Revise: Quadratic Functions

a = 1 → opens upward, minimum vertex.

y=x²−2x−3 → vertex (1,−4).

Sign of a: direction. |a|: width (bigger |a| = narrower).

y=3x² is narrower than y=0.5x².

D=b²−4ac. D>0: 2 roots. D=0: 1 root. D<0: none.

2x²+3x+5=0 → D=−31 → no real roots.

x = (−b ± √(b²−4ac)) / 2a.

2x²+3x−2=0 → x=0.5 or x=−2.

Find roots, then test a point in each interval.

x²−x−6>0 → x<−2 or x>3.

Upward parabola (a=1): minimum vertex at x=−b/2a.

y=x²−4x+1 → vertex (2,−3).

Downward parabola (a=−1): maximum vertex, same formula.

y=−x²+6x−5 → vertex (3,4), a maximum.

Sign of a sets direction; |a| sets narrowness.

y=−4x² opens down and is narrower than y=x².

Vertex x=−b/2a; substitute back for y; check sign of a for direction.

y=3x²+12x+7 → vertex (−2,−5).

D=b²−4ac tells the number of real roots before solving.

x²−6x+9=0 → D=0 → one repeated root.

x = (−b ± √(b²−4ac)) / 2a solves any quadratic.

x²−x−6=0 → x=3 or x=−2.

Roots split the number line; test one point per interval.

x²−9<0 → −3<x<3.