# How to explain why a parabola opens up or down

I am being ask to explain in 2 ways why is it that y=ax^2+bx+c parabola opens up if a is positive and why is it that y=ax^2+bx+c opens down when a is negative. One of the explanations has to be understood by beginning algebra student. I am unsure how I would explain it

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You could consider what happens as $|x|$ becomes very large on either side of the axis, or also alternately complete the square and see the sign. – Macavity Oct 31 '13 at 5:49
Plot $y=x^2$. Explain how $y=ax^2+bx+c$ is related, if $a\gt 0$ (scale, shift). See what reversing signs does. – André Nicolas Oct 31 '13 at 5:50

One more method:

You know the parabola $y = ax^2 + bx+c$ can be written as: $$y - y_0 = a(x-x_0)^2$$ Since this is just the parabola "shifted" to the axis. Now since $(x-x_0)^2$ is always positive, what determines what the parabola looks like is only $a$.

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Note that $x^2$ is positive for any x. That is $$x^2\ge0 \text{ for all x.}$$

This means that unless you put a minus sign before $x^2$, for the parabola $y=x^2$, $y$ is always positive and the parabola opens toward the positive values of $y$.

Now, why do we only care about the sign of $a$ in $y=ax^2+bx+c$? What would happen if b or c are negative?

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If $x$ is big and positive, and $a$ is positive, then $ax^2$ will be very big and positive, overwhelming any effect from $bx+c$.

If $x$ is big and negative, and $a$ is positive, then $ax^2$ will again be very big and positive.

So if $a$ is positive, the parabola opens upwards.

If $a$ is negative then if $x$ is big (positive or negative) the opposite occurs, and $ax^2$ will be very big and negative with the parabola opening downwards.

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I think this is the proper beginning algebra approach. It is a useful precursor to many ideas of limits at $\pm \infty$ – Ross Millikan Jan 7 '14 at 2:55

Hint: Show that the parabola has a unique minima or maxima depending on whether $a$ is positive or negative respectively using the second derivative idea. This is the calculus way. In the algebraic way, I guess plotting is one option.

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for telling a beginning algebra student. I first started by explaining that if a>o, y=ax^2, y will always be positive and it will open up because the first and second quadrant are where the y is always positive. But I dont know how to transition that into y=ax^2+bx+c – Maximiliano Oct 31 '13 at 6:50