Quadratic and Cubic Models

The quadratic model

A quadratic function has the form $f(x)=ax^2+bx+c$ with $a \ne 0$. Its graph is a parabola: opening upward when $a>0$ (a lowest point, a minimum) and downward when $a<0$ (a highest point, a maximum). Quadratics model any situation with a single turning point — the flight of a projectile, the trajectory of a jet of water, or a profit that rises then falls.

Key features: - Vertex (turning point) at $x=-\dfrac{b}{2a}$; substitute back to get the maximum or minimum output. This is the single most useful fact for optimization. - Axis of symmetry, the vertical line $x=-\dfrac{b}{2a}$, about which the parabola is mirror-symmetric. - $y$-intercept at $c$; $x$-intercepts (roots) where $f(x)=0$, found by factoring, the quadratic formula, or the GDC.

Projectile motion

Worked examp