Definition and General Form
Quadratic functions are polynomial functions of degree 2, expressed in the standard form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. This form directly answers the core question by highlighting that these functions model parabolic relationships, distinguishing them from linear (degree 1) or cubic (degree 3) functions.
Key Graphical Properties
The graph of a quadratic function is a parabola. If a > 0, it opens upwards (minimum point at vertex); if a < 0, it opens downwards (maximum point). The vertex is at x = -b/(2a), and the axis of symmetry is the vertical line through this x-value. The y-intercept is at (0, c), and roots (x-intercepts) are found using the quadratic formula x = [-b ± √(b² - 4ac)] / (2a), with the discriminant (b² - 4ac) indicating real roots (positive), one root (zero), or no real roots (negative).
Practical Example
Consider f(x) = 2x² - 4x + 1. Here, a = 2 > 0, so the parabola opens upwards. Vertex x = -(-4)/(2*2) = 1, and y = 2(1)² - 4(1) + 1 = -1, so vertex is (1, -1). Discriminant = (-4)² - 4*2*1 = 8 > 0, yielding two real roots. This example illustrates how properties predict the graph's shape and key points for applications like projectile motion.
Importance and Real-World Applications
Understanding quadratic properties is crucial in algebra for solving equations, optimizing problems, and modeling real scenarios like ball trajectories in physics or profit maximization in business. They address misconceptions, such as assuming all parabolas are identical—properties vary with a, b, c—enabling precise predictions and graphing without plotting every point.