https://file.aiquickdraw.com/mathbot/file/1725519774819mddn2fmj.jpg

The graph shows a parabola that opens upwards, indicating a quadratic function of the form \( y = ax^2 + bx + c \) or in vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

From the graph:
1. The vertex appears to be at \( (-2, -4) \).
2. The parabola passes through the point \( (0, 0) \) (the y-intercept).

### Step 1: Using the Vertex Form
We can use the vertex form of the quadratic equation:
\[
y = a(x - h)^2 + k
\]
Substituting the vertex \( (-2, -4) \) into this equation:
\[
y = a(x + 2)^2 - 4
\]

### Step 2: Using a Point to Solve for \(a\)
We know the parabola passes through \( (0, 0) \). Substituting this point \( (0, 0) \) into the equation to solve for \(a\):
\[
0 = a(0 + 2)^2 - 4
\]
\[
0 = 4a - 4
\]
\[
4a = 4 \quad \Rightarrow \quad a = 1
\]

### Step 3: Final Equation
Now that we have \( a = 1 \), the equation of the parabola is:
\[
y = (x + 2)^2 - 4
\]

This is the equation of the quadratic in vertex form. If you'd like it in standard form, we can expand it:

\[
y = (x + 2)(x + 2) - 4 = x^2 + 4x + 4 - 4 = x^2 + 4x
\]

Thus, the equation in standard form is:
\[
y = x^2 + 4x
\]

Would you like further details on how these forms are derived, or help with another step?

### Relative Questions:
1. How do you convert between standard form and vertex form of a quadratic equation?
2. What other points on the graph could help confirm the equation?
3. Can you find the axis of symmetry from the vertex form of the quadratic equation?
4. How can you determine if the parabola opens upward or downward?
5. How does changing the value of \(a\) affect the shape of the parabola?

### Tip:
When working with quadratic functions, identifying the vertex and intercepts from a graph can greatly simplify finding the equation.

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

Discover how to find the equation of a quadratic given its graph, with vertex at (-2, -4) and y-intercept at (0, 0). This problem includes the use of the vertex and standard forms of quadratic equations. Ideal for students in Grades 8-10.

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