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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 $y = x^2$"?

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up vote 0 down vote accepted

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).

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

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Geometrically excellent answer. Observe that $y=ax^2$ can be rewritten as $ay=(ax)^2$. So, for positive $a$, we are applying the same scaling to both variables. – André Nicolas Aug 11 '12 at 2:47

Multiplying by $a$ stretches the plot vertically.

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