Taller and Flatter

A number that multiplies the whole function changes heights, not positions. Take $y = 3f\left(x\right)$ with $f\left(x\right) = {x}^{2}$. 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: $y = \frac{1}{2}f\left(x\right)$ 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 $y = af\left(x\right)$, that point is a above the vertex instead of 1.