Step2: Analyze vertical asymptotes

Vertical asymptotes occur at the values of (x) that make the denominator zero (while the numerator is non - zero at those points). When (x=-4), the numerator (2x=2\times(-4)=-8\neq0). When (x = 4), the numerator (2x=2\times4 = 8\neq0).

Answer:

B. The function has two vertical asymptotes. The leftmost asymptote is (x=-4) and the rightmost asymptote is (x = 4)