Exploring the Equation of an EllipseHow 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 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?
Comment by Douglas Twitchell on Dec 30, 2012Thanks for sharing this lesson plan!
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