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Exploring the Equation of an Ellipse

How does the equation of an ellipse relate to its graph?

I Introduce the ellipse concept

"An ellipse looks like an oval. Pretty much a circle that is wider or taller in 1 of its dimensions. You can see pictures

of ellipses here"

II. Exploring the equation and its graph

(To do the following part of the exercise , please open up, an online graphing calculator that can graph

ellipses. We will explore the way that changing the equation affects the graph of the ellipse)

The general equation of an ellipse looks like x^2/a +y^2/b = 1 .

Now lets explore the roles of each of these 2 variables. First, let's make b =5 and try exploring different values for 'a'

Graph the following equations, making note of the graph of each:

x^2/1 +y^2/5 = 1 , a = 1
x^2/2 +y^2/5 = 1 , a = 2
x^2/3 +y^2/5 = 1 , a = 3
x^2/4 +y^2/5 = 1 , a = 4
x^2/5 +y^2/5 = 1 , a = 5

What effect does increasing 'a'have on the equation ?

Now, take a guess, what do you think will be the effect of changing the variable underneath the y^2 term? For instance, if the

term under y^2 starts at 1 and then increases, predict how the graph will change:

x^2/5 +y^2/1 = 1 , b = 1
x^2/5 +y^2/2 = 1 , b = 2
x^2/5 +y^2/3 = 1 , b = 3
x^2/5 +y^2/4 = 1 , b = 4
x^2/5 +y^2/5 = 1 , b = 5

Ok, was your prediction correct?

Concluding question, in terms of 'a' and 'b', what is true when the ellipse becomes a circle?

Member Comments

Comment by Douglas Twitchell on Dec 30, 2012
Thanks for sharing this lesson plan!

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