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I recently did a video about paraboloids.
If you'd like to watch it the link is below
but you don't have to,
the point is, in that video I proved
something for one specific parabola
and then I didn't bother proving it for all parabolas.
and that should have annoyed a lot of you.
Just because you've managed to prove something
for one specific shape does not necessarily mean
it generalises to all shapes.
In some cases it does, so all circles are similar.
Once you have an identity for one circle
you can then apply it to any other circle.
but let's say triangles, triangles come in all shapes
and sizes, they are not all similar,
just because you prove something for one triangle
does not necessarily mean it applies to another triangle
and if you look at parabolas they also
come in all shapes and sizes. What a variety.
How on Earth can I show something for one parabola
and assume it generalises to all of them?
Parabolas do all look different but
they're actually similar.
The notion of there being more than one parabola
is a lie. It's perpetuated by powerful
mathematics teachers who have a vested interest.
It turns out there is only one true parabola. [CELEBRATORY MUSIC STING]
[BACKING VOCALS] Yeah...
Behold! There is only one true parabola!
To prove that there is only one true parabola
let's start with its cousin, the circle.
All circles are similar.
So if I draw a circle on the board
close enough, in what way is that similar to, let's say,
the circle of a pound coin?
Well if I get my pound coin nice and close
You can see it's possible to line it up so it exactly
matches the circle on the board
For two shapes to be similar you have to be able to line
them up perfectly, only by translating them around
and making them bigger and smaller
Oh and you're allowed to turn them over
and rotate them, but that's it.
When it comes to changing the size of a shape to show
that it's similar to another one you have to be a little bit
careful. We need to scale it evenly
in every direction. We can see this with
three different rectangles. That 3 by 1 rectangle
is similar to the 6 by 2 rectangle
because it scales the same in each direction.
Not to labour the point but
we're gonna need this in a moment.
if you scale up 3 by k to get 6
and you scale up 1 by k to get 2
the whole system works. For the same k
in both cases, we're doubling it, you scale from
that shape, to this shape. They're similar,
whereas this guy doesn't work.
For these to be similar we would need 3 times some
scale factor k to be 4 and 1 by k to be 3.
And that can't work. There is no k that scales both
of these up to give you the new shape.
You can't go from this rectangle to that rectangle
without messing with its aspect ratio.
Okay.
Enough rectangles, let's get on to parabolas.
Translating parabolas is reasonably straightforward;
if you wanna move them up and down,
you simply add some term on the end,
which I'm going to call "C."
If I increase C it goes up,
If I have C as a negative number, if I'm subtracting
then it will go down.
Sideways is slightly more complicated.
Let's say we want to take our original y=x²
parabola and move it to the side
so it ends up sitting over here.
And we'll say our new parabola is sat at some point d.
So it's moved a distance of d across,
if you've done this at school recently or possibly
decades ago, you may remember you can adjust
this equation to be y=(x-d)² and that has the result of
just bumping it across d on the horizontal axis
We can even expand this out if we want to make it look
a bit more familiar; y=x²-2dx+d².
There's our new parabola that's been moved to the side.
The important thing to bear in mind in both of these
cases, going up-and-down and going side-to-side
combinations of which will get us anywhere we want
on the surface, is that neither of them change
that lead term. You always have a x² out the front
the other terms after the x² are what control
the position of it, on the plane.
What this means is, for a completely general parabola
y=ax²+bx+c, the end bit
the bx+c, is just a big bit of machinery
for moving it around. The only thing that
actually changes the shape is the ax² out the front.
Actually you know what? Maybe, maybe I've made a mistake.
I'm a little... because... even if you accept
that the +bx+c moves parabolas around the plane and
if you accept you can flip them and rotate them as well
all possible parabolas can be realigned
to have their turning point at the bottom
on the origin. But now
to show that they're similar we have to show
we can scale them evenly in every direction
and map any parabola onto any other parabola.
and here, I have put two as different as I really can,
I've got y=ax², a nice skinny one up there
and I've got y=bx², a much flatter one there.
And surely, to map on of these onto the other one
we're gonna have to squash it more
in one direction than the other.
We're gonna have to change the aspect ratio
of one of these to get it onto the other one.
I mean look, if I take one point here
on the first one so that is the point (Y1,X1)
that is a completely different point,
I mean, look at that rectangle,
compared to, let's have one out here
on the other parabola, that's gonna be X2,
that's gonna be Y2,
there's no way I can prove there is a common scale
that would take that rectangle, that point
and map it on to that point.
These parabolas cannot be similar.
Sorry that I've mislead you.
But hey, let's ... let's give it a go anyway
and see what happens.
Right, so we've got the point (X1,Y1) which is on
the parabola y=ax², that's the skinny red one.
We're now gonna scale both of them by exactly
the same factor, k.
so X2 is k times X1,
Y2 is k times Y1.
It's going the same dilation in every direction,
is (X2, Y2) on the wildly different parabola y=bx²?
Okay, let's start by taking our scale factors
all the way over here,
rearrange them to be X1=X2 on k, Y1 = Y2 on k
We now know the relationship from the first parabola,
that Y1=a times X1², we can substitute those in
Y2 on k = a times X2 on k, all squared,
this side can be cleaned up very easily, just by
squaring everything, so that now equals
a on k² times X2².
Oh, we got multiple k's going on, let's get rid of one of those
so if we multiply both sides by k
we have Y2 = a on just the one k times X2².
I mean that's nice. That shows us that
if we scale it evenly in all directions, our original
y=ax² parabola does give us another parabola.
But, is it the same as any other given second
y=bx² parabola?
Hang on, hang on hang on!
When we started our two wildly different parabolas
I just gave them a and b, and a and b could be
any two real numbers.
But down here, what if I now set k, our scale factor
to be a divided by b?
Which is just another real number.
If we scale the first parabola in every direction by a on b,
if I put that in there, it equals, well, it cancels out.
That would give us b times x2²
That means that (X2,Y2) are on the second
y=bx² parabola.
We can map any parabola onto any other parabola
by scaling it evenly in every direction
by a scale factor of a on b.
All parabolas are similar.
There is only one true parabola.
NOOOOOOOOOOOOOOOOOOOOO! [GOTHIC BELL MUSIC]
Gloria in x-squaris.
[GARBLED SPEECH FRAGMENT]