How to explain why a parabola opens up or down

Question

I am being ask to explain in two 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

Answer 1

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.

Answer 2

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.

Answer 3

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 $a$.