The graph shown suggests we need to find the rational function by analyzing the following key features:

1. Vertical asymptotes: These occur where the denominator of the rational function equals zero. Based on the graph, we can see vertical asymptotes at $x = -2$ and $x = 4$. This suggests factors in the denominator of the form $(x + 2)$ and $(x - 4)$.

2. Horizontal asymptote: There seems to be a horizontal asymptote at $y = 0$, which suggests the degree of the numerator is less than or equal to the degree of the denominator.

3. Behavior near vertical asymptotes:

   • At $x = -2$, the graph approaches positive infinity on the left and negative infinity on the right.
   • At $x = 4$, the graph approaches negative infinity on the left and positive infinity on the right.

4. Zeros of the function: The graph crosses the x-axis at $x = 2$, suggesting a zero at $x = 2$. This means the numerator contains a factor $(x - 2)$.

Possible Rational Function

From this, we can hypothesize that the function is of the form:

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

Determining the constant $a$:

• We also observe a point passing through $(x = 6, y = -1)$ on the graph. Substituting $x = 6$ and $y = -1$ into the function will allow us to solve for $a$.

Let's calculate the value of $a$.