What determines whether a parabola will open up or down

If the coefficient of the leading term is positive, it opens up. If the coefficient of the leading term is negative, it opens down. The leading term is the ax^2 term.

A parabola will usually have an equation like y = - 2x^2 + 3x + 2 (2x squared) What determines whether it is up or down is the sign of the coefficient of the x squared term in this case it is - 2. If the sign of the coefficient is negative then it the open part of the parabola will be facing down.