We are asked to write an equation for a rational function with the following characteristics:

1. Vertical asymptotes at $x = 1$ and $x = -3$.
2. x-intercept at $x = 4$.
3. y-intercept where $y = 5$ when $x = 0$.

Step 1: Setup the general form

The vertical asymptotes indicate the function has factors in the denominator that become zero at $x = 1$ and $x = -3$. Therefore, the denominator of the function is: $(x - 1)(x + 3)$ For the x-intercept at $x = 4$, there must be a factor in the numerator that becomes zero when $x = 4$. Therefore, the numerator contains the factor: $(x - 4)$ Thus, the rational function can be written as: $f(x) = \frac{k(x - 4)}{(x - 1)(x + 3)}$ where $k$ is a constant that we will determine using the y-intercept condition.

Step 2: Use the y-intercept condition

We are told that when $x = 0$, $y = 5$. Substituting $x = 0$ into the function: $5 = \frac{k(0 - 4)}{(0 - 1)(0 + 3)} = \frac{-4k}{-3} = \frac{4k}{3}$ Now solve for $k$: $5 = \frac{4k}{3} \implies k = \frac{5 \cdot 3}{4} = \frac{15}{4}$

Final equation

The rational function is: $f(x) = \frac{\frac{15}{4}(x - 4)}{(x - 1)(x + 3)}$

Let me know if you'd like further clarifications or details!

Here are five related questions to further explore this topic:

1. How do you find horizontal asymptotes for rational functions?
2. What happens to the graph of the function near the vertical asymptotes?
3. Can rational functions have oblique asymptotes? If yes, when?
4. How does changing the constant $k$ affect the behavior of the rational function?
5. What role do the factors of the numerator and denominator play in determining x-intercepts and asymptotes?

Tip: When writing a rational function, always check the x-intercepts and asymptotes first to structure your equation properly!