https://file.aiquickdraw.com/mathbot/file/1727817529483r0n3y66t.jpg

From the graph provided, we can observe several characteristics of the quadratic function:

1. The graph is a parabola that opens upward, indicating a positive leading coefficient.
2. The vertex is at \((1, -2)\).
3. The parabola crosses the y-axis at \( (0, 0) \), meaning the y-intercept is 0.

The standard form of a quadratic equation is:

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

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

Using the vertex \((1, -2)\), the equation becomes:

\[
y = a(x - 1)^2 - 2
\]

Next, we use the point where the parabola intersects the y-axis, \((0, 0)\), to find \(a\). Substituting \(x = 0\) and \(y = 0\) into the equation:

\[
0 = a(0 - 1)^2 - 2
\]
\[
0 = a(1) - 2
\]
\[
a = 2
\]

Thus, the equation of the parabola is:

\[
y = 2(x - 1)^2 - 2
\]

Would you like more details on how this equation was derived, or do you have any further questions?

Here are 5 questions to expand the understanding:

1. How can you convert the equation into the standard form \(y = ax^2 + bx + c\)?
2. What does the value of \(a = 2\) tell you about the graph's shape?
3. Can you describe how the vertex form of a quadratic relates to the graph?
4. How would the equation change if the parabola was shifted up or down?
5. What happens to the graph if the \(a\) value is negative?

**Tip**: The vertex form of a quadratic equation is useful for easily identifying the vertex and the direction in which the parabola opens!

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

Learn how to write a quadratic equation based on the graph with a vertex at (1, -2) and y-intercept at (0, 0). The problem covers key algebraic concepts such as vertex form and quadratic equations. Suitable for students in Grades 9-10.

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