https://file.aiquickdraw.com/mathbot/file/1725301410591gu67jekl.jpg

The graph appears to represent a parabola opening upwards, with its vertex below the x-axis. To find the equation of the parabola, we need to consider the standard form of a parabola:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola.

### Key observations from the graph:
1. **Vertex**: The vertex of the parabola is midway between the points where the parabola intersects the x-axis. The x-intercepts are approximately at \(-7\) and \(-2\).
   - The x-coordinate of the vertex \((h)\) is the midpoint of \(-7\) and \(-2\):
     \[
     h = \frac{-7 + (-2)}{2} = \frac{-9}{2} = -4.5
     \]
   - The y-coordinate of the vertex \((k)\) can be estimated from the graph. Since it is below the x-axis, \( k \) appears to be around \(-3.5\).

2. **X-intercepts**: The parabola crosses the x-axis at \( x = -7 \) and \( x = -2 \). These are the roots of the equation.

### Steps to determine the equation:
1. **Vertex Form**: Substitute \( h = -4.5 \) and \( k = -3.5 \) into the vertex form:
   \[
   y = a(x + 4.5)^2 - 3.5
   \]
2. **Find \( a \)**: Use one of the x-intercepts to find \( a \). Let's use \( x = -7 \) (where \( y = 0 \)):
   \[
   0 = a(-7 + 4.5)^2 - 3.5
   \]
   Simplifying:
   \[
   3.5 = a(-2.5)^2
   \]
   \[
   3.5 = a(6.25)
   \]
   \[
   a = \frac{3.5}{6.25} = 0.56
   \]
3. **Final Equation**:
   \[
   y = 0.56(x + 4.5)^2 - 3.5
   \]

This is the equation of the parabola.

Would you like more details, or do you have any other questions?

### Related Questions:
1. How do you convert the vertex form of a parabola to its standard form?
2. What is the significance of the coefficient \(a\) in a parabola's equation?
3. How can you determine the focus and directrix of a parabola given its equation?
4. What are the steps to find the axis of symmetry in a parabolic equation?
5. Can you explain how to find the roots of a quadratic equation graphically?

### Tip:
When identifying the vertex from a graph, always check the symmetry and ensure the vertex is correctly placed, as this is crucial for finding the correct equation.

Learn how to find the equation of a parabola from its graph. Detailed step-by-step solution for a parabola opening upwards with its vertex below the x-axis. Includes calculations for vertex, x-intercepts, and derivation of the parabola equation. Suitable for Grades 10-12.

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