Let's analyze the graph in the image and determine its equation in factored form as requested.

Step 1: Identify the Roots

From the graph, we can see that the curve crosses the x-axis at two points:

• $x = -2$
• $x = 2$

These are the roots of the polynomial function, so we can express the factors of the polynomial using these roots as follows:

$f(x) = a(x + 2)(x - 2)$

Step 2: Check the Behavior at the Roots

The graph crosses the x-axis at $x = -2$ and $x = 2$, and there is no bouncing at these points, indicating that these roots are of odd multiplicity (likely multiplicity 1).

Step 3: Determine the Leading Coefficient $a$

Next, we use another point on the graph to determine the value of $a$. The graph clearly passes through the point $(0, -4)$. We substitute this point into the equation to solve for $a$:

$f(0) = a(0 + 2)(0 - 2) = -4$ $a(2)(-2) = -4$ $-4a = -4 \quad \Rightarrow \quad a = 1$

Step 4: Write the Final Equation

Since $a = 1$, the equation in factored form is:

$f(x) = (x + 2)(x - 2)$

Thus, the factored form of the equation representing the graph is:

$f(x) = (x + 2)(x - 2)$

This is the equation of the graph.

Would you like more details or have any questions about this solution?

Here are 5 related questions for further understanding:

1. How does the multiplicity of a root affect the graph's behavior at that point?
2. What is the significance of using the point $(0, -4)$ to solve for $a$?
3. How would the equation change if the graph bounced off the x-axis at a root?
4. Can you explain why we only used two factors for this polynomial?
5. How would you convert this factored form equation into standard form?

Tip: Always check the behavior of the graph at the roots to identify their multiplicity (whether the graph crosses or touches the x-axis).