To construct a rational function with the given properties, let’s break it down step by step.

---

1. General form of a rational function:

A rational function $f(x)$ has the general form:

$f(x) = \frac{N(x)}{D(x)}$

where $N(x)$ is the numerator (containing the x-intercepts) and $D(x)$ is the denominator (containing the vertical asymptotes). Let's build these based on the conditions.

---

2. Vertical asymptotes:

• Vertical asymptotes occur when the denominator becomes zero.
• Since the vertical asymptotes are at $x = -5$ and $x = 1$, the factors of the denominator must be:

$D(x) = (x + 5)(x - 1)$

---

3. x-intercepts:

• The numerator determines the x-intercepts.
• Since the x-intercepts are at $(-4, 0)$ and $(-1, 0)$, the numerator must contain these as factors:

$N(x) = (x + 4)(x + 1)$

---

4. Putting it together:

So far, the rational function is:

$f(x) = \frac{A (x + 4)(x + 1)}{(x + 5)(x - 1)}$

where $A$ is a constant that ensures the function passes through the y-intercept at $(0, 7)$.

---

5. Finding the constant $A$:

• Use the y-intercept at $(0, 7)$ to find $A$.
• Substituting $x = 0$ and $f(0) = 7$:

$7 = \frac{A (0 + 4)(0 + 1)}{(0 + 5)(0 - 1)}$

$7 = \frac{A \cdot 4 \cdot 1}{5 \cdot (-1)}$

$7 = \frac{-4A}{5}$

$-35 = -4A \quad \Rightarrow \quad A = \frac{35}{4}$

---

6. Final equation:

Thus, the rational function is:

$f(x) = \frac{\frac{35}{4} (x + 4)(x + 1)}{(x + 5)(x - 1)}$

Or equivalently:

$f(x) = \frac{35 (x + 4)(x + 1)}{4 (x + 5)(x - 1)}$

---

Would you like further explanation or a graph of this function?

---

Here are 5 related questions:

1. What happens to the function as $x \to \infty$?
2. How do horizontal asymptotes differ from vertical asymptotes?
3. How do you find slant asymptotes for rational functions?
4. How would the equation change if the x-intercepts had multiplicity 2?
5. How can you determine whether the function has any holes?

Tip: When solving for constants using known points, be mindful of negative signs—small mistakes can affect the solution significantly.