Completing the square, $$ ax^2+bx+c = a\left( x + \frac{b}{2a} \right)^2 + \frac{4ac-b^2}{4a} $$ The last term contains the discriminant, which only tells you when the parabola can intersect the $x$-axis: the first term is a square, so is always nonnegative.
The first term is a shift in the variable $x$, and we see that it looks like a rescaled shift of $x^2$ if $a>0$, and $-x^2$ if $a<0$. Hence the "upwardness" or "downwardness" of the parabola is determined by the sign of $a$.
Intuitively, $c$ can't tell you anything about how the parabola behaves as $x$ varies, since it only occurs in a term that does not depend on $x$.