Quadratic Functions and Graphs
What the exam tests
Quadratic Functions and Graphs tests the connection between algebra and geometry: vertex form, axes of symmetry, discriminants, ranges, and maxima or minima on restricted domains. Kyoto problems often place a quadratic inside a parameter or inequality question.
Core rules
- Complete the square: ax^2 + bx + c = a(x - h)^2 + k, where h = -b/(2a) and k = c - b^2/(4a). The vertex is (h, k).
- If a > 0, the parabola opens upward and k is the global minimum; if a < 0, it opens downward and k is the global maximum.
- The axis of symmetry is x = h. Roots satisfy a(x - h)^2 + k = 0; the discriminant is D = b^2 - 4ac.
- On a closed interval, compare the vertex value (if inside) with the endpoint values. On an open interval, the vertex may give an extremum but endpoints may not be attained.
Worked example
For f(x) = 2x^2 - 8x + 5, complete the square: 2(x^2 - 4x) + 5 = 2[(x - 2)^2 - 4] + 5 = 2(x - 2)^2 - 3. The vertex is (2, -3), and since a = 2 > 0 it is a minimum on the whole real line. On the interval [0, 1], the function is decreasing because the vertex lies to the right, so the maximum is f(0) = 5 and the minimum is f(1) = 2 - 8 + 5 = -1.
Traps
Do not assume the vertex lies in the given interval. Watch the sign of a when deciding maximum versus minimum. Check arithmetic when completing the square, especially the constant subtracted inside the bracket. A positive discriminant gives two real roots but not necessarily rational or integer roots.