To identify the equation of the graph, let's analyze the properties of the polynomial:

1. Intercepts:

   • The graph crosses the x-axis at points approximately $x = -4$, $x = -2$, and $x = 1$. This suggests the polynomial has roots at these values.

2. Behavior at Each Root:

   • At $x = -4$, the graph appears to cross the x-axis in a "normal" linear fashion, indicating a simple root of multiplicity 1.
   • At $x = -2$, the graph also crosses in a linear fashion, suggesting this root also has multiplicity 1.
   • At $x = 1$, however, the graph "bounces" off the x-axis, implying that this root has a higher multiplicity, likely 3, as it resembles a cubic behavior around that point.

3. Equation Formulation:

   • Based on these observations, we can infer that the equation should be of the form: $y = (x + 4)(x + 2)(x - 1)^3$

Conclusion:

The correct equation for the graph is: $y = (x + 4)(x + 2)(x - 1)^3$

This matches option: "y = (x + 4)(x + 2)(x - 1)^3".

Would you like more details on any of these steps or have additional questions?

Related Questions:

1. How do we determine the multiplicity of a root from a graph?
2. What is the significance of the root multiplicities in a polynomial's equation?
3. How do we find the degree of a polynomial based on its graph?
4. Can we determine the y-intercept from the equation of this polynomial?
5. How would the graph change if one of the roots had a different multiplicity?

Tip:

To analyze polynomial graphs, focus on where the graph intersects or "bounces" at the x-axis to understand the multiplicities of the roots.