Graphs and Graphing
Graphs and Graphing
In mathematics, computer science, chemistry, and other sciences, a graph is a geometric representation, a picture, of a relation or function. In other words, it is a set of objects called points or vertices that are connected by lines or edges so as to visibly show an overall pattern or trend of information.
A relation is a subset of the set of all ordered pairs (x,y) for which each x is a member of some set X and each y is a member of another set Y. A specific relationship between each x and y determines which ordered pairs are in the subset. A function is a similar set of ordered pairs, with the added restriction that no two ordered pairs have the same first member. A graph, then, is a pictorial representation of the ordered pairs that comprise a relation or function. At the same time, it is a pictorial representation of the relationship between the first and second elements of each of the ordered pairs.
Representing ordered pairs
In 1637, Rene Descartes (1594-1650), the French mathematician and philosopher, published a book entitled Géométrie, in which he applied algebraic methods to the study of geometry. In the book, Descartes described a system (now called the rectangular coordinate system or the Cartesian coordinate system) for using points in a plane to represent ordered pairs. Given any two sets X and Y, the Cartesian product (written X × Y) of these two sets is the set of all possible ordered pairs (x,y) formed by choosing an element x from the set X and pairing it with an element y from the set Y. A relation between two sets X and Y, is a subset of their Cartesian product. To graph a relation, it is first necessary to represent the Cartesian product geometrically. Then, the graph of a particular relation is produced by highlighting that part of the representation corresponding to the points contained in the relation.
Geometrically, the Cartesian product of two sets is represented by two perpendicular lines, one horizontal, one vertical, called axes. The point where the axes intersect is called the origin. Members of the set X are represented in this picture by associating each member of X with points on the horizontal axis (called the x-axis). Members of the set Y are represented by associating each member of Y with points on the vertical axis (called the y-axis). It is interesting to note that this picture is easily extended to three dimensions by considering the Cartesian product of the sets (X × Y) and Z. Z is then represented by a third axis perpendicular to the plane that represents the ordered pairs in the set (X × Y). Having established a picture of the set of all possible ordered pairs, the next step in producing a graph is to represent the subset of ordered pairs that are contained in a given relation. This can be done in a number of ways. The most common are the bar graph, the scatter graph and the line graph.
A bar graph is used to picture the relationship between a relatively small number of objects, such as information listed in tabular form. Tables often represent mathematical relations, in that they consist of ordered pairs (listed in rows) for which the first and second elements of each pair (listed in separate columns) are related in a specific way. For example, a department store receipt is a relation defined by a table. It lists each item purchased together with its retail price. The first element of each ordered pair is the item, and the second element is that item’s purchase price. This type of relation lends itself well to the bar graph, because it contains information that is not strictly numeric, and because there are relatively few ordered pairs. In this example the prices are represented by points on the vertical axis, while the items purchased are represented by short line segments centered about the first few positive integers on the horizontal axis. To create the graph, the price of each item is located on the vertical axis, and a bar of that height is filled in directly above the location of the corresponding item on the horizontal axis. The advantage of the bar graph is that it allows immediate comparison of the relative purchase prices, including identification of the most expensive and least expensive items. It also provides a visual means of estimating the average cost of an item, and the total amount of money spent.
The scatter graph is similar to the bar graph in that it is used to represent relations containing a small number of ordered pairs. However, it differs from the bar graph in that both axes can be used to represent sets of real numbers. Since it is not feasible to represent pairs of real numbers with bars that have some width, ordered pairs of the relation are plotted by marking the corresponding point with a small symbol, such as a circle, or square. Since the scatter graph represents relations between sets of real numbers, it may also include negative as well as positive numbers. In producing a scatter graph, the location of each point is established by its horizontal distance from the y-axis and its vertical distance from the x-axis. Scatter graphs are used extensively in picturing the results of experiments. Data is generated by controlling one variable (called the independent variable) and measuring the response of a second variable (called the dependent variable). The data is recorded and then plotted, the independent variable being associated with the x-axis and the dependent variable with the y-axis.
Very often, a function is defined by an equation relating elements from the set of real numbers to other elements, also from the set of real numbers. When this is the case, the function will usually contain an infinite number of ordered pairs. For instance, if both X and Y correspond to the set of real numbers, then the equation y = 2x + 3 defines a function, specifically the set of ordered pairs (x, 2x+3). The graph of this function is represented in the rectangular coordinate system by a line. To graph this equation, locate any two points in the plane and, then, connect them together. As a check, a third point should be located, and its position on the line verified. Any equation whose graph is a straight line, can be written in the form y = mx + b, where m and b are constants called the slope and y-intercept, respectively. The slope is the ratio of vertical change (rise) to horizontal change (run) between any two points on the line. The y-intercept is the point where the graph crosses the y-axis. This information is very useful in determining the equation of a line from its graph. In addition to straight lines, many equations have graphs that are curved lines. Polynomials, including the conic sections, and the trigonometric functions (sine, cosine, tangent, and the inverse of each) all have graphs that are curves. It is useful to graph these kinds of functions in order to picture their behavior. In addition to graphing equations, it is often very useful to find the equation from the graph. This is how mathematical models of nature are developed. With the aid of computers, scientists draw smooth lines through a few points of experimental data, and deduce the equations that define those smooth lines. In this way they are able to model natural occurrences, and use the models to predict the results of future occurrences.
There are many practical applications of graphs and graphing. In the sciences and engineering, sets of numbers represent physical quantities. Graphing the relationship between these quantities is a useful tool
Cartesian product —The Cartesian product of two sets X and Y is the set of all possible ordered pairs (x, y) formed by taking the first element of the pair from the set X and the second element of the pair from the set Y.
Function —A function is a relation for which no two ordered pairs have the same first element.
Ordered pair —A pair of elements (x,y) such that the pair (y,x) is not the same as (x,y) unless x = y.
Relation —A relation between two sets X and Y is a subset of all possible ordered pairs (x,y) for which there exists a specific relationship between each x and y.
Variable —A variable is a quantity that is allowed to have a changing value, or that represents an unknown quantity.
for understanding nature. One specific example is the graphing of current versus voltage, used by electrical engineers, to picture the behavior of various circuit components. The rectangular coordinate system can be used to represent all possible combinations of current and voltage. Nature, however, severely limits the allowed combinations, depending on the particular electrical device through which current is flowing. By plotting the allowed combinations of current and voltage for various devices, engineers are able to picture the different behaviors of these devices. They use this information to design circuits with combinations of devices that will behave as predicted.
See also Variance.
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