91Ó°ÊÓ

The radius is an important aspect of a circle, as it defines its size. It is the distance from the center of the circle to any point along the edge of the circle. The radius is constant for all points on the circle, meaning each point on the circle is the same distance from the center.
In the equation of a circle \((x - h)^2 + (y - k)^2 = r^2\), the radius \(r\) is represented by the square root of the number on the right side of the equation. In our exercise, the circle's equation is given as \(x^2 + y^2 = 9\). By rewriting it to match the general equation, \(r^2 = 9\), taking the square root gives us \(r = 3\). This tells us that the radius of our circle is 3 units. Knowing the radius helps in accurately drawing the circle and understanding its size.

The center of a circle is a fixed point from which every point on the circle is equidistant. In the standard equation of a circle \((x - h)^2 + (y - k)^2 = r^2\), the center is the point \( (h, k) \). This is the reference point that helps in plotting the circle on a graph.
In the exercise provided, comparing the equation \(x^2 + y^2 = 9\) to the general form shows the equation \((x - 0)^2 + (y - 0)^2 = 3^2\). Here, both \(h\) and \(k\) are 0, so the center of the circle is at \( (0, 0) \), also called the origin in a coordinate plane.

• The circle is symmetric around its center.
• The center does not appear explicitly in the given equation but can be derived by recognizing the completed square format.

Identifying the center is vital for graphing the circle accurately.

Graphing circles involves plotting points and drawing curves to visually represent the equation of the circle. Understanding how to graph circles helps in visualizing the relationship between the algebraic equation and its geometric representation.
For the circle given by the equation \(x^2 + y^2 = 9\), which we found has a center at \( (0, 0) \) and a radius of 3, the graphing process can be done as follows:

• Start by plotting the center of the circle at the origin \( (0, 0) \) on the graph.
• From this point, mark points that are 3 units away in all cardinal directions (up, down, left, and right), leading to points \( (3, 0), (-3, 0), (0, 3), and (0, -3) \).
• These points are the farthest points on the circle along the x and y axes.

Once these points are marked, draw a smooth curve to connect them, forming the edge of the circle. Having these visual markers assists in ensuring the circle is drawn proportionally and accurately on the coordinate plane.