Lesson 3:


By Carly Coffman




Throughout this lesson, we will be exploring circles and the equations of circles.  

First, open a Microsoft Word document, title it as “Circle”, and type your name and period under the title.  Remember, any questions asked in this lesson should be answered in complete sentences in your Word document.  You do not have to type the questions.  Also, feel free to add any facts or findings that you come across during the lesson.


Now we’re ready to begin.  We will begin with a circle centered at the origin.  Look at what happens when the constant on the right side of the equation is changed.  Click on the play button at the bottom of the screen when you open the following link.


Exploration 1


1)        Record the values of the constant on the right side of the equation, n, when the

radius of the circle is 4, 3, 2, and 1.  What is the relationship?


Let’s explore what happens if we change the constant to a negative number.  Open exploration 1 and click on the n button.  Change the window to have n go from -16 to 0 in 40 steps.

2)        What happens when the constant on the right side of the equation (n) is negative? Why does this happen?  (Hint:  look at the rest of the equation and think of possible answers)

Now, let’s explore what changing h and k does to the circle.  We will leave the radius at 3 throughout this part of our investigation.  Let’s begin by making both variables 0 and changing only h.  Look at what happens.


3)        What happens to the circle when h increases?  What part of the circle does h affect?


Look at the following exploration to explore other values of h.


Exploration 2


4)        Create a table and list the values of h from the picture above #3 and some negative

values of h from the exploration.  In the other column list the coordinates of the centers of each circle.  What do you notice?



Now, let’s look at k.  We will leave h at 0 and the radius as 3.  Click on the play button on the bottom and feel free to change the values of k around by clicking on the k button and adjusting the lowest and highest values.


Exploration 3


5)        Create another table and list values of k in one column (include positive and negative

values) and list the coordinates of the center of each circle in the second column.  What do you notice?



So, now you should be able to determine the center and radius of a circle just by looking at the equation.  You can check each answer by clicking on an exploration or opening Graphing Calculator (also known as NuCalc) and typing in the equation.


      Determine the center and radius for each of the following equations.  Remember to answer in

complete sentences and include each equation (you can copy and paste the equations as











If you are given an equation that is not in the correct form, you can complete the square to find the center and radius.  Look at this link to learn how.

Completing the Square


Write each equation in the form  by using the method above.  Then determine the center and radius for each circle.






*Print this document and place in your notebook or portfolio with “Conics” and “Ellipse” papers.


Congratulations, you are finished with the circle lesson!


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