91Ó°ÊÓ

The radius of a circle is a fundamental concept in geometry, which determines the size of the circle. It is the distance from the center of the circle to any point on its circumference.
For any circle represented by the equation \(x^2 + y^2 = r^2\), the radius \(r\) can be directly obtained.
In our exercise, the circle is defined by the equation \(x^2 + y^2 = 9\). By comparing it with the standard form, we immediately find that the radius is 3, because \(r^2 = 9\) implies \(r = \sqrt{9} = 3\).
Understanding the radius helps to visualize the circle and understand its properties, like where its boundary lies based on its center.

Solving for \(y\) is a common step in rearranging equations to isolate a specific variable. This is particularly useful for converting an equation into a function that can be graphically represented.
In equations like \(x^2 + y^2 = 9\), solving for \(y\) involves rearranging terms to express \(y\) in terms of \(x\).
By subtracting \(x^2\) from each side, we have \(y^2 = 9 - x^2\). Taking the square root of both sides gives us \(y = \pm\sqrt{9 - x^2}\). The \(\pm\) symbol indicates that there are two potential values for \(y\) — one for the top half of the circle (positive) and one for the bottom half (negative).
Understanding this helps in situations where you need to determine one branch of a curve, such as the top half of a circle.

The graph of a function is a visual representation that showcases the relationship between its variables. When dealing with circles, or any other shape, transforming an equation into a function form enables graphing specific parts of the shape.
In the context of the original exercise, once we solve \(y = \sqrt{9 - x^2}\) for the top half of the circle, we now have a function that can be graphed on a coordinate plane.
Graphing this function, you'll see a semi-circle that rises from left to right as \(x\) moves from -3 to 3. This visually translates the abstract mathematical equation into a curve that can be understood and analyzed easily.

The domain of a function refers to all the possible values of \(x\) for which the function is defined. It is essential for determining how far left and right a function's graph extends in the coordinate plane.
For the function \(y = \sqrt{9 - x^2}\), the values under the square root, \(9 - x^2\), must be non-negative (i.e., greater than or equal to zero) for the function to be real and valid.

• The inequality \(9 - x^2 \geq 0\) leads to \(-3 \leq x \leq 3\), meaning the domain of this function is all real numbers \(x\) from -3 to 3.

This ensures the function captures the shape we want — specifically, the top half of the circle corresponding to the original equation \(x^2 + y^2 = 9\). Understanding the domain helps confirm this part of the graph is plotted correctly.