Quadratic Functions

A quadratic function is a function of the form:

\( f(x) = ax^2 + bx + c \), where \( a, b, c \) are real numbers and \( a \neq 0 \).

Key Features of Quadratic Functions:

Graph Shape: A parabola (U-shaped curve).

Direction:

• • If \( a > 0 \), parabola opens upward.
  • If \( a < 0 \), parabola opens downward.

Axis of Symmetry: Vertical line \( x = -\frac{b}{2a} \).

Vertex (Turning Point): \(\text{Vertex: } \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) \)

Y-intercept: \( (0, c) \).

X-intercepts (Roots): Solutions of \( ax^2 + bx + c = 0 \).

Forms of Quadratic Functions:

• Standard Form: \( y = ax^2 + bx + c \)
• Factorized Form: \( y = a(x – p)(x – q) \), where p and q are roots.
• Vertex Form: \( y = a(x – h)^2 + k \), where (h, k) is the vertex.

Axis of Symmetry and Vertex Formula:

\( x = -\frac{b}{2a}, \quad y = f\left(-\frac{b}{2a}\right) \)

Nature of Roots (Discriminant):

\(\Delta = b^2 – 4ac \)

• If \( \Delta > 0 \): Two distinct real roots.
• If \( \Delta = 0 \): One real root (repeated).
• If \( \Delta < 0 \): No real roots (complex roots).