graphing

Graphing is the process of representing mathematical relationships or data visually by plotting points, curves, or shapes on a coordinate system. It reveals patterns and relationships that may be hidden in raw numbers.

In two dimensions, graphs typically use a Cartesian coordinate system with a horizontal x-axis and a vertical y-axis. Curves illustrate equations such as y = f(x). Other systems include polar coordinates (r, θ) and three-dimensional coordinates (x, y, z) for surfaces. Graphs may be implicit or parametric, with x(t) and y(t) tracing the path.

Graphing can be done by hand or with digital tools. Manual plotting involves selecting input values and plotting the corresponding points. Line graphs emphasize continuous change; bar graphs and histograms summarize discrete categories or distributions; scatter plots show relationships between two variables. Graphing calculators and software automate plotting and render 2D and 3D graphs, with options for solving, fitting, and annotating.

Interpreting graphs involves identifying intercepts, slopes, asymptotes, domain and range, and regions of increase or decrease. Proper labeling and scaling are essential, as axis units and intervals affect readability. Graphs can be misleading if scales are distorted, data are noisy, or the representation omits important context.

Graphing is foundational across mathematics, science, and data analysis, supporting hypothesis generation, model validation, and communication of results. In graph theory, the term graph denotes a different concept: a set of vertices connected by edges, used to study relational structures rather than plots.