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.

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

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

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.

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Yu:

ReplyDeleteAfter 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.

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

ReplyDeletenot quite! "similar" means translation and scaling (i.e. x and y scaled by the same factor)

ReplyDeleteX = ax + b/2

Y = ay + b^2/4 - ac

=> Y = X^2

thus all (vertical) parabolas are similar canonical parabola y=x^2

Sensible. YU

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

ReplyDeleteNow 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

vertical only?

ReplyDeleterotated parabolas are similar too. right?

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.

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ReplyDelete