The question is about finding intercepts and asymptotes for the rational function:

$f(x) = \frac{3x}{x - 5}$

Let's break down the missing answer step by step:

1. Y-Intercept:

   • To find the y-intercept, set $x = 0$ and evaluate $f(0)$: $f(0) = \frac{3 \cdot 0}{0 - 5} = 0$ Therefore, the y-intercept is $0$.

2. X-Intercept (already answered):

   • To find the x-intercept, set $f(x) = 0$: $\frac{3x}{x - 5} = 0 \Rightarrow x = 0$ So, the x-intercept is $x = 0$.

3. Vertical Asymptote (already answered):

   • The vertical asymptote occurs where the denominator is zero: $x - 5 = 0 \Rightarrow x = 5$ So, the vertical asymptote is $x = 5$.

4. Horizontal Asymptote (missing answer):

   • For horizontal asymptotes, compare the degrees of the numerator and denominator.
   • Both numerator and denominator have degree 1, so divide the leading coefficients: $y = \frac{3}{1} = 3$ Thus, the horizontal asymptote is $y = 3$.

So, the missing answer in the box for the horizontal asymptote should be: $y = 3$

Would you like more details or have any questions?

Follow-up Questions:

1. How would the graph of $f(x) = \frac{3x}{x - 5}$ change if the numerator was different?
2. What does the existence of a vertical asymptote at $x = 5$ tell us about the function's behavior?
3. How does the horizontal asymptote affect the end behavior of $f(x)$?
4. What would happen to the intercepts if we changed the function to $f(x) = \frac{3x + 1}{x - 5}$?
5. How can we verify intercepts and asymptotes using a graphing calculator?

Tip: For rational functions, the horizontal asymptote depends on the degrees of the numerator and denominator.