What does the graph of a quadratic function look like?

The graph is always a parabola — a smooth U-shaped (or inverted U-shaped) curve. It has exactly one vertex (the minimum or maximum point), one axis of symmetry, and can cross the x-axis 0, 1, or 2 times.

Where are quadratic functions used in real life?

Quadratic functions model many real-world phenomena: the path of a thrown ball (projectile motion), profit optimization in business, the shape of satellite dishes and suspension bridges, and the area of rectangles with a fixed perimeter.

How do I find the vertex of a quadratic function?

The x-coordinate of the vertex is x = -b/(2a). Substitute this into f(x) to get the y-coordinate. In vertex form f(x) = a(x-h)² + k, the vertex is simply (h, k).

VIZMath · Function Guide

Quadratic Function

f(x) = ax² + bx + c

The parabola-shaped curve behind projectile motion, optimization, and physics.

Quadratic Function

A quadratic function is a polynomial function of degree 2, written in the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a U-shaped (or inverted U-shaped) curve called a parabola.

The direction of the parabola depends on the sign of a: when a > 0, it opens upward; when a < 0, it opens downward. The lowest or highest point of the parabola is called the vertex, and the parabola is symmetric about the vertical line through the vertex.

Standard Form

f(x) = ax² + bx + c

Leading coefficient — controls width and direction (a ≠ 0)

Linear coefficient — shifts the axis of symmetry

Constant — the y-intercept of the parabola

Key Properties

Domain

All real numbers: (-∞, +∞)

Range

[vertex y-value, +∞) if a > 0 or (-∞, vertex y-value] if a < 0

Vertex

x = -b/(2a), y = f(-b/(2a))

Axis of Symmetry

x = -b/(2a)

Y-intercept

(0, c)

X-intercepts

Solved by the quadratic formula: x = (-b ± √(b²-4ac)) / 2a

Examples

f(x) = x²

Basic parabola with vertex at origin, opens upward.

f(x) = -x² + 4

Inverted parabola, vertex at (0, 4), opens downward.

f(x) = 2(x-1)² - 3

Vertex form: vertex at (1, -3), stretched vertically.

Visualize the Quadratic Function

Adjust a, b, c sliders and watch the parabola change in real time

Try in VIZMath →

FAQ

A function is quadratic when its highest power of x is exactly 2. The standard form is f(x) = ax² + bx + c with a ≠ 0. If a were 0, it would become a linear function.

Related Functions

Linear Function Exponential Function