To solve this problem, we need to construct a rational function $f(x)$ with the following properties:

1. A vertical asymptote at $x = 2$. This means the denominator of the function should be zero when $x = 2$.
2. A horizontal asymptote at $y = -4$. This means as $x \to \infty$, the function $f(x)$ should approach $-4$.

Step-by-Step Solution:

1. Vertical Asymptote

A vertical asymptote at $x = 2$ implies that the denominator has a factor of $(x - 2)$. Thus, the function will be of the form:

$f(x) = \frac{N(x)}{D(x)} = \frac{N(x)}{x - 2}$

where $N(x)$ is the numerator.

2. Horizontal Asymptote

For a horizontal asymptote at $y = -4$, the degrees of the numerator $N(x)$ and denominator $D(x)$ must be the same. Moreover, the leading coefficient of the numerator divided by the leading coefficient of the denominator should equal $-4$. If the denominator is $x - 2$, the numerator $N(x)$ must be $-4x + c$, where $c$ is a constant.

Thus, a possible function is:

$f(x) = \frac{-4x + c}{x - 2}.$

3. Simplifying

We don't need $c$ to create a horizontal asymptote; it only shifts the function vertically. The simplest possible function satisfying both conditions is:

$f(x) = \frac{-4x}{x - 2}.$

Let’s rewrite it to include the horizontal asymptote explicitly:

$f(x) = \frac{-4x}{x - 2}.$

Verification:

1. Vertical Asymptote at $x = 2$: When $x = 2$, the denominator becomes zero, creating a vertical asymptote.

2. Horizontal Asymptote at $y = -4$: As $x \to \infty$, the terms of highest degree dominate. For large $x$, $f(x) \approx -4$, matching the desired horizontal asymptote.

Would you like a more detailed breakdown of the reasoning or have any specific questions about this function?

Follow-up Questions:

1. What happens to the function behavior as $x \to -\infty$?
2. How does adding constants to the numerator affect the horizontal asymptote?
3. Can there be more than one rational function satisfying these conditions?
4. What is the domain of the constructed function?
5. How do vertical and horizontal asymptotes affect the graph?

Tip:

When constructing rational functions, always check the degrees of the numerator and denominator for horizontal asymptotes and the factors of the denominator for vertical asymptotes.