The graph shown in the image depicts a rational function with vertical asymptotes and horizontal asymptotes. Here's how we can determine the equation of the function:

Observations:

1. Vertical Asymptotes:

   • The graph has vertical asymptotes at $x = -3$ and $x = 2$.

2. Horizontal Asymptote:

   • The graph approaches a horizontal line as $x \to \infty$. It appears to be approaching $y = 1$.

3. Intercepts and Behavior:

   • The graph crosses the x-axis around $x = -1$ and the y-axis around $y = -2$.

Based on these observations, the function can be modeled by a rational function of the form: $y = \frac{a(x + 1)}{(x + 3)(x - 2)} + b$

Here, $a$ is a constant that scales the function vertically and adjusts the intercept, and $b$ is a constant that shifts the function vertically.

Equation Form:

1. Vertical Asymptotes: The factors in the denominator $(x + 3)(x - 2)$ ensure the vertical asymptotes at $x = -3$ and $x = 2$.

2. Horizontal Asymptote: For the horizontal asymptote $y = 1$, the function should be shifted up by 1. Hence, $b = 1$.

3. Intercepts: To match the intercept at $x = -1$ and behavior, we need to find the correct value of $a$.

Function Derivation:

Let’s start with: $y = \frac{a(x + 1)}{(x + 3)(x - 2)} + 1$

To find the value of $a$:

• We can check the y-intercept by substituting $x = 0$: $y = \frac{a(0 + 1)}{(0 + 3)(0 - 2)} + 1 = \frac{a}{-6} + 1$ If the y-intercept is $-2$, then: $-2 = \frac{a}{-6} + 1$ $-3 = \frac{a}{-6}$ $a = 18$

So, the equation for the function is: $y = \frac{18(x + 1)}{(x + 3)(x - 2)} + 1$

Final Answer:

$y = \frac{18(x + 1)}{(x + 3)(x - 2)} + 1$

Would you like further details or have any questions?

Here are some related questions to explore further:

1. What is a rational function, and how do its asymptotes affect its graph?
2. How do you determine the vertical asymptotes of a rational function?
3. What is the significance of the horizontal asymptote in a rational function?
4. How do intercepts help in determining the equation of a function?
5. Can you explain the behavior of the function near its asymptotes?
6. What are the general steps to finding the equation of a rational function given its graph?
7. How does the constant in the numerator affect the graph of a rational function?
8. How do you verify the equation of a function by comparing it to its graph?

Tip: Always check both the intercepts and asymptotic behavior when deriving the equation of a rational function from its graph.