We are given the function $f(x) = \frac{7x}{x - 4}$, and we are tasked with finding the value of the function at specific values of $x$, as well as plotting points beyond the x-intercepts and vertical asymptotes.

1. Vertical Asymptote

A vertical asymptote occurs where the denominator is zero. For the given function $f(x) = \frac{7x}{x - 4}$, the denominator $x - 4 = 0$ when $x = 4$. Thus, there is a vertical asymptote at $x = 4$.

2. X-Intercept

To find the x-intercept, set the numerator equal to zero: $7x = 0$ $x = 0$ So, the x-intercept is at $(0, 0)$.

3. Finding the Values of the Function

We are asked to compute the values of the function at the given points.

• $f(-6) = \frac{7(-6)}{-6 - 4} = \frac{-42}{-10} = 4.2$
• $f(-5) = \frac{7(-5)}{-5 - 4} = \frac{-35}{-9} \approx 3.89$
• $f(3) = \frac{7(3)}{3 - 4} = \frac{21}{-1} = -21$
• $f(5) = \frac{7(5)}{5 - 4} = \frac{35}{1} = 35$
• $f(7) = \frac{7(7)}{7 - 4} = \frac{49}{3} \approx 16.33$

4. Summary of Values:

• $f(-6) = 4.2$
• $f(-5) \approx 3.89$
• $f(3) = -21$
• $f(5) = 35$
• $f(7) \approx 16.33$

Conclusion

The vertical asymptote is at $x = 4$, and the x-intercept is at $(0, 0)$. The function values at the specified $x$-coordinates have been calculated.

Would you like to go over any part in more detail, or should I help you plot this function?

---

Here are 5 related questions you can explore:

1. What happens to the function as $x$ approaches the vertical asymptote?
2. How do you find horizontal asymptotes for rational functions?
3. How do x-intercepts relate to the numerator of a rational function?
4. How does the sign of the function change around its intercepts and asymptotes?
5. Can a rational function have more than one vertical asymptote?

Tip: When dealing with rational functions, always check for both vertical asymptotes and horizontal asymptotes to understand their long-term behavior.