https://file.aiquickdraw.com/mathbot/file/1730696121356jt31nstw.jpg

The exercise provides a rational function:

\[
f(x) = \frac{6x}{x^2 - 16}
\]

and asks several questions about it. Let’s go through each part as presented in the image:

1. **X-Intercept**:
   - To find the x-intercept, set \( f(x) = 0 \).
   - Since \( f(x) = \frac{6x}{x^2 - 16} \), the x-intercept occurs when the numerator is zero.
   - Thus, \( 6x = 0 \Rightarrow x = 0 \).
   - So, the x-intercept is at \( x = 0 \).

2. **Vertical Asymptotes**:
   - Vertical asymptotes occur where the denominator is zero and the numerator is not zero.
   - The denominator is \( x^2 - 16 = 0 \Rightarrow x^2 = 16 \Rightarrow x = \pm 4 \).
   - Therefore, there are vertical asymptotes at \( x = 4 \) and \( x = -4 \).

3. **Horizontal Asymptote**:
   - For rational functions, if the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.
   - The degrees are both 1 (for \(6x\) in the numerator and \(x^2\) in the denominator).
   - Hence, the horizontal asymptote is \( y = 0 \).

4. **Function Values for Specific x Values**:
   - To evaluate \( f(x) \) at the given values, substitute each \( x \) into \( f(x) = \frac{6x}{x^2 - 16} \).
   
   Let’s compute these:

   - \( f(-6) = \frac{6(-6)}{(-6)^2 - 16} = \frac{-36}{36 - 16} = \frac{-36}{20} = -\frac{9}{5} \)
   - \( f(-5) = \frac{6(-5)}{(-5)^2 - 16} = \frac{-30}{25 - 16} = \frac{-30}{9} = -\frac{10}{3} \)
   - \( f(-1) = \frac{6(-1)}{(-1)^2 - 16} = \frac{-6}{1 - 16} = \frac{-6}{-15} = \frac{2}{5} \)
   - \( f(1) = \frac{6(1)}{1^2 - 16} = \frac{6}{1 - 16} = \frac{6}{-15} = -\frac{2}{5} \)
   - \( f(6) = \frac{6(6)}{6^2 - 16} = \frac{36}{36 - 16} = \frac{36}{20} = \frac{9}{5} \)
   - \( f(8) = \frac{6(8)}{8^2 - 16} = \frac{48}{64 - 16} = \frac{48}{48} = 1 \)

Let me know if you need further details or calculations!

### Related Questions:
1. What are the conditions for a rational function to have a slant asymptote?
2. How do we determine the domain of a rational function?
3. How can we identify points of discontinuity in rational functions?
4. What are some techniques to sketch rational functions accurately?
5. How does the end behavior of a rational function relate to its asymptotes?

### Tip:
When identifying asymptotes, always check if factors cancel out in the fraction. Canceled factors indicate holes, not asymptotes.

Follow the seven-step strategy to graph the rational function f(x) = 6x / (x^2 - 16). Find the x-intercepts, vertical and horizontal asymptotes, and calculate the function values for specific x values.

Learn how to graph the rational function f(x) = 6x / (x^2 - 16) using the seven-step strategy. This problem covers finding x-intercepts, vertical and horizontal asymptotes, and evaluating specific function values. Ideal for high school students studying rational functions and asymptotes.

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