https://file.aiquickdraw.com/mathbot/file/1724724225491ffu3jakm.jpg

The graph in the image depicts a parabola. To find the equation of the parabola, we need to determine key characteristics such as the vertex and any other points on the graph.

From observation:
- The vertex of the parabola appears to be at the point \((0, 2)\).
- The parabola opens upwards, which suggests that the coefficient of the squared term is positive.

Given that the vertex is \((0, 2)\), we can use the vertex form of a quadratic equation, which is:

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

Where:
- \((h, k)\) is the vertex,
- \(a\) is a constant that determines the "width" and direction of the parabola.

Substituting the vertex \((0, 2)\) into the equation, we have:

\[
y = a(x - 0)^2 + 2
\]
\[
y = ax^2 + 2
\]

To find the value of \(a\), we need another point on the graph. From observation, a point that lies on the parabola is \((-1, 3)\).

Substituting \((-1, 3)\) into the equation:

\[
3 = a(-1)^2 + 2
\]
\[
3 = a + 2
\]
\[
a = 1
\]

Thus, the equation of the parabola is:

\[
y = x^2 + 2
\]

Would you like to explore this further, or do you have any questions?

Here are 5 related questions:
1. How would the equation change if the vertex were at a different point?
2. What would happen if the parabola opened downwards instead of upwards?
3. Can you identify the axis of symmetry from the equation?
4. How would you find the x-intercepts of this parabola?
5. What role does the coefficient \(a\) play in determining the "width" of the parabola?

**Tip:** The vertex form of a quadratic equation is very useful when you know the vertex, allowing you to quickly identify the parabola's position and shape.

Learn how to find the equation of a parabola when given its vertex and a point on the graph. This problem involves using the vertex form of a quadratic equation and solving for the coefficient 'a'. Suitable for high school students.

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