• If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

  • You already know Dokkio is an AI-powered assistant to organize & manage your digital files & messages. Very soon, Dokkio will support Outlook as well as One Drive. Check it out today!


E3 (adv)

This version was saved 16 years, 10 months ago View current version     Page history
Saved by PBworks
on June 10, 2007 at 11:09:20 am

E3 (adv) - write the equations of circles and ellipses in transformational form and as mapping rules to visualize and sketch graphs.







Circle: a single, closed curved line


Ellipse: a single, closed curved line that has one axis longer than the other (x-axis or y-axis) *Example below has a longer y-axis*



Writing an equation for a circle: To write an equation for a circle, it first needs to be put on a graph. Let's make it easy and have the centre of the circle at the origin of the graph (0,0).


The equation of a circle looks like this: [1/r (x-h)]^2 + [1/r (y-k)]^2 = 1  This is called Transformational Form. The r stands for radius, h is the horizontal translation and k is the vertical translation. h and k is also the centre of the circle.


The equation for the circle above circle is: [1/5 (x-0)]^2 + [1/5 (y-0)]^2 = 1


If we only had the equation and not the diagram of the circle, we would need a mapping rule to make the diagram. A mapping rule looks like this:

(x,y) -> (x+h, ay+k) Notice how the h and k are now positive? This becomes important later. The a stands for the stretch, but it not needed here.


The mapping rule for the above circle is: (x,y) -> (x+0, y+0)



Let's make it harder and change the circle on the graph:


The equation for this circle is: [1/5 (x-0)]^2 + [1/5 (y-3)]^2 =1  All we did with this circle was move it vertically by +3. This +3 is equal to k. This is what happened: [1/5 (x-(0)]^2 + [1/5 (y-(+3)]^2 =1 We did not move the circle horizontally from our last circle so it stays at 0.


The mapping rule now is: (x,y) -> (x+0, y+3) In the mapping rule, the h and k are positive, unlike the equation where they are negative. This is to show that the circle moved +3 on the y-axis. It makes it easier to draw.



Writing an equation for an ellipse: The equation for an ellipse is the same as an equation for a circle except we now have to use a stretch. We'll start off with an ellipse with a centre at the origin (0,0).


The equation of an ellipse is: [1/a (x-h)]^2 + [1/b (y-k)]^2 =1 This is also called transformational form. In this equation, the only difference from the circle equation is the a and b. Both a and b stand for stretch. a is usually the longer stretch but it does not matter.


The equation for the above ellipse is: [1/3 (x-0)]^2 + [1/6 (y-0)]^2 =1


The mapping rule for the above ellipse is: (x,y) -> (x+0, 2y+0) Because this ellipse has a longer y-axis, the stretch goes with the y in the mapping rule. In this ellipse, the y-axis is twice the length of the x-axis, giving it a stretch of 2.




Comments (0)

You don't have permission to comment on this page.