Quadratic Functions

Algebra
definition

Also known as: parabola, x-squared

Grade 9-12

View on concept map

A polynomial function of degree 2 whose graph is a U-shaped parabola that opens up or down. Appears in physics, economics, and any context with acceleration.

This concept is covered in depth in our Quadratic Equations Guide, with worked examples, practice problems, and common mistakes.

Definition

A polynomial function of degree 2 whose graph is a U-shaped parabola that opens up or down.

💡 Intuition

The path of a thrown ball — rising then falling — traces a parabola opening downward.

🎯 Core Idea

Quadratics model acceleration, projectiles, and optimization.

Example

f(x) = x^2 - 4x + 3 — a parabola opening up with vertex at (2, -1).

Formula

f(x) = ax^2 + bx + c \quad \text{or} \quad f(x) = a(x-h)^2 + k

Notation

a is the leading coefficient (determines opening direction), (h, k) is the vertex, and x = -\frac{b}{2a} is the axis of symmetry.

🌟 Why It Matters

Appears in physics, economics, and any context with acceleration.

💭 Hint When Stuck

Make a table of x-values from -3 to 3 and compute y for each to see the parabola's shape.

Formal View

A quadratic function f: \mathbb{R} \to \mathbb{R} has the form f(x) = ax^2 + bx + c with a \neq 0. Its zero set is \{x \in \mathbb{R} \mid ax^2 + bx + c = 0\}, with |\text{zeros}| \in \{0, 1, 2\} determined by \operatorname{sgn}(b^2 - 4ac).

🚧 Common Stuck Point

Vertex form vs standard form—each reveals different information.

⚠️ Common Mistakes

  • Sign errors in factoring
  • Forgetting \pm in quadratic formula

Frequently Asked Questions

What is Quadratic Functions in Math?

A polynomial function of degree 2 whose graph is a U-shaped parabola that opens up or down.

Why is Quadratic Functions important?

Appears in physics, economics, and any context with acceleration.

What do students usually get wrong about Quadratic Functions?

Vertex form vs standard form—each reveals different information.

What should I learn before Quadratic Functions?

Before studying Quadratic Functions, you should understand: linear functions, exponents.

How Quadratic Functions Connects to Other Ideas

To understand quadratic functions, you should first be comfortable with linear functions and exponents. Once you have a solid grasp of quadratic functions, you can move on to quadratic formula and polynomials.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula →
← Back to Explorer

Visualization

Static

Visual representation of Quadratic Functions