Learning Resources
Outcomes
In this lesson you will
- examine the equation of a circle using coordinate geometry
- develop the equation of the circle from a right triangle in the coordinate plane
By the end of this section students will be able to:
- analyze and translate between symbolic, graphic, and written representations of circles and ellipses
- translate between different forms of equations of circles and ellipses
- write proofs using various axiomatic systems and assess the validity of deductive arguments
- write the equations of circles and ellipses in transformational form and as mapping rules to visualize and sketch graphs
- demonstrate an understanding of the transformational relationship between a circle and an ellipse
- apply properties of circles
- solve problems involving the equations and characteristics of circles and ellipses
Introduction
Mathematical Modeling, Book 3 p.252 - 267
You have been working with circles and circle properties in various ways throughout this unit. Now it is time to describe the circle and its transformations algebraically.
In this section, you will explore the equations of circles, apply transformations to the basic circular equation and graph the results.
As with other equations for graphs, the circle can be represented in standard form, general form, and transformational form. You will learn how to convert from any one form to the other.
Finally by controlling certain characteristics of the circular equation you will see how a circle can be transformed into an ellipse and its resulting equations in all forms.
This section should take you between 5.5 and 6.5 hours to complete.
Prerequisites
To be successful in this lesson, it would be helpful to know the following:
- Pythagorean theorem
- distance formula