Question

find all asymptotes, x - intercepts, and y - intercepts for the graph of the rational function and sketch the graph of the function. f(x) = 5x/(x² - 8x + 16). find all vertical asymptotes. select the correct choice below and, if necessary, fill in the answer box within your choice. a. x = 4 (simplify your answer. use a comma to separate answers as needed.) b. there are no vertical asymptotes. find the horizontal or oblique asymptote. select the correct choice below and, if necessary, fill in the answer box within your choice. a. y = 0 (simplify your answer.) b. there is no horizontal nor oblique asymptote. find 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) x = . (simplify your answer. type an ordered pair. use a comma to separate answers as needed.) b. there are no x - intercepts.

Explanation:

Step1: Find vertical asymptotes

Set the denominator equal to zero. Given $f(x)=\frac{5x}{x^{2}-8x + 16}=(x - 4)^{2}$. Solving $(x - 4)^{2}=0$ gives $x = 4$.

Step2: Find horizontal asymptote

Since the degree of the numerator ($n = 1$) is less than the degree of the denominator ($m=2$), the horizontal asymptote is $y = 0$.

Step3: Find x - intercepts

Set the numerator equal to zero. Solving $5x=0$ gives $x = 0$.

Step4: Find y - intercepts

Set $x = 0$ in the function. $f(0)=\frac{5\times0}{0^{2}-8\times0 + 16}=0$.

Answer:

Vertical asymptote: $x = 4$
Horizontal asymptote: $y = 0$
x - intercept: $(0,0)$
y - intercept: $(0,0)$