https://file.aiquickdraw.com/mathbot/file/1729712249596b6tsfk8o.jpg

Looking at the graph you provided, we can observe that the quadratic function has a vertex and a clear parabolic shape.

### Key observations:
- The vertex of the parabola is at \((2, -1)\), which suggests that the equation is in **vertex form**.
- The vertex form of a quadratic equation is:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. From the graph, \((h, k) = (2, -1)\).

- We also notice that the parabola opens upward, which indicates that the coefficient \(a > 0\).
  
#### Step 1: Substituting the vertex coordinates:
\[
y = a(x - 2)^2 - 1
\]

#### Step 2: Determine the value of \(a\):
To find \(a\), we can use another point from the graph. From the graph, we can see that the parabola passes through the point \((1, 0)\).

- Substituting \(x = 1\) and \(y = 0\) into the equation to solve for \(a\):
\[
0 = a(1 - 2)^2 - 1
\]
\[
0 = a(1)^2 - 1
\]
\[
0 = a - 1
\]
\[
a = 1
\]

#### Step 3: Final equation:
Now that we have \(a = 1\), the equation of the parabola becomes:
\[
y = (x - 2)^2 - 1
\]

This is the equation for the quadratic graph.

Would you like more details on any part of this process or have additional questions?

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Here are 5 questions related to this:
1. What is the general form of a quadratic equation, and how can we convert from vertex form?
2. How can we determine the axis of symmetry for a quadratic function?
3. How do we find the x-intercepts (roots) of a quadratic equation?
4. What are the implications of the value of \(a\) on the shape of a parabola?
5. How can we graph a quadratic equation using only its vertex and one additional point?

**Tip:** In quadratic equations, the sign of the coefficient \(a\) determines whether the parabola opens upward (\(a > 0\)) or downward (\(a < 0\)).

Write an equation (any form) for the quadratic graphed below.

Learn how to derive the equation of a quadratic function from its graph with a vertex at (2, -1). This problem involves using the vertex form of a quadratic equation and determining the 'a' value with an additional point. Suitable for students in Grades 9-11.

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