To find the polynomial function $f(x)$ that matches this graph, let’s analyze the visible characteristics of the function:

1. Roots: The graph crosses or touches the x-axis at specific points. Observing these intercepts will help us determine the roots of the polynomial:

   • At approximately $x = -4$
   • At $x = 0$
   • At $x = 3$

   Since the problem specifies that $f(x)$ should be written in factored form, we can assign factors based on these roots. These roots suggest factors like: $(x + 4), \quad x, \quad \text{and} \quad (x - 3)$

2. Multiplicity of Roots: The behavior at each root determines the multiplicity:

   • The graph passes through $x = -4$ and $x = 3$, suggesting these are likely roots of multiplicity 1.
   • At $x = 0$, the graph just touches the x-axis and bounces back, indicating a root of even multiplicity (likely multiplicity 2 for simplicity).

   Therefore, a likely factored form for $f(x)$ is: $f(x) = a \cdot (x + 4) \cdot x^2 \cdot (x - 3)$

3. Determining $a$: We are given that $a = 1$, so no additional scaling is needed.

Thus, the polynomial function that matches the graph is: $f(x) = (x + 4) \cdot x^2 \cdot (x - 3)$

Would you like more details on this approach, or do you have any other questions?

Related Questions:

1. How do different multiplicities of roots affect the shape of a polynomial graph?
2. What happens to a polynomial graph when the leading coefficient changes?
3. How can you determine the degree of a polynomial based on its graph?
4. What is the relationship between turning points and the degree of a polynomial?
5. How do changes in the signs of coefficients affect the direction of a polynomial graph?

Tip: The multiplicity of a root changes the way the graph behaves near that root—roots with even multiplicity touch and turn, while roots with odd multiplicity cross the x-axis.