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OK, where are we? We started with the geometric definition of a parabola. Then we jumped straight to the machinery, and we never attempted to connect the two. But that is what we’re going to do now.
Remember what we did with circles? We started with our geometric definition. We picked an arbitrary point on the circle, called it ( $x$ , $y$ ), and wrote an equation that said “you, Mr. ( $x$ , $y$ ), are exactly 5 units away from the origin.” That equation became the equation for the circle.
Now we’re going to do the same thing with a parabola. We’re going to write an equation that says “you, Mr. ( $x$ , $y$ ), are the same distance from the focus that you are from the directrix.” In doing so, we will write the equation for a parabola, based on the geometric definition. And we will discover, along the way, that the distance from the focus to the vertex really is $\frac{1}{\mathrm{4a}}$ .
That’s really all the setup this assignment needs. But they will need a lot of help doing it. Let them go at it, in groups, and walk around and give hints when necessary. The answers we are looking for are:
Warn them that this will be on the test!
Which brings me to…
Our first test on conics. I go back and forth as to whether I should give them a bunch of free information at the top of the test—but it’s probably a good idea to give them a chart, sort of like the one on top of my sample.
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