## Wednesday, May 2, 2007

### Conan O'Scribin

Hello everyone! Today Mr.K became one with the sword, he became samurai... Then with a dazzling display of speed, his beautifully forged, ancient.... meter stick, cut through the cone that was placed in front of Richard. With each slice glorious knowledge poured out! The horizontal slice yielded a circle. The slice on a slight angle gave us an ellipse. The slice with the more intense angle gave us a parabola. Then the vertical slice gave us a hyperbola.

Then we move onto this mess....

This all started with a piece of ordinary white paper...
The we drew a dot in the middle (horizontally) and a wee bit up from the bottom of the page...
We then proceeded to draw roughly 9 more dots on the bottom of the page spaced evenly, or about evenly...
They we folded the paper each time overlapping a different dot over our center dot...
Then we darkened the innermost line in respect to the middle dot that we created and guess what! It was a parabola....
We knew that doing a little bit of origami and coloring wasn't the point of this...
So we proceeded to name all the points...
We then plotted two more points, P and Q, and proceeded to draw a line perpendicular to the bottom edge of the paper going to each of the points....
Then we drew a line from the points to point F...
Measured and....
They were the same!....
We then drew a line straight down from F....
The two sections of that were also the same....
Starting to see a pattern?....
So we then learned a special name for these points on the parabola....
Locus points, "A set of points that follow a certain rule."....
These locus points follow the line of the parabola and this gives us the pattern that any point on the parabola can have two segments drawn from it, to the directrix and the focus (explained a lil' bit later) and those two segments will have the same length....
Here they are....
We found out that F stands for Focus, kinda related to the focal point in physics I'd assume...
Then we found out that the line the D's were on had a special name, the directrix....
So now that all those patterns had been pointed out we decided that we needed to find relationships between the coordinates of the points...
We found a bunch....
Vertex (h,k).....
P (x,y)....
D1 (x, k-p) because it has the same x-co as p and a y-co of the vertex's minus P...
Focus (h, k + p) because it has the same x-co as V and is located p units above V...
So now that we had all this information we embarked on setting a couple of lengths that had the same lengths equal to each other, I can't remember what the initial reasoning for doing that was but we ended up with the standard form for the equation of a parabola....
Then using the distance formula and simplifying we got our standard form, as shown below...

Well that's my scribe for the night....
Now to go study a bit for chem....
Then try to bring closer to completion Miles' and my comp. sci. project deep into the morning....
*sigh*....

Our next scribe will be.....

TIMMY!