If the discriminant is positive,that means that there are roots that are real and therefore, when y=0, the roots are there and for the quadratic to meet it has to go down (or up if in ax²+bx+c a is negative). If the discriminant is 0, that means that since the x value of the vertex is -b/2a and if you have
x=
and b^2-4ac=0, we're left with 0 and we are therefore left with -b/2a as the vertex and the vertex being the x intercept. If the discriminant is negative, that means that there is no y intercept and the vertex therefore has to be over y=0
x=