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)=\frac{5x}{x^{2}-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=
(simplify your answer.)
b. there is no horizontal nor oblique asymptote.

Explanation:

Step1: Factor the denominator

The denominator $x^{2}-8x + 16=(x - 4)^{2}$.

Step2: Find vertical asymptotes

Set the denominator equal to zero: $(x - 4)^{2}=0$, so $x = 4$ is the vertical - asymptote.

Step3: Find horizontal asymptote

The degree of the numerator $n = 1$ and the degree of the denominator $m=2$. Since $n

Step4: Find x - intercepts

Set the numerator equal to zero: $5x=0$, so $x = 0$ is the x - intercept.

Step5: Find y - intercepts

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

Answer:

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