The question asks to write a function that matches the graph shown in the image. Let me extract and analyze the key features of the graph:

Key Observations from the Graph:

1. Type of Graph: The graph resembles a cubic function based on its general "S" shape with two turning points.
2. Intercepts:
   • X-Intercepts: The graph crosses the x-axis at $x = -3$, $x = 1$, and $x = 4$.
   • Y-Intercept: The graph crosses the y-axis at approximately $y = 0$ (though this requires exact confirmation).
3. Behavior:
   • The graph increases for large positive $x$ and decreases for large negative $x$, suggesting a positive leading coefficient.
   • It has a local maximum and minimum, typical of cubic functions.

Possible Function:

Given the x-intercepts $x = -3$, $x = 1$, and $x = 4$, the function can be written in factored form as: $f(x) = a(x + 3)(x - 1)(x - 4)$ where $a$ is a scaling factor that can adjust the vertical stretch or compression of the graph.

To further refine the function, let me know if you'd like to clarify or confirm specific points from the graph, such as a specific point on the curve (e.g., the y-intercept). Would you like further details or a step-by-step approach to fitting this function exactly?

Related Questions:

1. What is the significance of the scaling factor $a$ in adjusting the graph's shape?
2. How can we use another known point on the graph to determine the value of $a$?
3. Why does the number of turning points in a graph help identify the degree of the function?
4. How do the intercepts relate to the factors of a polynomial function?
5. What would happen to the graph if we changed the leading coefficient $a$ to a negative value?

Tip:

For a cubic function $f(x) = a(x - r_1)(x - r_2)(x - r_3)$, the roots $r_1, r_2, r_3$ represent the x-intercepts, and you can determine $a$ using a point $(x, y)$ that lies on the graph!