Describing the effect on $ax^2$ by manipulating $a$

Question

Please take, for example, $y = x^2$ and $y = 2x^2$.

Graphs: Wolfram Alpha

What is the most appropriate way to describe the effect of $a$? "$a$ causes the parabola to open at $1/a$ the rate of $y = x^2$"?

Answer 1

First of all, the sign of $a$ impacts if it opens up or down.

And the way I think about how it affects the shape is that $ax^2$ is $x^2$ stretched vertically by a factor of $|a|$. So on $y=x^2$ there is the point (1,1), on $y=2x^2$ the $y$ value is scaled by a factor of 2 so the graph includes the point (1,2).

Answer 2

Multiplying by $a$ stretches the plot vertically.

http://www.wolframalpha.com/input/?i=y%3Dpower%28x%2C+2%29%2C+y%3D2+power%28x%2C+2%29

Answer 3

All parabolas are similar, so (assuming $a > 0$) one can obtain $y=ax^2$ from $y=x^2$ through a scaling of $1/a$.