Graphs of mathematical functions

05/05/2026

The online simulations of graphs of mathematical functions on this page will allow you to visualize graphically different mathematical functions and check how the graphs change by altering some of their parameters

This Thematic Unit is part of our Mathematics collection

What are graphs of mathematical functions

The graph of a mathematical function is the visual representation of all possible solutions of that function. This is most commonly done in a two-dimensional system, where each solution is expressed as a point in the plane. However, there are also three-dimensional graphs that allow representing functions of two independent variables, where the values are located in a three-dimensional space. The graphical representation of functions is a powerful tool in mathematics that allows visualizing and understanding the behavior of a relationship between two or more variables.

Coordinate systems

The graphical representation of a function is done in a coordinate system. There are different systems, each of which offers specific tools best suited to different types of analysis and geometric problems.

Cartesian coordinate system

The Cartesian coordinate system is the most commonly used coordinate system and is composed of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). In this system, points are represented by ordinate pairs that indicate their location in the plane, which facilitates both the analysis and the graphical representation of mathematical functions. To plot the graph of a function, values are assigned to the independent variable (x) and the corresponding values of the dependent variable (y) are calculated using the function rule. The obtained points (x, y) are plotted on the Cartesian plane and connected by continuous lines or smooth curves. This set of points forms the graphical representation of the function.

Polar system

In the polar system, each point is defined by an angle and a distance from a point of origin called the pole. This system is especially useful for describing curves and figures that have radial symmetry, such as circles and spirals.

Cylindrical coordinate system

The cylindrical coordinate system extends the polar system with a third coordinate to describe the height of a point in three-dimensional space. It is useful in physics and geometry problems involving cylinders or circular structures.

Spherical coordinate system

The spherical coordinate system, on the other hand, is used primarily in three-dimensional problems where points are described in terms of a radius and two angles.

Analysis of graphs of mathematical functions

The graph of a function can have several characteristics, such as slope, concavity, points of intersection with the axes, as well as horizontal or vertical asymptotes. These features provide valuable information about the behavior of the function.

By analyzing the graph of a function, properties such as growth or decay of the function, maxima and minima, inflection points and symmetry can be determined. In addition, the graphical representation allows the visualization of patterns, trends and relationships between variables, which facilitates the interpretation of the results.

Application and technology of graphs of mathematical functions

The graphical representation of functions is widely used in various areas, such as physics, economics, engineering and biology. It helps to model phenomena, predict behavior and make decisions based on data.

Current technology, such as computer plotting programs and graphing calculators, greatly facilitates the graphing of functions. These tools allow accurate and detailed graphing of complex functions, which helps to explore and better understand their behavior.

The online simulations of graphs of mathematical functions on this page are a great tool that can help you in many different ways – don’t miss them!

Explore the exciting STEM world with our free, online, simulations and accompanying companion courses! With them you’ll be able to experience and learn hands-on. Take this opportunity to immerse yourself in virtual experiences while advancing your education – awaken your scientific curiosity and discover all that the STEM world has to offer!

Simulations of graphs of mathematical functions

Graphs of linear functions

Explore the world of lines. Investigate the relationships between linear equations, slope, and line graphs. Test yourself in the line game!

Graphs of quadratic functions

Discover how changing the coefficients changes the shape of a curve. View graphs of individual terms (e.g., y = bx) to see how they add up to generate the polynomial curve. Generate definitions for vertex, roots, and axis of symmetry. Compare different forms of a quadratic function. Define a curve by its focus and directrix.

Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

Gottfried Wilhelm Leibniz

1646

–

1716

Gottfried Wilhelm Leibniz was one of the founders of infinitesimal calculus, created modern mathematical notation, and contributed to mechanics and optics with groundbreaking theories

“Music is the hidden exercise of arithmetic of the soul, which does not know it is calculating.”

Joseph-Louis Lagrange

1736

–

1813

Joseph-Louis Lagrange formulated the Lagrangian equations, key in classical mechanics to describe motion. He also founded variational calculus, applying it to algebra and mathematical physics

“As soon as a mathematical truth is established, it always serves as a starting point for new truth”

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John von Neumann

1903

–

1957

John von Neumann developed the modern computer architecture, contributed to game theory, quantum mechanics, and statistics, transforming mathematics and computer science

“Truth is, at best, a hypothesis”

Carl Friedrich Gauss

1777

–

1855

Carl Friedrich Gauss developed fundamental methods in algebra, number theory, and statistics, consolidating the foundation of modern mathematics.

“Mathematics is the queen of the sciences and number theory its queen”

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Test your knowledge

What is a mathematical graph, and why is it an essential tool for representing relationships between variables?

A mathematical graph is a visual representation that shows how two or more variables relate to each other, usually through points, lines or curves placed on a coordinate system. Its importance lies in the fact that it allows us to interpret patterns, trends and behaviors at a glance—something that would be much harder to detect using only numbers or tables. Graphs help us understand how one variable changes in response to another, identify maxima and minima, observe intervals of increase or decrease and distinguish between linear and nonlinear relationships. They are fundamental in science, economics, statistics and physics, where data analysis often depends on clear visual representations that support decision‑making and interpretation. Their ability to condense complex information into a single, intuitive image makes graphs indispensable both for learning and for professional analysis.

How do axes, scales and plotted points in a graph help interpret the information it contains?

The axes provide the reference framework in which the variables are placed: the horizontal axis typically represents the independent variable, while the vertical axis represents the dependent one. Scales determine how values are distributed along each axis and allow distances, increments and proportions to be interpreted consistently. The plotted points, lines or bars represent the actual data and visually reveal how the variables behave. Together, these elements make it possible to identify trends, compare values, detect abrupt changes and understand functional relationships. Without clear axes, appropriate scales and accurately placed points, a graph would lose precision and could easily lead to misinterpretation. These components ensure that the visual information is reliable, meaningful and useful for analysis.

Why use a graph if I already have the data in a table? It feels like repeating the same thing.

A table gives you the numbers, but a graph lets you see how they behave. With a graph, you can instantly notice whether something rises, falls, stays stable or changes suddenly—patterns that can be hard to spot in a list of values. Comparing data or understanding trends becomes much faster when everything is visual. It’s not repetition: it’s turning raw numbers into a picture your brain can interpret much more easily.

Why are some graphs lines while others are bars or points? Shouldn’t there be just one type?

Different types of graphs highlight different kinds of information. Line graphs are great for showing continuous changes, like temperature throughout the day. Bar graphs work better for comparing categories, such as sales by product. Scatter plots help analyze relationships between two variables without assuming they connect smoothly. There isn’t a single “best” type because not all data behaves the same way. Choosing the right graph makes the information clearer and easier to understand.

Why do some graphs look misleading? I’ve seen ones that make changes look bigger than they are.

A graph can give the wrong impression if the scale is poorly chosen or if the axes are cropped in a way that exaggerates differences. It’s not that the graph is lying, but its design can influence how we interpret the data. That’s why it’s important to check the axis values and the scale being used. A well‑designed graph represents information faithfully; a poorly designed one can distort it, even if the numbers are technically correct.