graph

/graf, grahf/, n.

1. a diagram representing a system of connections or interrelations among two or more things by a number of distinctive dots, lines, bars, etc.

2. Math.

a. a series of points, discrete or continuous, as in forming a curve or surface, each of which represents a value of a given function.

b. Also called linear graph. a network of lines connecting points.

3. a written symbol for an idea, a sound, or a linguistic expression.

v.t.

4. Math. to draw (a curve) as representing a given function.

5. to represent by means of a graph.

[1875-80; short for graphic formula; see GRAPHIC]

Syn. 1. See map.

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Visual representation of a data set or a mathematical equation, inequality, or function to show relationships or tendencies that these formulas can only suggest symbolically and abstractly.

Though histograms and pie charts are also graphs, the term usually applies to point plots on a coordinate system. For example, a graph of the relationship between real numbers and their squares matches each real number on a horizontal axis with its square on a vertical axis. The resulting set of points in this case is a parabola. A graph of an inequality is usually a shaded region on one side of a curve, whose shape depends not only on the equation or inequality but on the coordinate system chosen.

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▪ mathematics

      pictorial representation of statistical data or of a functional relationship between variables. Graphs have the advantage of showing general tendencies in the quantitative behaviour of data, and therefore serve a predictive function. As mere approximations, however, they can be inaccurate and sometimes misleading.

      Most graphs employ two axes, in which the horizontal axis represents a group of independent variables, and the vertical axis represents a group of dependent variables. The most common graph is a broken-line graph, where the independent variable is usually a factor of time. Data points are plotted on such a grid and then connected with line segments to give an approximate curve of, for example, seasonal fluctuations in sales trends. Data points need not be connected in a broken line, however. Instead they may be simply clustered around a median line or curve, as is often the case in experimental physics or chemistry.

      If the independent variable is not expressly temporal, a bar graph may be used to show discrete numerical quantities in relation to each other. To illustrate the relative populations of various nations, for example, a series of parallel columns, or bars, may be used. The length of each bar would be proportional to the size of the population of the respective country it represents. Thus, a demographer could see at a glance that China's population is about 30 percent larger than its closest rival, India.

      This same information may be expressed in a part-to-whole relationship by using a circular graph, in which a circle is divided into sections, and where the size, or angle, of each sector is directly proportional to the percentage of the whole it represents. Such a graph would show the same relative population sizes as the bar graph, but it would also illustrate that approximately one-fourth of the world's population resides in China. This type of graph, also known as a pie chart, is most commonly used to show the breakdown of items in a budget.

      In analytic geometry, graphs are used to map out functions of two variables on a Cartesian coordinate system, which is composed of a horizontal x-axis, or abscissa, and a vertical y-axis, or ordinate. Each axis is a real number line, and their intersection at the zero point of each is called the origin. A graph in this sense is the locus of all points (x,y) that satisfy a particular function.

      The easiest functions to graph are linear, or first-degree, equations, the simplest of which is y = x. The graph of this equation is a straight line that traverses the lower left and upper right quadrants of the graph, passing through the origin at a 45-degree angle. Such regularly-shaped curves as parabolas, hyperbolas, circles, and ellipses are graphs of second-degree equations. These and other nonlinear functions are sometimes graphed on a logarithmic grid, where a point on an axis is not the variable itself but the logarithm of that variable. Thus, a parabola with Cartesian coordinates may become a straight line with logarithmic coordinates.

      In certain cases, polar coordinates (q.v.) provide a more appropriate graphic system, whereby a series of concentric circles with straight lines through their common centre, or origin, serves to locate points on a circular plane. Both Cartesian and polar coordinates may be expanded to represent three dimensions by introducing a third variable into the respective algebraic or trigonometric functions. The inclusion of three axes results in an isometric graph for solid bodies in the former case and a graph with spherical coordinates for curved surfaces in the latter.

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