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This image shows a matching exercise where you are asked to associate each given equation of a circle with its corresponding graph.

### Equations:
1. \((x + 3)^2 + (y - 2)^2 = 4\)
2. \((x - 5)^2 + (y + 5)^2 = 25\)
3. \((x + 5)^2 + (y - 5)^2 = 25\)
4. \((x - 3)^2 + (y + 2)^2 = 4\)

### General Form of a Circle Equation:
The equation of a circle in the form \((x - h)^2 + (y - k)^2 = r^2\) describes a circle with center \((h, k)\) and radius \(r\).

### Analysis of the Equations:
1. **Equation 1:** \((x + 3)^2 + (y - 2)^2 = 4\) has center \((-3, 2)\) and radius 2 (since \(r = \sqrt{4} = 2\)).
2. **Equation 2:** \((x - 5)^2 + (y + 5)^2 = 25\) has center \((5, -5)\) and radius 5 (since \(r = \sqrt{25} = 5\)).
3. **Equation 3:** \((x + 5)^2 + (y - 5)^2 = 25\) has center \((-5, 5)\) and radius 5.
4. **Equation 4:** \((x - 3)^2 + (y + 2)^2 = 4\) has center \((3, -2)\) and radius 2.

### Matching with the Graphs:
Let's examine the graphs one by one:

- **Graph (a):** This circle appears to be centered at a positive \(x\)-value and negative \(y\)-value and has a relatively larger radius. The center seems to be around \((3, -2)\), which corresponds to Equation 4: \((x - 3)^2 + (y + 2)^2 = 4\).
  
- **Graph (b):** This graph shows a circle with a small radius, likely centered at \((-3, 2)\). This matches Equation 1: \((x + 3)^2 + (y - 2)^2 = 4\).

- **Graph (c):** A small circle positioned near \((0, 0)\), likely centered at \((-5, 5)\), which matches Equation 3: \((x + 5)^2 + (y - 5)^2 = 25\).

- **Graph (d):** The circle is large, and the center is close to \((5, -5)\). This matches Equation 2: \((x - 5)^2 + (y + 5)^2 = 25\).

### Final Matching:
- Graph (a): Equation 4
- Graph (b): Equation 1
- Graph (c): Equation 3
- Graph (d): Equation 2

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Would you like more details or clarification? Here are five related questions for further practice:
1. How does changing the value of \(r^2\) affect the size of the circle?
2. How do you determine the center of a circle from its equation?
3. What happens if you change the signs inside the parentheses in the circle equation?
4. Can a circle's equation ever have a negative radius squared value? Why or why not?
5. How can you graph a circle without knowing its equation?

**Tip:** Always remember that the center coordinates in the equation \((x - h)^2 + (y - k)^2 = r^2\) are determined by the values \((h, k)\) with signs reversed from the equation form.

Match each graph with the correct equation. The given equations describe circles, and the task is to identify which graph corresponds to each equation.

Solve the problem of matching circle equations to graphs by identifying the center and radius. This problem involves the general formula for the equation of a circle and is suitable for students in grades 8-10.

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