The graph of a quadratic function, y=ax²+bx+c (a≠0), is a smooth curve called parabola that is symmetric with respect to a vertical line called the axis of symmetry. The axis intersects the parabola at a point called the vertex, turning point of the curve, where the curve takes the maximum or minimum value.
To find whether the curve "opens up" or "opens down", change the standard form above into a vertex form such as y=a(x-d)²+e (a≠0), and examine the values of y on both sides of the axis of symmetry. The line x=d is the axis of symmetry, and coodinates of the vertex is x=d and y=e,(d,e).
Notice the first term a(x-d)², where (x-d)²≥0. So, if a>0, then a(x-d)²≥0 and y increases as x increases or decreases on both sides of the axis of symmetry. Thus the parabola opens up(holds water) and takes the minimum, y=e, at the vertex, x=d.
If a<0, then a(x-d)²<0 and y decreases as x increases or decreases on both sides of the axis of symmetry. Thus, the parabola opens down(spills water) and takes the maximum, y=e,at the vertex.
Therefore, the parabola: y=ax²+bx+c, opens up if a<0, or opens down if a<0.
