The given graph of $f(x)$ represents a quadratic function in standard parabolic form. To determine the equation of $f(x)$ in factored form, we need to find its:

1. Roots (x-intercepts): The points where the graph crosses the x-axis.
2. Vertex: For verification, the lowest point of the parabola can confirm the vertical stretch factor.

Step-by-step Solution:

1. Roots (x-intercepts): From the graph, the x-intercepts appear to be $x = -2$ and $x = 2$.

   • This gives us the factors: $(x + 2)$ and $(x - 2)$.

2. General factored form of a quadratic function: $f(x) = a(x + 2)(x - 2)$

3. Determine the leading coefficient $a$: Use the y-coordinate of a known point on the parabola, such as the vertex. From the graph, the vertex appears to be at $(0, -8)$. Substituting $x = 0$ and $f(0) = -8$: $-8 = a(0 + 2)(0 - 2)$ $-8 = a(2)(-2)$ $-8 = -4a \quad \Rightarrow \quad a = 2$

4. Final Factored Form: Substituting $a = 2$: $f(x) = 2(x + 2)(x - 2)$

Simplified Form for Verification:

Expanding $f(x)$: $f(x) = 2(x^2 - 4) = 2x^2 - 8$ This matches the graph perfectly, confirming the solution.

Final Answer:

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

Would you like further clarification on this? Let me know!

Here are five questions to expand on this topic:

1. How do we convert a quadratic function from factored form to standard form?
2. Why does the sign of $a$ determine the parabola's direction?
3. What is the connection between the roots and the axis of symmetry?
4. How can you find the vertex directly using the quadratic equation?
5. What are some real-world applications of quadratic functions?

Tip: Always verify your solution by substituting known points into the equation for accuracy.