https://file.aiquickdraw.com/mathbot/file/1727450165011hcj7zute.jpg

The graph shown is a parabola, and we are tasked with finding the quadratic function \( f(x) \). From the graph, we can observe the following key points:

1. The vertex of the parabola is at \((-1, -4)\), which gives us important information for writing the function in **vertex form**:
   \[
   f(x) = a(x - h)^2 + k
   \]
   where \((h, k)\) is the vertex. In this case, \(h = -1\) and \(k = -4\), so the equation becomes:
   \[
   f(x) = a(x + 1)^2 - 4
   \]
   
2. Next, we use the additional point \((-2, -2)\) provided in the graph to solve for \(a\). Substitute \(x = -2\) and \(f(x) = -2\) into the equation:
   \[
   -2 = a(-2 + 1)^2 - 4
   \]
   Simplifying:
   \[
   -2 = a(1)^2 - 4
   \]
   \[
   -2 = a - 4
   \]
   Solving for \(a\):
   \[
   a = 2
   \]

3. Therefore, the equation of the quadratic function is:
   \[
   f(x) = 2(x + 1)^2 - 4
   \]

Would you like more details or have any further questions?

Here are 5 related questions to expand this topic:

1. What is the **standard form** of the quadratic function from the vertex form we found?
2. How do you **find the axis of symmetry** from the equation of a quadratic function?
3. What is the importance of the **discriminant** in quadratic functions?
4. Can a parabola be translated without changing its **vertex**?
5. How does the value of \(a\) affect the **direction** and **width** of the parabola?

**Tip:** Remember that the sign of \(a\) in a quadratic equation determines whether the parabola opens upward or downward!

Find the equation of the quadratic function f whose graph is shown below.

Learn how to find the equation of a quadratic function by identifying key points from its graph, such as the vertex (-1, -4) and another point (-2, -2). This guide provides a step-by-step approach using the vertex form of quadratic equations, suitable for students in Grades 9-10.

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