Taller and Flatter

A number that multiplies the whole function changes heights, not positions. Take y = 3f(x) with f(x) = x^2. Every output of f is tripled. The vertex (0, 0) stays put, because 3 times 0 is 0. But one step to the right, where f gives 1, the new graph gives 3: the point (1, 1) becomes (1, 3).

Tripling every height makes the curve climb three times as fast.

So the curve is narrower. A factor between 0 and 1 does the opposite: y = 1/2 f(x) halves every height, and one step from the vertex it has risen only 1/2, so it looks flatter.

So two points fix the curve: the vertex says where it sits, and the point one step to its right says how fast it climbs. On y = a f(x), that point is a above the vertex instead of 1.