https://file.aiquickdraw.com/mathbot/file/1727235887452u24qqur5.jpg

The graph you uploaded appears to be a quadratic function. To find the factored form of the equation of a quadratic, we need to use the x-intercepts and the vertex.

From the graph:

- The x-intercepts (roots) are \((0, 0)\) and \((6, 0)\).
- The vertex is \((3, 3)\), which gives us the peak of the parabola.

The factored form of a quadratic equation is:

\[
y = a(x - r_1)(x - r_2)
\]

where \(r_1\) and \(r_2\) are the x-intercepts.

### Step 1: Using the x-intercepts
The intercepts are \((0, 0)\) and \((6, 0)\), so the equation becomes:

\[
y = a(x - 0)(x - 6) = a x (x - 6)
\]

\[
y = a x (x - 6)
\]

### Step 2: Using the vertex to solve for \(a\)
Substitute the vertex \((3, 3)\) into the equation to solve for \(a\):

\[
3 = a (3)(3 - 6)
\]
\[
3 = a (3)(-3)
\]
\[
3 = -9a
\]
\[
a = -\frac{1}{3}
\]

### Step 3: Final equation
Now that we know \(a = -\frac{1}{3}\), the factored form of the quadratic equation is:

\[
y = -\frac{1}{3}x(x - 6)
\]

This is the factored form of the quadratic equation based on the graph.

Would you like further details or have any questions?

### 5 Related Questions:
1. How would the equation change if the vertex was lower or higher on the graph?
2. How can we convert this factored form to standard form?
3. What is the significance of the value of \(a\) in the graph's shape?
4. Can the quadratic equation have other forms besides factored form?
5. How would we find the vertex if it wasn't provided on the graph?

### Tip:
When identifying the factored form of a quadratic equation, always start by locating the x-intercepts, then use the vertex to find the value of \(a\).

Determine the factored form equation of the quadratic relation shown in the graph, where the x-intercepts are (0, 0) and (6, 0) and the vertex is at (3, 3).

Learn how to find the factored form of a quadratic equation given the x-intercepts (0, 0) and (6, 0), and the vertex at (3, 3). This example illustrates the key concepts of quadratic functions, including using the vertex to find the coefficient a.

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