https://file.aiquickdraw.com/mathbot/file/17277641561008tqmu478.jpg

To solve this problem, let's analyze the graph.

The graph shows a parabola opening upwards, with its vertex at the point \((2, -3)\). From this, we can derive the general form of the quadratic function:

\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. For this parabola, the vertex is \((2, -3)\), so the equation becomes:

\[
y = a(x - 2)^2 - 3
\]

Now, we need to determine the value of \(a\). The graph passes through the point \((0, 1)\), so we can substitute \(x = 0\) and \(y = 1\) to find \(a\):

\[
1 = a(0 - 2)^2 - 3
\]
\[
1 = a(4) - 3
\]
\[
1 + 3 = 4a
\]
\[
4 = 4a \quad \Rightarrow \quad a = 1
\]

Thus, the quadratic equation representing the parabola is:

\[
y = (x - 2)^2 - 3
\]

Since the graph shows the area below the parabola shaded, we are dealing with a quadratic inequality where the values of \(y\) are less than or equal to the equation of the parabola. Therefore, the inequality is:

\[
y \leq (x - 2)^2 - 3
\]

This is the quadratic inequality represented by the graph.

Would you like further clarification or details on this? Here are some related questions to expand on this concept:

1. What are the different forms of a quadratic function, and how can they be used in inequalities?
2. How do you determine whether to use \(\leq\) or \(\geq\) in quadratic inequalities based on the graph?
3. How can you find the vertex of a parabola given its quadratic equation?
4. How does the value of \(a\) affect the shape of a parabola?
5. How can you determine the axis of symmetry for a quadratic graph?

**Tip:** When analyzing quadratic graphs, always check the direction in which the parabola opens (upwards or downwards) to determine the appropriate inequality.

Which quadratic inequality does the graph represent?

Learn how to derive the quadratic inequality y <= (x - 2)^2 - 3 from the graph of a parabola. This detailed explanation covers how to find the vertex form of the quadratic equation and determine the inequality.

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