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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.

Determine the factored form equation of each of the following quadratic relations: b) from the graph provided.

Learn how to determine the factored form of a quadratic equation from its graph. This solution walks through identifying the x-intercepts and vertex, then solving for the factored form of the quadratic relation. Suitable for high school students studying algebra and quadratic functions.

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