To determine the factored form equation of the given quadratic relation from the graph, we need to identify the roots (x-intercepts) and the vertex of the parabola.

From the graph:

• The x-intercepts (roots) appear to be at $x = 1$ and $x = 4$.
• The vertex appears to be at $(2.5, -14)$.

Using the roots, we can express the quadratic equation in its factored form: $y = a(x - 1)(x - 4)$

To find the value of $a$, we use the vertex. We know that at $x = 2.5$, $y = -14$: $-14 = a(2.5 - 1)(2.5 - 4)$ $-14 = a(1.5)(-1.5)$ $-14 = a(-2.25)$ $a = \frac{-14}{-2.25}$ $a = \frac{14}{2.25}$ $a = \frac{56}{9}$

Therefore, the factored form of the quadratic equation is: $y = \frac{56}{9}(x - 1)(x - 4)$

Would you like more details or have any questions?

Here are five related questions that might help expand your understanding:

1. How do you convert a quadratic equation from factored form to standard form?
2. How can you determine the vertex of a quadratic function given its factored form?
3. What is the significance of the leading coefficient in a quadratic equation?
4. How can you verify the roots of a quadratic equation graphically?
5. How does the value of $a$ affect the shape and direction of a parabola?

Tip: When finding the factored form of a quadratic equation from a graph, ensure you accurately determine the roots and vertex for precise calculation.