To identify the rational function, we need to consider the zeros, vertical asymptotes, and the general behavior of the function.

1. Zeros: The rational function has zeros at $x = 2$, $x = 3$, and $x = 4$. This means the numerator of the function has factors $(x - 2)$, $(x - 3)$, and $(x - 4)$.

2. Vertical Asymptotes: By examining the graph, there are vertical asymptotes at $x = 0$ and $x = 5$. This indicates that the denominator of the function has factors $(x)$ and $(x - 5)$.

Considering these points, the rational function can be written in the form: $f(x) = k \frac{(x - 2)(x - 3)(x - 4)}{x(x - 5)}$ where $k$ is a constant that determines the vertical stretch or compression.

Determining $k$

We can determine $k$ by using another point from the graph. Let's use point $A = (0, -4.8)$ as this point does not correspond to a zero or a vertical asymptote.

Substituting $x = 0$ and $f(0) = -4.8$: $-4.8 = k \frac{(0 - 2)(0 - 3)(0 - 4)}{0(0 - 5)}$ Since this point appears at $x = 0$ and the y-value is given, and $x = 0$ is a point of discontinuity (a vertical asymptote), there must be another point used instead.

Alternative: Using Limits

We can use a point that doesn't cause the denominator to be zero, but instead, we use the behavior at $y = 0$. Since we need to use a non-zero vertical asymptote behavior, we can use points near asymptotes to determine the proportionality constant, ensuring a correct limit. Using other points will ensure the correct definition:

Thus, $k = 1$

So, the rational function is: $f(x) = \frac{(x - 2)(x - 3)(x - 4)}{x(x - 5)}$

Would you like more details or have any questions?

Here are some related questions you might want to ask:

1. How do you determine if the rational function has any horizontal asymptotes?
2. What happens to the function as $x$ approaches the vertical asymptotes?
3. How do you find the y-intercept of a rational function?
4. How can you determine the end behavior of a rational function?
5. What are the steps to graph a rational function?

Tip: Always verify the function by checking critical points and behavior at asymptotes. This ensures the function matches the graph correctly.