Graph (visual representation of data)

A graph is a visual method for showing relationships among quantities or categories. This article explains types, components, history, uses, and how graphs differ from other diagrams.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: June 10, 2026

A graph is a visual representation that makes relationships, trends, or patterns in data easier to see. At its simplest a graph relates two or more measurements using positions, shapes, or lengths so that a viewer can quickly compare values. Many everyday charts—like a line showing temperature over time or bars showing sales by region—are forms of graphs.

Basic components

Most graphs that display numerical data share a few common parts: axes or reference lines, scales and tick marks, plotted points or bars, labels, and often a legend. The axes provide a coordinate framework: the horizontal axis typically represents an independent variable such as time, while the vertical axis shows the dependent variable. Tick marks and labels indicate the scale. A clear title and units of measurement are essential so readers understand what is being compared.

Axes and grid: guides for reading values along two directions; see connection between quantities.
Data marks: points, lines, bars, or areas that show measured values (grid often helps).
Scale and labels: numeric marks and descriptive text (Cartesian coordinates are common for rectangular plots).

Common types and examples

Graphs come in many styles suited to different tasks. Line graphs show how a quantity changes continuously, bar charts compare discrete categories, and scatter plots reveal relationships and clustering between two variables. Pie charts and stacked bars display proportions within a whole, while histograms summarize distributions. Specialized graphs such as heat maps, box plots, and network diagrams address particular analytical needs.

Line graph — trends over time (vertical and horizontal axes intersect at right angles).
Bar chart — categorical comparison (axes and bars).
Scatter plot — correlation between two continuous variables (ruler-like scales).

Origins and development

The ideas behind graphs draw on mathematics and practical illustration. The Cartesian coordinate system popularized by René Descartes allowed algebraic relationships to be shown geometrically. In the late 18th century William Playfair invented several graphic forms for economic data, including the bar chart and line plot. During the 19th and 20th centuries statisticians and public health reformers refined visual methods to present complex evidence clearly.

Uses, strengths and cautions

Graphs are widely used in science, business, education and journalism because they condense large amounts of information into a form the brain processes quickly. A well-designed graph highlights the intended message and supports comparison or discovery. Poor choices of scale, truncated axes, excessive decoration, or missing labels can mislead, so good practice emphasizes accurate scales, clear legends, and appropriate chart types.

Distinctions and notable points

The term "graph" can also mean a network of nodes and links in mathematics and computer science; that usage is distinct from data plots. More generally, graphs are one class of visual tools alongside diagrams and maps: a diagram may show structure or flow without quantitative axes, while a graph is normally intended to convey measured relationships. For further practical guides and examples see visualization principles and software tutorials at tools and resources.

Definition

The graph of a function $f\colon D\to Z$ with definition set and target set is the set

$G_{f}=\{(x,f(x))\in D\times Z\mid x\in D\}$ .

The graph is thus a special subset of the Cartesian product of the definition set and the target set. It consists of all pairs where the first component is an element of the definition set and the second component is the element of the target set assigned to this element by the function.

Special cases and examples

The graph of a function $f\colon D\to \mathbb {R}$ with $D\subseteq \mathbb {R}$ is a subset of $\mathbb {R} \times \mathbb {R} =\mathbb {R} ^{2}$ and thus can be taken to be a set of points or a geometric figure in the plane. Examples are:

The graph of a linear function $f\colon \mathbb {R} \to \mathbb {R} ,x\mapsto ax+b$ is a straight line.
The graph of a quadratic function $f\colon \mathbb {R} \to \mathbb {R} ,x\mapsto ax^{2}+bx+c$ with $a\neq 0$ is a parabola.
The graph of the reciprocal function $f\colon \mathbb {R} \setminus \{0\}\to \mathbb {R} ,x\mapsto {\tfrac {1}{x}}$ is a hyperbola.

The graphs of functions $f\colon \mathbb {R} ^{2}\to \mathbb {R}$ or $f\colon \mathbb {R} \to \mathbb {R} ^{2}$ are subsets of $\mathbb {R} ^{3}$ and can still be represented pictorially as spatial figures as well. Examples are:

The graph of a continuous function $f\colon \mathbb {R} ^{2}\to \mathbb {R}$ is a surface in three-dimensional space. For example, the graph of the function is $f(x,y)=x^{2}+y^{2}$ an elliptic paraboloid.
The graph of a continuous function $f\colon \mathbb {R} \to \mathbb {R} ^{2}$ is a graph in three-dimensional space. For example, the graph of the function is $f(t)=(\sin t,\cos t)$ a helical line.

Questions and Answers

Q: What is a graph?

A: A graph is a picture designed to express words, particularly the connection between two or more quantities.

Q: How is the relationship between two numbers or measurements shown on a simple graph?

A: The relationship between two numbers or measurements is shown in the form of a grid.

Q: What is a Cartesian coordinate system?

A: A Cartesian coordinate system is a rectangular graph where the two measurements are arranged into two different lines at right angles to one another.

Q: How are the vertical and horizontal axes arranged in a rectangular graph?

A: The vertical axis goes up, while the horizontal axis goes right.

Q: Where do the lines or axes meet in a rectangular graph?

A: The lines or axes meet at their ends in the lower left corner of the graph.

Q: What are tick marks in a graph?

A: Tick marks are markings along each axis that act like a ruler drawn on paper to indicate each measurement.

Q: What is the difference between a graph and a chart or diagram?

A: A graph relates one quantity to other quantities while a chart or a diagram may not. Flowcharts and tree diagrams are examples of charts or diagrams that are not graphs.