Quadratic Functions
Quadratic Functions covers graphs of y=ax^2+bx+c, vertex form, maxima/minima, quadratic equations and inequalities, and the discriminant's link to the number of real roots.
Core rules
- Complete the square: y=a(x-h)^2+k gives vertex (h,k). If a>0 the minimum is k; if a<0 the maximum is k.
- For ax^2+bx+c=0, D=b^2-4ac: D>0 two distinct real roots, D=0 one repeated real root, D<0 no real roots.
- For inequalities, find roots and test sign intervals; the sign of a determines whether the parabola opens upward or downward.
- The axis of symmetry is x=-b/(2a), and the vertex value is obtained by substituting this x into the function.
Worked example
Solve x^2-3x+2>0. Factor (x-1)(x-2)>0. The roots are 1 and 2. The upward parabola is positive outside the roots, so x<1 or x>2. If the inequality were <0, the answer would be 1<x<2.
Traps
Do not assume D<0 means no solutions at all; it means no real solutions. Reverse the interval logic when a<0. In word problems, restrict x by context, such as nonnegative length. Check whether endpoints are included: > and < exclude roots, ≥ and ≤ include them. Also do not confuse the vertex x-coordinate -b/(2a) with the maximum/minimum value itself.