To determine the equation for the function graphed, let's examine the key characteristics, particularly the vertical and horizontal asymptotes, as well as the behavior near these asymptotes.

Step 1: Identifying Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, usually due to a zero in the denominator. From the graph:

• There is a vertical asymptote at $x = -2$.
• There is another vertical asymptote at $x = 3$.

This suggests that the denominator has factors $(x + 2)$ and $(x - 3)$.

Step 2: Identifying Horizontal Asymptote

As $x$ approaches large positive or negative values, the graph approaches $y = 0$, which indicates a horizontal asymptote at $y = 0$.

This horizontal asymptote suggests that the degree of the numerator is less than or equal to the degree of the denominator. Since we aim to use the smallest degrees possible, we will try a degree 1 numerator.

Step 3: Determining the Numerator

Since there is no horizontal asymptote other than $y = 0$, we can assume the simplest form for the numerator is a constant, say $A$.

Step 4: Putting It All Together

Based on the identified factors, we can propose a rational function of the form: $y = \frac{A}{(x + 2)(x - 3)}$

Step 5: Determining $A$ Using a Point on the Graph

If a specific point on the graph was given, we could substitute it in to solve for $A$. Without a specific point, $A$ remains a constant multiplier.

Therefore, the simplest form of the function, assuming $A = 1$, would be: $y = \frac{1}{(x + 2)(x - 3)}$

Would you like further details on this solution?

Here are five related questions:

1. How does the degree of the numerator affect the horizontal asymptote?
2. What would happen to the equation if the numerator was a higher-degree polynomial?
3. Can rational functions have multiple horizontal asymptotes?
4. How do you find slant asymptotes in rational functions?
5. How does the sign of $A$ (positive or negative) impact the direction of the graph?

Tip: In rational functions, vertical asymptotes correspond to factors in the denominator, while horizontal asymptotes depend on the relationship between the degrees of the numerator and denominator.