determine which of the given points are on the graph of the equation. Equation: x^2 + y^2 = 4 Points: (0, 2);( - 2, 2); (/2, /2)

To determine whether the given points lie on the graph of the equation \(x^2 + y^2 = 4\), we will substitute the coordinates of each point into the equation and check if the equation holds true.

### Step 1: Analyze the equation
The equation \(x^2 + y^2 = 4\) represents a circle centered at the origin \((0, 0)\) with a radius of 2.

### Step 2: Check each point

**Point 1: \((0, 2)\)**
\[
x = 0, \, y = 2
\]
Substitute into the equation:
\[
0^2 + 2^2 = 0 + 4 = 4
\]
Since \(4 = 4\), the point \((0, 2)\) **is on the graph**.

**Point 2: \((-2, 2)\)**
\[
x = -2, \, y = 2
\]
Substitute into the equation:
\[
(-2)^2 + 2^2 = 4 + 4 = 8
\]
Since \(8 \neq 4\), the point \((-2, 2)\) **is not on the graph**.

**Point 3: \((\sqrt{2}, \sqrt{2})\)**
\[
x = \sqrt{2}, \, y = \sqrt{2}
\]
Substitute into the equation:
\[
(\sqrt{2})^2 + (\sqrt{2})^2 = 2 + 2 = 4
\]
Since \(4 = 4\), the point \((\sqrt{2}, \sqrt{2})\) **is on the graph**.

### Conclusion:
- The points \((0, 2)\) and \((\sqrt{2}, \sqrt{2})\) **are on the graph**.
- The point \((-2, 2)\) **is not on the graph**.

Would you like more details or have any questions?

### Related Questions:
1. What is the radius of the circle defined by \(x^2 + y^2 = 9\)?
2. How would the equation change if the circle was centered at \((1, 1)\)?
3. Can you find the points where the graph intersects the x-axis?
4. What are the coordinates of the points where this circle intersects the y-axis?
5. How do you determine the distance of a point from the center of a circle?

**Tip:** To determine if a point lies on a circle, always substitute its coordinates into the circle's equation and verify if the equation holds true.

Learn how to determine whether given points lie on the circle defined by the equation x^2 + y^2 = 4. Detailed step-by-step analysis with examples including (0, 2), (-2, 2), and (√2, √2). Suitable for high school students.

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