In this question, you're asked to show that $a = 4$ for the function $f(x) = \frac{ax^2 + bx + c}{x^2 - c}$, with the given conditions.

The graph of the function is shown, and we're provided with some important features:

• The graph has a horizontal asymptote at $y = 2$.
• It has a behavior resembling a rational function with an asymptote.

Let's break down how to proceed:

Step 1: Horizontal Asymptote

The function's horizontal asymptote can help us find relationships between the constants. For large values of $x$, the term $x^2$ dominates both in the numerator and denominator. Thus, we can analyze the behavior of $f(x)$ for $x \to \infty$.

For large $x$, the function becomes approximately:

$f(x) \approx \frac{ax^2}{x^2} = a$

Since we are told the horizontal asymptote is $y = 2$, this means that for large $x$, $f(x) \to 2$. Therefore, $a = 2$.

Step 2: Compare with Graph Features

However, the question asks you to show that $a = 4$, which suggests there might be a different part of the graph we need to consider to clarify this.

The horizontal asymptote being $y = 2$ at $x \to \infty$ should be checked against the behavior of the function's other terms. Let me know if you want to proceed with a more detailed breakdown of how the other constants relate!