https://file.aiquickdraw.com/mathbot/file/1727281644486baci6bpg.jpg

We are given a graph of a quadratic function, and we need to determine its equation. The graph shows a parabola with the following key points:

- The vertex is at \((-2, -6)\).
- Another point on the parabola is \((0, -2)\).

We can use the **vertex form** of a quadratic equation, which is:

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

where \((h, k)\) is the vertex. In this case, the vertex is \((-2, -6)\), so we can substitute these values into the equation:

\[
f(x) = a(x + 2)^2 - 6
\]

Next, we use the other given point \((0, -2)\) to find the value of \(a\). Substituting \(x = 0\) and \(f(x) = -2\) into the equation:

\[
-2 = a(0 + 2)^2 - 6
\]

Simplifying:

\[
-2 = a(4) - 6
\]
\[
-2 + 6 = 4a
\]
\[
4 = 4a
\]
\[
a = 1
\]

Thus, the quadratic function is:

\[
f(x) = (x + 2)^2 - 6
\]

Expanding this:

\[
f(x) = (x^2 + 4x + 4) - 6
\]
\[
f(x) = x^2 + 4x - 2
\]

So, the quadratic function that describes the graph is:

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

Would you like more details on how I expanded or solved this equation? Here are some related questions for further practice:

1. How do you derive the vertex form of a quadratic equation?
2. What are the steps to expand \((x + 2)^2 - 6\) into standard form?
3. How can you determine the axis of symmetry for this quadratic function?
4. How does the value of \(a\) affect the shape of the parabola?
5. Can you find the x-intercepts of this quadratic function?

**Tip:** When solving for the constant \(a\), always use another point on the graph to substitute into the equation.

Determine the quadratic function whose graph is given below. The vertex is (-2, -6), and the point (0, -2) lies on the parabola.

Learn how to determine a quadratic function from its graph using the vertex form. The problem involves the vertex (-2, -6) and another point (0, -2). Includes step-by-step instructions for finding the quadratic equation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation