>>16759674
From pic related, I started with the slope formula.
Any line must have a constant slope or gradient
from point to point, so I take two random points
to find m.

Then with a bit of algebra to rearrange
it all, and renaming parts of the formula to account
for generality on the second point and collapse
the constants down to a single letter, I have
proved that any line must have the form y=mx+b
(for whatever level of "proof" I did seems enough).

To similarly prove in this manner for other curves
somehow we need to make use of a key part of
that function that makes it unique (slopes for lines,
periodicity for sine curves, etc.), generalize the
second point, and make any algebraic motions
or substitutions needed. The line was easy for
me to do, but the other curves might take work.