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!