Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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

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.

share|cite|improve this answer
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

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 $a$.

share|cite|improve this answer

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.

share|cite|improve this answer
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

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?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.