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In the following figure I understand the $bx+c$ part. It is simply the equation of a line. But I don't understand where did $ax^2$ came from? What exactly is it? What does $a$ tell us about a parabola?

enter image description here

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  • $\begingroup$ Can you clarify the question? It's not clear which part of the figure you don't understand, or what you need answered. $\endgroup$
    – user142299
    Jun 4, 2014 at 21:52
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    $\begingroup$ I don't understand where did $ax^2$ come from? What exactly is it? What does $a$ tell us in parabola? $\endgroup$ Jun 4, 2014 at 22:03

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When you have $y=ax^2+bx+c$, the $a$ affects the shape of the parabola. An $a$ with large absolute value causes the parabola to be narrow and steep. An $a$ near zero causes the parabola to flatten out. Positive $a$ causes the parabola to open upward, while negative $a$ causes it to open downward.

In addition to what you noticed, the $bx$ causes the parabola to move around: positive $b$ cause the parabola to move down and left, while negative $b$ cause the parabola to move up and right. Finally, $c$ moves the parabola up (positive $c$) and down (negative $c$).

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  • $\begingroup$ Does $a$ has any relation with focus and directrix? Can that be shown on the figure? $\endgroup$ Jun 4, 2014 at 22:09
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    $\begingroup$ Yes. The focus and directrix are both $1/(4a)$ from the vertex of the parabola. $\endgroup$
    – Teepeemm
    Jun 4, 2014 at 22:13
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The blue line whose slope is $b$ and intercept is $c$ has $y=bx+c$ for its equation. The equation $y=ax^2+bx+c$ is describing the downward-pointing black parabola which lies below the blue line. Because it's downward pointing (and because it lies below the blue line), the coefficient $a$ must be negative, just as $b$ and $c$ are, according to the picture, both positive.

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