https://file.aiquickdraw.com/mathbot/file/1727297216663lyh7bpup.jpg

The graph in the image shows a parabola, and the task is to determine the quadratic function that describes it. Let's analyze the graph:

- The parabola is opening upwards, meaning the coefficient of the quadratic term \(a\) is positive.
- The vertex of the parabola is given as \((-2, -16)\). The vertex form of a quadratic function is:
  \[
  f(x) = a(x - h)^2 + k
  \]
  where \((h, k)\) is the vertex. Here, \(h = -2\) and \(k = -16\), so the equation becomes:
  \[
  f(x) = a(x + 2)^2 - 16
  \]
- We need to determine \(a\). To do this, we can use another point on the graph. From the graph, we can see that the parabola passes through the point \((0, -4)\). Substituting \(x = 0\) and \(f(x) = -4\) into the equation gives:
  \[
  -4 = a(0 + 2)^2 - 16
  \]
  Simplifying this:
  \[
  -4 = a(4) - 16
  \]
  \[
  -4 + 16 = 4a
  \]
  \[
  12 = 4a
  \]
  \[
  a = 3
  \]
- Therefore, the quadratic function is:
  \[
  f(x) = 3(x + 2)^2 - 16
  \]

Would you like more details or further clarification?

Here are 5 related questions to explore further:
1. How would the equation change if the parabola opened downwards?
2. What would happen to the equation if the vertex were at \((3, 5)\)?
3. How do we convert from vertex form to standard form of a quadratic function?
4. What is the effect of changing the value of \(a\) on the shape of the parabola?
5. How can we find the x-intercepts of a parabola given its equation?

**Tip:** When identifying a quadratic function from a graph, the vertex and one additional point are crucial for determining the complete equation.

Determine the quadratic function whose graph is given below.

Learn how to find the quadratic equation for a parabola given its graph. The vertex form of the equation is used, and points on the graph are utilized to determine the value of 'a'. This example includes a graph with vertex (-2, -16) and passes through (0, -4).

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