What is the difference between a(x^2 - b) and (ax^2 - b)?

In quadratics, cubics, quartics, etc. what is the difference between a(x^2 - b) and (ax^2 - b)? Does one affect translation or turning point/point of inflection? I kind seem to find it anywhere in my textbook and I'm really confused.
Update: Okay so I understand that you get a different result when you expand the brackets. I'm really asking when is the a used to find the coordinates versus just the dilation?
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