https://file.aiquickdraw.com/mathbot/file/1733679640869dkvsgr4m.jpg

Let’s analyze the graph and find the equation of the parabola step-by-step.

### Observations:
1. The **vertex** of the parabola is at \( (5, -5) \). This indicates the equation is in **vertex form**:
   \[
   y = a(x - h)^2 + k
   \]
   Here:
   - \( h = 5 \) (the x-coordinate of the vertex),
   - \( k = -5 \) (the y-coordinate of the vertex).

   So the equation becomes:
   \[
   y = a(x - 5)^2 - 5
   \]

2. The parabola opens upwards, indicating \( a > 0 \).  
   To determine \( a \), we use another point on the parabola. From the graph, the point \( (6, -3) \) lies on the parabola.

### Substituting \( (6, -3) \) into the equation:
\[
y = a(x - 5)^2 - 5
\]
\[
-3 = a(6 - 5)^2 - 5
\]
\[
-3 = a(1)^2 - 5
\]
\[
-3 + 5 = a
\]
\[
a = 2
\]

### Final Equation:
The equation of the parabola is:
\[
y = 2(x - 5)^2 - 5
\]

### Answer:
The correct choice is **d) \( 2(x - 5)^2 - 5 \)**.

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Would you like a detailed breakdown of another problem or additional questions related to parabolas? Here are some related questions for practice:

1. How do you determine the vertex of a parabola from its equation?
2. What happens to the graph if the value of \( a \) in the equation changes?
3. Can a parabola have a negative \( a \) value? What does that look like?
4. How can we convert a quadratic equation from standard form to vertex form?
5. What are the key characteristics of a parabola in the context of its graph?

**Tip:** Always use the vertex and another point to confirm the equation of a parabola in vertex form.

Find the equation of the parabola based on the provided graph.

Learn how to determine the equation of a parabola from its graph. This problem uses the vertex form equation y = a(x - h)^2 + k and demonstrates step-by-step how to calculate the parameter 'a' using a given point. Suitable for students in Grades 9-11.

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