While looking at the graph of a parabola on Geogebra, I noticed that you could "flatten it out" by changing the

*x*-scale on the axes

*or*by zooming in (changing

*x*- and

*y*-scales proportionally) on the vertex. This is intuitive: if you zoom in on any smooth curve, it straightens out - in calculus you call that local linearity.

But this means that any two parabolas can be made to look identical just by zooming (as opposed to scaling only one axis), which is how two similar objects can be made to look identical.

We say two object are similar if their corresponding lengths form proportions (pairs of equal fractions); can we prove that two parabolas are similar in this way?

Say you have two parabolas,

*y = ax*² and

*y = bx*². To show that two shapes are similar, we find a

*scale factor*(

*k*) so that any length on the first shape equals

*k*times the corresponding length on the second shape. For parabolas, let's use the pair of lengths

*x*and

*y*for convenience.

On the first parabola, this pair of lengths is

*x1*and

*ax1*²

*.*On the second parabola, the pair of lengths is

*x2*and

*bx2*²

*.*

We must find a

*k*so that

*x*2 =

*kx1*and

*bx2*² =

*kax1*².

Substituting for

*x2*, we have

*b(kx1)*² =

*kax1*².

So

*bk*²

*x1*² =

*kax1*².

Dividing, we have

*bk*=

*a*, or

*k*=

*a*/

*b*.

So the scale factor between any two parabolas

*y = ax*² and

*y = bx*² is

*a*/

*b*, conveniently.

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