Quadratic Functions

• Standard: $f(x) = ax^2 + bx + c$. Y-intercept: $c$.
• Vertex: $f(x) = a(x-h)^2 + k$. Vertex: $(h,k)$.
• Factored: $f(x) = a(x-p)(x-q)$. Roots: $p$ and $q$.

• Vertex: $(h,k)$ or $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$.
• Axis of symmetry: $x = -\frac{b}{2a}$.
• Discriminant: $\Delta = b^2 - 4ac$.
  • $\Delta > 0$: two real roots. $\Delta = 0$: one repeated root. $\Delta < 0$: no real roots.

1. Factoring. 2. Completing the square. 3. Quadratic formula: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$.

$f(x) = 2x^2 - 8x + 3$.

Vertex: $x = \frac{8}{4} = 2$. $f(2) = 8 - 16 + 3 = -5$. Vertex: $(2, -5)$.

$\Delta = 64 - 24 = 40 > 0$ → two real roots.