What does it mean when the a value and the b value in ax^2+bx+c have opposite signs? How does that affect the graphed parabola?

Answer 1

It means that a is positive and b is negative, or that a is negative and b is positive.

If a is positive, the parabola opens upward; if negative, it opens downward
The sign of b determines whether the graph is shifted left or right.

find some handy online graphing site, and play around with a few values to see what happens.