## Wednesday, March 31, 2010

### Fox 270

1. Yu:
After translations all parabolas, y=ax^2+bx+c, can be changed to the same form Y=kX^2, k≠0.
.: all parabolas are similar. A different parabola has a different constant of proportionality, k.

2. you go to show that for every parabola, k=f(a,b,c) exists with no exception. what's the function for k?

3. not quite! "similar" means translation and scaling (i.e. x and y scaled by the same factor)
X = ax + b/2
Y = ay + b^2/4 - ac
=> Y = X^2
thus all (vertical) parabolas are similar canonical parabola y=x^2

4. Sensible. YU

5. Any parabola can be translated so that its vertex is at the origin, making its equation y = kx^2.

Now replace y by y / k to stretch vertically by a factor of k, and replace x by x / k to stretch horizontally by a factor of k.

This makes the equation y / k = k (x / k)^2, or y = x^2.

The parabola was stretched by the same factor in both directions so it didn't change shape when we transformed it into y = x^2, so every parabola is the same shape as y = x^2.

Parabolas which look wider (or narrower) than y = x^2 are simply what you see when you zoom in (or out).
-Bob Ryden

6. vertical only?
rotated parabolas are similar too. right?

7. true, but my demonstration was only valid in the context of a sub-family of parabolas, namely the vertical parabolas (y=ax^2+bx+c). Otherwise the definition of "similar" would have been: translation+scaling+rotation. Note that symmetry is not considered here, because parabolas have internal axis-symmetry.