https://file.aiquickdraw.com/mathbot/file/1727317863854r3rhm7uf.jpg

The graph represents a parabola, and we are tasked with finding the quadratic equation based on the provided information. Let's break it down.

### Key Points from the Graph:
1. **Vertex**: The highest point of the parabola is at \(( -4, 16 )\). This indicates that the parabola is in vertex form, and its vertex is \((-4, 16)\).
2. **X-Intercepts (Roots)**: The parabola crosses the x-axis at \(x = -8\) and \(x = 0\).

### Equation Setup:
The general factored form of a quadratic equation is:

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

Where \(r_1\) and \(r_2\) are the roots (x-intercepts), and \(a\) is a coefficient that determines the vertical stretch or compression.

- From the graph, we have the roots \(r_1 = -8\) and \(r_2 = 0\). So, the factored form becomes:

\[
f(x) = a(x + 8)(x)
\]

### Finding the Value of \(a\):
The vertex is at \((-4, 16)\). We can use this point to find the value of \(a\).

Substitute \(x = -4\) and \(f(x) = 16\) into the equation:

\[
16 = a(-4 + 8)(-4)
\]

Simplify:

\[
16 = a(4)(-4)
\]
\[
16 = -16a
\]
\[
a = -1
\]

### Final Equation:
Thus, the quadratic equation is:

\[
f(x) = -1(x + 8)(x)
\]

Or, expanded:

\[
f(x) = -(x^2 + 8x) = -x^2 - 8x
\]

Therefore, the equation of the parabola is:

\[
f(x) = -x^2 - 8x
\]

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

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Here are 5 related questions:
1. How do you determine the vertex from a quadratic equation in standard form?
2. What role does the coefficient \(a\) play in the shape of a parabola?
3. Can a quadratic equation have more than two roots?
4. How do you factor a quadratic equation that doesn’t have clear x-intercepts?
5. What is the significance of the x-intercepts in a real-world scenario?

**Tip**: When finding the equation of a parabola from a graph, always check the vertex and x-intercepts, as they provide valuable information for constructing the equation.

Use the graph to assemble the quadratic equation from the provided factors.

Learn how to derive the quadratic equation from a graph using the vertex and x-intercepts. This solution breaks down the process, including finding the factored form and standard form. Suitable for Grades 8-10 students studying algebra and quadratic equations.

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