Working with Circles [ Math B ]
Objectives
● To derive the equation of a circle from its graph.
● To use techno logy to graph a circle from its equation.
Materials
● TI-83 or equivalent
● Compass
Overview
Using prior knowledge of circles and algebra , students will investigate the
relationship between the graph of a circle and its equation. Students will also
tackle the dilemma of graphing a circle on their TI-83s. Student should already
have basic understanding of graphing functions on the TI-83.
“Do Now” Questions
Students will take a few minutes prior to the start of the lesson to read/think
about/answer the following:
● What is a function?
● How can you tell if something is a function just by looking at its graph?
● Sketch the graph of a circle. Is it a function?
Procedure
1. Ask class, “What is a circle?” List students’ responses on the board. “The
locus of points equidistant to a fixed point” should be included in
discussion.
2. A brief history of conic sections : Apollonius, a Greek mathematician,
studied the conic sections many years ago. Like him, we too find the topic
of circles quite fascinating.
3. Why are circles important in our lives? What are some applications of
circles in the world today?
4. There are four conic sections: circle, ellipse , parabola , and hyperbola. But
in our presentation, we will discuss only the circle.
5. Ask the fol lowing two questions : “What is the general equation for a
circle?” and “What does each letter represent?” Students at this level are
expected to recall this information from previous lessons.
Note: (x-a) ^2+(y-b) ^2=r^2, where (a, b) is the center of the circle and
r is
the radius.
6. Refer students to handout.
7. On a set of axis, instruct students to select a point.
8. Using a compass, students should then draw the locus of points
equidistant to their fixed point.
9. Have students answer the first three questions on their
own. Teacher(s)
will circulate to confirm that students are following the procedures.
10. Ask class, “How do we graph the line y =2x+1 on our TI-83?” “What form
does the TI-83 require in order to plot equations ?”
11. Have students look at the equation for their circles and realize that it
cannot be inputted as is. There is a problem. However, there is also a
solution. “How can we make our circle equations into the form that the TI-
83 will accept?”
12. Briefly discuss “do now” segment and have student understand that the
TI-83 graphs functions and that a circle is not a function because it does
not pass the “vertical-line test”.
13. As a class, come to the conclusion that the equation must be solved for “y”.
Discuss the + and – solution for y . {1, -1} is the proper command for
entering ± in the TI-83.
Working with Circles
Objectives
• To derive the equation of a circle from its graph.
• To use technology to graph a circle from its equation.
Materials
• TI-83 or equivalent
• Compass
Overview
Using your knowledge of circles and algebra, we are going investigate the
relationship
between the graph of a circle and its equation. We are then going to see how the
general
equation for a circle can be modified to allow it to be graphed on your
calculator.
Circles
• What is the definition of a circle?
• What is the general equation for a circle?
• What does each variable represent ?
• The TI-83 plots equations which are in what form?
Using the space provided below, write the general equation
for a circle in the form specified
above.
1. On the graph provided below, draw and label a set of
axes as instructed by your teacher.
2. Pick any point on the graph, and label that point.
3. Using the compass provided, draw a circle of any radius, centered at the
point you picked. (I
recommend that you choose a radius which will keep the circle on the graph.)
• What is the radius of the circle above?
• What are the coordinates of the center of the circle?
• What is the equation of the circle above in standard
form?
• What is the equation of the circle above in the form
required for the TI-83?
Graphing on the TI-83
• Press the mode key and ensure that your TI-83 is in Func mode.
• Press the window key on your TI-83. Enter the following window
settings
Xmin =
Xmax =
Xscl =
Ymin =
Ymax =
Yscl =
• Graph the circle on your TI-83. Does the calculator
graph match the graph above? Why
or why not?
