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The problem asks you to find the equation for the upper half of the circle defined by:

\[
x^2 + y^2 = 64
\]

To express this in the form \( y = g(x) \), follow these steps:

### Step-by-Step Solution:

1. Start with the equation of the circle:
   \[
   x^2 + y^2 = 64
   \]

2. Solve for \( y^2 \) by subtracting \( x^2 \) from both sides:
   \[
   y^2 = 64 - x^2
   \]

3. To solve for \( y \), take the square root of both sides:
   \[
   y = \pm \sqrt{64 - x^2}
   \]

4. Since we are asked to find the equation for the **upper half** of the circle, we take the positive square root:
   \[
   y = \sqrt{64 - x^2}
   \]

Thus, the equation for the upper half of the circle in the form \( y = g(x) \) is:

\[
y = \sqrt{64 - x^2}
\]

Would you like a further breakdown or any clarifications on the steps?

### Relative Questions:
1. How do you find the equation of the lower half of the circle?
2. What is the domain of the function \( y = \sqrt{64 - x^2} \)?
3. How would you modify the equation if the circle had a different radius, say \( r \)?
4. Can this function be expressed as a piecewise function?
5. What is the geometric interpretation of the positive and negative square roots in this context?

### Tip:
For circles, the upper half is described by the positive square root, while the lower half corresponds to the negative square root of the equation.

Find the equation for the upper half of the circle x^2 + y^2 = 64.

This problem helps you find the equation of the upper half of a circle using the formula x^2 + y^2 = r^2. Learn how to express the equation in the form y = sqrt(r^2 - x^2), suitable for high school students in Grades 10-12.

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