Shape and key features
A quadratic function is f(x) = ax² + bx + c (a ≠ 0). Its graph is a parabola. If a > 0 the parabola opens upward and has a minimum point; if a < 0 it opens downward and has a maximum point. The turning point is the vertex, and the vertical line through it is the axis of symmetry, x = −b/2a.
Completing the square (vertex form)
Writing the function as a(x − h)² + k reveals the vertex (h, k) at once. The value k is the minimum (a > 0) or maximum (a < 0) value, and x = h is the axis of symmetry. The y-intercept is always c, since f(0) = c. The discriminant b² − 4ac also describes the graph: if it is positive the parabola cuts the x-axis at two points, if zero it touches the axis at the vertex, and if negative it never reaches the axis. Combining the sign of a with the discriminant lets you decide quickly whether a quadratic is always positive, always negative, or changes sign.
Key formula
Vertex form: f(x) = a(x − h)² + k, vertex (h, k). Axis of symmetry: x = −b/2a. Maximum/minimum value = k.
Worked example
Express f(x) = x² − 4x + 7 in vertex form and state its minimum value.
x² − 4x + 7 = (x² − 4x + 4) + 3 = (x − 2)² + 3.
So the vertex is (2, 3), the axis of symmetry is x = 2, and because a = 1 > 0 the minimum value is 3.
Remember
- a > 0 gives a minimum; a < 0 gives a maximum.
- The axis of symmetry passes through the vertex: x = −b/2a.
- In a(x − h)² + k the vertex is (h, k) and the max/min value is k.