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{-1}{(x + 3)^2} find all vertical asymptotes. select the correct choice below and, if necessary, fill in the answer box within your choice. a. x=-3 (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) . (simplify your answer. type an ordered pair. use a comma to separate answers as needed.) b. there are no x - intercepts. find the y - intercept. select the correct choice below and, if necessary, fill in the answer box within your choice. a. the y - intercept is . (simplify your answer. type an ordered pair.) b. there is no y - intercept.

Explanation:

Step1: Find vertical asymptote

Set denominator equal to 0. $(x + 3)^2=0$, solving gives $x=-3$.

Step2: Find horizontal asymptote

Degree of numerator is 0 and of denominator is 2. Since degree of denominator > degree of numerator, $y = 0$ is the horizontal asymptote.

Step3: Find x - intercept

Set $y = 0$, so $\frac{-1}{(x + 3)^2}=0$. But - 1≠0 for any real x, so no x - intercepts.

Step4: Find y - intercept

Set $x = 0$, then $y=\frac{-1}{(0 + 3)^2}=-\frac{1}{9}$. The y - intercept is the point $(0,-\frac{1}{9})$.

Answer:

Vertical asymptote: $x=-3$
Horizontal asymptote: $y = 0$
x - intercept: There are no x - intercepts
y - intercept: $(0,-\frac{1}{9})$