https://file.aiquickdraw.com/mathbot/file/17266263355993fddnusn.jpg

The graph provided depicts a parabola that opens downward, and we need to match it with one of the given quadratic equations.

The standard form of a parabola is:

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

Where:
- \(a\) determines the direction and width of the parabola.
  - If \(a < 0\), the parabola opens downward.
  - If \(a > 0\), the parabola opens upward.
  - The magnitude of \(a\) controls how "wide" or "narrow" the parabola is.
- \((h, k)\) is the vertex of the parabola.

### Analysis of the graph:
- The vertex appears to be at \((-3, 3)\).
- The parabola opens downward, so \(a\) must be negative.
- The parabola is relatively wide, so the value of \(a\) should be small in magnitude (close to 0).

### Given equations:
1. \(y = -0.2(x + 3)^2 + 3\)
2. \(y = 0.2(x - 3)^2 + 3\)
3. \(y = 0.2(x + 3)^2 + 3\)
4. \(y = -0.2(x - 3)^2 + 3\)

### Matching:
- The vertex \((-3, 3)\) is matched when \((x + 3)\) is squared (since \(x + 3 = 0\) at \(x = -3\)).
- The parabola opens downward, so \(a\) must be negative, meaning we need \(a = -0.2\).

Thus, the correct equation is:

\[
y = -0.2(x + 3)^2 + 3
\]

Would you like more details or have any questions?

### Related Questions:
1. What does the parameter \(a\) in a quadratic equation represent geometrically?
2. How does the value of \(h\) in the equation \(y = a(x - h)^2 + k\) shift the parabola horizontally?
3. What is the effect of the \(k\) value in a quadratic equation on the graph's vertical position?
4. How do you find the axis of symmetry of a parabola from its equation?
5. What does the vertex form of a quadratic equation reveal about the graph?

### Tip:
The sign of the coefficient \(a\) in the quadratic equation always tells you whether the parabola opens upwards (positive) or downwards (negative).

Which equation best matches the graph shown below?

Explore how to match a graph of a parabola to its corresponding quadratic equation. This problem covers vertex form equations and properties of parabolas, suited for high school students (Grades 9-12).

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