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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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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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 |
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