Problem 17 - Algebra A Combined Function

Question

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

Center: (0, 0); Radius: 3.

Step 1: Identify the Standard Form

The given equation is \(x^2 + y^2 = 9\). This is already in the standard form of the equation of a circle \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Determine the Center

Compare the given equation \(x^2 + y^2 = 9\) with the standard form \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h = 0\) and \(k = 0\), as there are no \(x\) or \(y\) shifts in the equation. Thus, the center of the circle is \((0, 0)\).

Step 3: Determine the Radius

In the equation \(x^2 + y^2 = 9\), we have \(r^2 = 9\). To find \(r\), take the square root of both sides: \(r = \sqrt{9} = 3\). Thus, the radius is 3.

Step 4: Graph the Circle

To graph the circle, plot the center at the point \((0, 0)\). Then, from this center, measure a distance of 3 units in all directions to plot the edge of the circle. Connect these points to form the circle.

Key Concepts

Center of a Circle

Understanding the center of a circle is crucial in graphing and analyzing its properties. The standard form of a circle's equation is \[(x-h)^2 + (y-k)^2 = r^2\]where the center is described by the point \((h, k)\). The center is simply where the circle is centered on a coordinate plane. In the equation \(x^2 + y^2 = 9\), notice that there are no terms added to \(x\) or \(y\). This means that \(h = 0\) and \(k = 0\).

• The center is the point from which all the points on the circle are equidistant.
• In this exercise, the center is at the origin \((0, 0)\).

Having no shifts noted in the equation helps us quickly identify that the circle's center is at the origin.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the edge of the circle. It is pivotal in understanding the circle's size. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the term \(r^2\) provides information about the radius.
To find the actual value of the radius \(r\), one must take the square root of \(r^2\). For the given equation \(x^2 + y^2 = 9\):

• \(r^2 = 9\)
• Take the square root: \(r = \sqrt{9} = 3\)

So, the radius is 3 units. This distance is measured from the center to any point along the circle's edge, helping in plotting the circle on a graph accurately. Knowing the length of the radius is essential for correctly graphing and understanding the spread of the circle.

Graphing Circles

Graphing a circle involves plotting its center and using the radius to draw its boundary. The steps are straightforward once you've determined the center and the radius.

• Plot the center of the circle on a graph. For this circle, the center is at \((0, 0)\), the origin of the coordinate system.
• From the center, use the radius to measure outwards in all directions. In this case, extend 3 units away from the center in every direction: right, left, up, and down.
• Draw the circle by connecting these edge points smoothly to form the circumference.

Graphing visualizes the circle and helps you understand its position and size in the coordinate plane. Using the center and radius makes it easy to picture where the circle lies and how big it is, enhancing comprehension of the geometric figures represented by equations.