Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $f(x)=9e^{-0.5x^{2}}$. b. there are no x - intercepts. find the y - intercepts of $f(x)$. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the y - intercept(s) is/are at $y = 9$. (type an integer or a decimal. use a comma to separate answers as needed.) b. there are no y - intercepts. find any horizontal asymptotes of $f(x)$. 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.) c. there are no horizontal asymptotes.

Explanation:

Step1: Find the y - intercept

The y - intercept of a function \(y = f(x)\) is found by setting \(x = 0\).
For \(f(x)=9e^{-0.5x^{2}}\), when \(x = 0\), we have \(f(0)=9e^{-0.5\times0^{2}}\).
Since \(e^{0}=1\), then \(f(0)=9\times1 = 9\).

Step2: Find the horizontal asymptote

We use the limit \(\lim_{x
ightarrow\pm\infty}f(x)\).
For \(y = 9e^{-0.5x^{2}}\), we know that \(\lim_{x
ightarrow\pm\infty}- 0.5x^{2}=-\infty\).
And \(\lim_{u
ightarrow-\infty}e^{u}=0\). Let \(u=-0.5x^{2}\), so \(\lim_{x
ightarrow\pm\infty}9e^{-0.5x^{2}}=9\times\lim_{x
ightarrow\pm\infty}e^{-0.5x^{2}} = 0\).

Answer:

For the y - intercept: A. The y - intercept(s) is/are at \(y = 9\).
For the horizontal asymptote: A. The function has one horizontal asymptote, \(y = 0\).