Now that we know what the equation of a circle means, we can use it to identify the center and the radius and sketch the graph of the circle in the plane. But just for a refresher, let’s restate the definition of the equation of a circle.

**DEFINITION:** The equation of a circle with center \(\left( {h,k} \right)\) and radius \(r\) is given by

\({\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\)

Let’s implement this information now.

**EXAMPLE:** Identify the center and radius of the circle. Then sketch the graph of the circle.

\({\left( {x - \frac{1}{2}} \right)^2} + {\left( {y + \frac{1}{2}} \right)^2} = 4\)

**SOLUTION:** We manipulate this equation slightly to get it exactly in the form above:

\({\left( {x - \frac{1}{2}} \right)^2} + {\left( {y + \frac{1}{2}} \right)^2} = 4\)

\({\left( {x - \frac{1}{2}} \right)^2} + {\left( {y - \left( { - \frac{1}{2}} \right)} \right)^2} = {2^2}\)

Now this equation is in the general form. From here we can conclude that

\({\text{Radius}} = 2\)

\({\text{Centeris}}\left( {\Large \frac{1}{2}, - \Large \frac{1}{2}} \right)\)

Here is the graph:

**EXAMPLE:** Identify the center and radius of the circle. Then sketch the graph of the circle.

\({\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = 7\)

**SOLUTION:** Again we manipulate the equation into the general form. We have

\({\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = 7\)

\({\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = {\left( {\sqrt 7 } \right)^2}\)

Then the radius of the circle is \(\sqrt 7 \approx 2.6\), and the center is \(\left( {2,\frac{7}{2}} \right)\). The graph is below.

Below you can **download** some **free** math worksheets and practice.

Graph each equation.

**Example of one question:**

**Watch bellow how to solve this example:**

Identify the center and radius of each. Then sketch the graph.

**Example of one question:**

**Watch bellow how to solve this example:**

Identify the center and radius of each. Then sketch the graph.

**Example of one question:**

**Watch bellow how to solve this example:**