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 http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php"

II. Exploring the equation and its graph

(To do the following part of the exercise , please open up www.meta-calculator.com, 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 x²/a +y²/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:

x²/1 +y²/5 = 1 , a = 1
x²/2 +y²/5 = 1 , a = 2
x²/3 +y²/5 = 1 , a = 3
x²/4 +y²/5 = 1 , a = 4
x²/5 +y²/5 = 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 y² term? For instance, if the

term under y² starts at 1 and then increases, predict how the graph will change:


x²/5 +y²/1 = 1 , b = 1
x²/5 +y²/2 = 1 , b = 2
x²/5 +y²/3 = 1 , b = 3
x²/5 +y²/4 = 1 , b = 4
x²/5 +y²/5 = 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?