Did you know that all parabolas are the same shape?

I mean this in the sense that all circles and all squares are the same shape. You can always go from one circle or square to another by rescaling. While it is obvious that all circles and squares are the same shape, it may seem counterintuitive for parabolas. I know it seemed that way to me when I first found out.

Part of the problem is that while circles and squares are of finite size, parabolas are infinite. How can you say that two things that are infinitely large have the same shape.

Here is a way of looking at it. Consider the curve y=x^2 where the scale is in mm. So we have points (20mm, 400mm) and (30mm, 900mm). If we rescale to cm, the same points would be (2cm, 40cm) and (3cm, 90cm). The points are in the exact same place, but in cm they satisfy y=10x^2. In general, rescaling by a factor of 1/k converts the equation y=x^2 to y=kx^2.

Imagine a wide parabola drawn on the ground. If you go up in an airplane, the changing view is equivalent to rescaling. The same point that was at 10 feet may now appear to be at only 1 foot. The scalings will have factors of 1/k. If the original equation of the parabola was y=x^2, what you see will be the narrower parabola y=kx^2.

If you were originally a doubter, have I managed to convince you?