The Shape of Mathematics ... Act 1 - Parabolas ... Scene 3 - 'a' stands 'b' moves

Above we see a family of paraobolas, functions of the form

y = ax2 + bx + c

where 'a' is equal to 1, and 'c' equal to 0.

'b' is varied between -4, (the magenta curve on the right)

through 0, (the heavier green curve at center)

to 4, (the violet curve on the left)

Since the shape of a parabola is set by 'a', which is steady, that shape does not change.

Modifying 'b' has the effect of sliding the parabola on the origin.

When 'b' is 0, the parabola will cross the origin where its slope is 0 -- at its center, where it is momentarily flat.

When 'b' is 1, it will cross the origin where its slope is 1; which is to say, 45 degrees. .At that moment, y rises as fast as x travels.

when 'b' is 2, it will cross where the slope is 2, (63.434 degrees). .At that moment y rises twice as fast as x travels.

And so on. .At some point each parabola is going to cross the origin. .It will cross where the slope equals 'b'.

Since the parabola is, in effect, sliding up over and down again on a pin -- the path it takes is going to be parabolic.

This is shown by the upside-down grayish curve, which runs through each parabola's relative minimum, (low point).

The function of that curve is y = -x2 .