https://file.aiquickdraw.com/mathbot/file/1725597065696sywmzcz6.jpg

The image you provided contains a graph of a parabola, and the task is to find the equation of the parabola based on the graph.

To analyze:

- The graph shows a parabola opening upwards.
- The vertex of the parabola appears to be at the point \((0, -4)\), which suggests that the equation has a vertex form of \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, \((h, k) = (0, -4)\), so the equation becomes:
  \[
  y = a(x - 0)^2 - 4 \quad \text{or} \quad y = ax^2 - 4
  \]
- To find the value of \(a\), we can use another point from the graph. From the graph, one point is \((1, -3)\). Substituting \(x = 1\) and \(y = -3\) into the equation \(y = ax^2 - 4\):

  \[
  -3 = a(1)^2 - 4
  \]
  \[
  -3 = a - 4
  \]
  \[
  a = 1
  \]

Thus, the equation of the parabola is:
\[
y = x^2 - 4
\]

The multiple-choice answers should confirm this equation.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do we derive the vertex form of a quadratic equation?
2. What happens to the graph when the value of \(a\) changes in the equation \(y = ax^2 + bx + c\)?
3. Can the equation \(y = x^2 - 4\) be written in standard form?
4. How do you find the vertex of a parabola from the general form of a quadratic equation?
5. What are the effects of shifting the graph horizontally or vertically?

**Tip:** Always check for key points on the graph, like the vertex and intercepts, to derive an accurate equation.

Write the equation of the parabola shown in the graph below.

Learn how to find the equation of a parabola by analyzing its graph. This example covers the use of the vertex form of a quadratic equation, with the vertex located at (0, -4). Suitable for students in Grades 7-9.

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