Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ \mathrm{f}(\mathrm{x})=\frac{8 \mathrm{x}}{\mathrm{x}^{2}-1} $$ A. The y -intercept(s) is/are at $$ \mathrm{y}= $$ (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 $$ \mathrm{f}(\mathrm{x}) $$. 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, $$ \square $$. (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is $$ \square $$ and the bottom asymptote is $$ \square $$. (Type equations.) C. There are no horizontal asymptotes.

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1. **Finding the y-intercept**

The y-intercept occurs when $$ x=0 $$. Substituting $$ x=0 $$ into $$ f(x) $$:

$$
   f(0) = \frac{8(0)}{0^2-1} = \frac{0}{-1} = 0
   $$

**Answer A:** The y-intercept is at $$ y=0 $$.

2. **Determining the horizontal asymptote**

For the function

$$
   f(x)=\frac{8x}{x^2-1},
   $$

compare the degrees of the numerator and the denominator. The numerator is degree 1 and the denominator is degree 2. Since the degree of the numerator is less than that of the denominator, the horizontal asymptote is given by

$$
   y=0.
   $$

**Answer:** The function has one horizontal asymptote, $$ y=0 $$.

3. **Additional Pertinent Information**

- **Domain:** The function is undefined where the denominator is zero. Solve

$$
     x^2-1=0 \quad \Rightarrow \quad (x-1)(x+1)=0,
     $$

which gives $$ x=1 $$ and $$ x=-1 $$. These are vertical asymptotes.

- **x-intercept:** Found by setting $$ f(x)=0 $$. Since the numerator is $$ 8x $$, setting $$ 8x=0 $$ gives $$ x=0 $$ (which confirms the y-intercept at $$ (0,0) $$ is also the x-intercept).

- **Vertical Asymptotes:** $$ x = 1 $$ and $$ x = -1 $$.

4. **Sketching the Graph**

- **Intercepts:** The graph passes through the origin $$ (0,0) $$.
   - **Vertical Asymptotes:** Draw dashed lines at $$ x=1 $$ and $$ x=-1 $$.
   - **Horizontal Asymptote:** Draw a dashed line along $$ y=0 $$.
   - **Behavior:** 
     - For large positive or negative $$ x $$, $$ f(x) $$ approaches 0.
     - As $$ x $$ nears 1 and -1, the function value increases or decreases without bound.

The complete answers are:

**A.** The y-intercept is at $$ y=0 $$.  
**Horizontal Asymptote:** The function has one horizontal asymptote, $$ y=0 $$.
**A.** The y-intercept is at $$ y=0 $$. **Horizontal Asymptote:** The function has one horizontal asymptote, $$ y=0 $$.

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