Question

h is the graph of a quadratic equation that has a positive discriminant?

On a coordinate plane, a parabola opens up. It goes through (negative 1, 4), has a vertex at (1, 0), and goes through (3, 4).

Answer 1

No. The parabola has equation y = a(x − 1)^2 (vertex form). Using (3,4):

4 = a(3 − 1)^2 = 4a ⇒ a = 1,

so y = (x − 1)^2 = x^2 − 2x + 1. The discriminant is b^2 − 4ac = (−2)^2 − 4·1·1 = 4 − 4 = 0. Thus the discriminant is 0 (not positive): the parabola is tangent to the x-axis at its vertex.