Problem 20 - 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. $$ (x-3)^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

The center of the circle is (3, 0) and the radius is 3.

Step 1: Identify the Standard Form of a Circle

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

Step 2: Compare Given Equation to Standard Form

Compare the given equation \((x-3)^2+y^2=9\) to the standard form. Here, \((h, k) = (3, 0)\) and \(r^2 = 9\).

Step 3: Determine the Center of the Circle

From the comparison, recognize \(h = 3\) and \(k = 0\). Thus, the center of the circle is at the point \((3, 0)\).

Step 4: Determine the Radius of the Circle

From \(r^2 = 9\), calculate the radius \(r\) by taking the square root. \(r = \sqrt{9} = 3\).

Step 5: Sketch the Graph of the Circle

Using the center \((3, 0)\) and radius \(3\), draw a circle centered at \((3, 0)\) with a radius of 3 units.

Key Concepts

Understanding the Standard Form of a Circle

The standard form of a circle's equation is crucial for identifying the circle's key characteristics. This form is expressed as \((x - h)^2 + (y - k)^2 = r^2\). Here, \( (h, k) \) represents the center of the circle, while \( r \) denotes the radius. Using this form makes it easy to determine both the circle's center and its size. This format serves as a straightforward tool for graphing or analyzing circles within coordinate planes.

• A circle's equation in standard form directly shows both the center and the radius squared.
• When \( h \) and \( k \) are zero, the circle is centered at the origin.

Understanding this form is a foundational skill in geometry and necessary for graphing or transforming circles.

Pinpointing the Center of a Circle

The center of a circle is a pivotal concept in understanding and working with circular graphs. In the standard form \((x - h)^2 + (y - k)^2 = r^2\), the center of the circle is at \((h, k)\). This means:

• \( h \) is the horizontal distance from the origin along the x-axis.
• \( k \) is the vertical distance from the origin along the y-axis.

In our example, the equation \((x-3)^2+y^2=9\) reveals that \( h = 3 \), and \( k = 0 \). Thus, the center is located at \((3, 0)\). By identifying this point, you establish the circle's exact position on the graph.

Calculating the Radius of a Circle

The radius is half of the diameter and measures from the center to any point on the circle. In the equation \((x - h)^2 + (y - k)^2 = r^2\), \( r^2 \) is the radius squared. To find \( r \), simply take the square root of \( r^2 \).

• If \( r^2 = 9 \), then \( r = \sqrt{9} = 3 \).
• The radius determines how far the circle extends from its center.

For our given example, with a center at \((3, 0)\) and \( r = 3 \), the circle reaches 3 units away from the center in all directions.

Forming the Equation of a Circle from Key Elements

Creating the equation of a circle involves using the center and radius. With a known center \((h, k)\) and radius \( r \), you substitute these into the standard form equation: \((x - h)^2 + (y - k)^2 = r^2\).

• This structure allows the circle to be precisely plotted on a graph.
• The given circle equation \((x-3)^2+y^2=9\) illustrates this process.

By understanding both the geometric and algebraic aspects, one can seamlessly translate between a circle's visual representation and its algebraic formula.