https://file.aiquickdraw.com/mathbot/file/17274213442078zzu081q.jpg

The problem shown in the image involves determining the equation of a quadratic function given its graph. Let's walk through each step based on what’s visible:

### Step 1: 
- **Vertex identification:**  
  From the graph, the vertex is given as \((-3, -2)\). This is correct, and this point will be useful for finding the standard form of the quadratic equation.

### Step 2:
- **Quadratic function standard form:**  
  The standard form of a quadratic function is:
  \[
  f(x) = a(x - h)^2 + k
  \]
  Where \((h, k)\) is the vertex of the parabola. Since the vertex is \((-3, -2)\), we substitute \(h = -3\) and \(k = -2\):
  \[
  f(x) = a(x + 3)^2 - 2
  \]

### Step 3:
- **Finding the value of \(a\):**  
  To find \(a\), we use the additional point \((0, 7)\) on the graph. Substituting \(x = 0\) and \(f(x) = 7\) into the equation:
  \[
  7 = a(0 + 3)^2 - 2
  \]
  Simplifying:
  \[
  7 = a(9) - 2
  \]
  Add 2 to both sides:
  \[
  9 = 9a
  \]
  Solving for \(a\):
  \[
  a = 1
  \]

Thus, the quadratic function in standard form is:
\[
f(x) = (x + 3)^2 - 2
\]

### Final Answer:
The quadratic function that describes the parabola is:
\[
f(x) = (x + 3)^2 - 2
\]

Let me know if you'd like more details, or if you have any questions!

Here are some follow-up questions:
1. How can you verify that the quadratic function is correct?
2. What is the effect of the parameter \(a\) on the shape of the parabola?
3. How would the equation change if the parabola opened downward?
4. Can you rewrite the quadratic function in expanded form?
5. What are the x-intercepts of the parabola?

**Tip:** The value of \(a\) determines the "width" and "direction" of the parabola. If \(a > 0\), the parabola opens upward; if \(a < 0\), it opens downward.

Write the standard form of the quadratic function whose graph is the parabola shown.

Learn how to write the standard form of a quadratic function given its graph. This tutorial covers finding the vertex and determining the coefficient 'a' using a point on the graph. Suitable for students in Grades 9-11.

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