Quadratic Functions

Definition

A quadratic function is a second-degree polynomial function of the form $f(x) = ax^2 + bx + c$ , where $a \neq 0$ . Its graph is a symmetric curve known as a parabola.

Why It Matters

The properties of a parabola—specifically the vertex and axis of symmetry—are the primary tools for finding “optimal” states in any curved system. Without them, you are blind to the “turning points” in physical and economic systems.

Core Concepts

Vertex Form: $f(x) = a(x-h)^2 + k$ $f (x) = a (x - h)^{2} + k$ , where $(h, k)$ $(h, k)$ is the vertex.
- How to read: “The F of x equals a times the quantity x minus h squared plus k, where h comma k is the vertex.”
- Meaning / when to use: Rewrites the quadratic in shifted form so the vertex (turning point) is immediately visible at (h, k). Preferred for graphing and optimization.
Vertex Coordinates: In standard form, $h = -\frac{b}{2a}$ $h = - \frac{b}{2 a}$ , $k = f(h) = c - \frac{b^2}{4a}$ $k = f (h) = c - \frac{b ^{2}}{4 a}$ .
- How to read: “The H equals negative b divided by 2 a; the k equals f of h which equals c minus b squared divided by 4 a.”
- Meaning / when to use: The x-coordinate of the vertex (axis of symmetry) for any quadratic. The y-coordinate is the max or min value.
Maximum/Minimum Value Formula: For $f(x) = ax^2 + bx + c$ $f (x) = a x^{2} + b x + c$ , the extreme value occurs at $x = -\frac{b}{2a}$ $x = - \frac{b}{2 a}$ .
- How to read: “The extreme value of f of x occurs at x equals negative b divided by 2 a.”
- Meaning / when to use: Quick way to find the optimal input without rewriting in vertex form. Central to quadratic optimization problems.
Concavity: Opens up if $a > 0$ , down if $a < 0$ .
Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$ .
Intercepts: $y$ -intercept is $c$ ; $x$ -intercepts found via the quadratic formula.

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes