Quadratic Functions

Algebra
definition

Also known as: parabola, x-squared

Grade 9-12

View on concept map

A quadratic function is a polynomial function of degree 2, written as f(x) = ax^2 + bx + c with a \neq 0, whose graph is a U-shaped curve called a parabola that opens upward when a > 0 or downward when a < 0. Quadratic functions model acceleration, projectile motion, and profit optimization in economics.

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

Definition

A quadratic function is a polynomial function of degree 2, written as f(x) = ax^2 + bx + c with a \neq 0, whose graph is a U-shaped curve called a parabola that opens upward when a > 0 or downward when a < 0.

💡 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

Quadratic functions model acceleration, projectile motion, and profit optimization in economics. Engineers use them to design parabolic antennas and bridges. They are the simplest nonlinear functions and the gateway to understanding polynomial behavior.

💭 Hint When Stuck

When you see a quadratic, first identify a, b, and c in the standard form ax^2 + bx + c. Then find the vertex using x = -b/(2a) and compute y at that point. Finally, determine the direction of opening from the sign of a and plot a few points on either side of the vertex.

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 when factoring — always double-check by expanding your factors back out
  • Forgetting \pm in the quadratic formula, which causes you to miss one of the two solutions
  • Confusing the vertex coordinates: the vertex x-value is -b/(2a), not b/(2a)

Frequently Asked Questions

What is Quadratic Functions in Math?

A quadratic function is a polynomial function of degree 2, written as f(x) = ax^2 + bx + c with a \neq 0, whose graph is a U-shaped curve called a parabola that opens upward when a > 0 or downward when a < 0.

What is the Quadratic Functions formula?

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

When do you use Quadratic Functions?

When you see a quadratic, first identify a, b, and c in the standard form ax^2 + bx + c. Then find the vertex using x = -b/(2a) and compute y at that point. Finally, determine the direction of opening from the sign of a and plot a few points on either side of the vertex.

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