Graphs and Graphing

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, 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.