Graphs
Graphs
A graph is a pictorial representation of the relationship between two quantities. A graph can be anything from a simple bar graph that displays the measurements of various objects to a more complicated graph of functions in two or three dimensions. The former shows the relationship between the kind of object and its quantity; the latter shows the relationship between input and output. Graphing is a way to make information easier for a viewer to absorb.
Types of Graphs
The simplest graphs show the number of many objects. For example, a bar graph might name the months of the year along a horizontal axis and show numbers (for the number of days in each month) along a vertical axis. Then a rectangle (or bar) is drawn above each month. The height of the bar might indicate the number of days in that month on which it rained, or on which a person exercised, or on which the temperature rose above 90 degrees. See the generic example of a bar graph below (top right).
Another simple kind of graph is a circle graph or pie graph, which shows fractions or percentages. In this kind of graph, a circle is divided into pieshaped sectors. Each sector is given a label and indicates the fraction of the total area that goes with that label. See the generic example of a pie graph below (top middle).
A pie chart might be used to display the percentages of a budget that are allotted to various expenditures. If the sector labeled "medical bills" takes up twotenths of the area of the circle, that means that twotenths, or 20 percent, of the budget is devoted to medical expenses. Usually the percentages are written on the sectors along with their labels to make the graph easier to read. In both bar graphs and pie graphs, the reader can immediately pick out the largest and smallest categories without having to search through a chart or list, making it easy to compare the relative sizes of many objects simultaneously.
Often the two quantities being graphed can both be represented numerically. For example, student scores on examinations are often plotted on a graph, especially if there are many students taking the exam. In such a graph, the numbers on the horizontal axis represent the possible scores on the exam, and the numbers on the vertical axis represent numbers of students who earned that score. The information could be plotted as a simple bar graph. If only the top point of each bar is plotted and a curve is drawn to connect these points, the result is a line graph. See the generic example of a line graph on the pevious page (top left). Although the points on the line between the plotted points do not correspond to any pieces of information, a smooth line can be easier to understand than a large collection of bars or dots.
Graphs for Continuous Data
Graphs become slightly more complicated when one (or both) of the quantities in the graph can have continuous values rather than a discrete set. A common example of this is a quantity that changes over time. For example, a scientist might be observing the rate of growth of bacteria. The rates could be plotted so that the horizontal axis displays units of time and the vertical axis displays numbers (how many bacteria exist).
Then, for instance, the point (3,1000) would mean that at "time 3" (which could mean three o'clock, or three seconds after starting, or various other times, depending on the units being used and the starting point of the experiment) there were one thousand bacteria in the sample. The rise and fall of the graph show the increases and decreases in the number of bacteria.
In this case, even though only a finite set of points represent actual data, the remaining points do have a natural interpretation. For instance, suppose that in addition to the point (3,1000), the graph also contains the point (4,1500) and that both of these points correspond to actual measurements. If the scientist joins all of the points on the graph by a line, then the point (3.5,1200) might lie on the graph, or perhaps the point (3.5,1350).
There are many different lines that can be drawn through a collection of points. Looking at the overall shape of the data points helps the scientist decide which line is the most reasonable fit. In the previous example, the scientist could estimate that at time 3.5, there were 1200 (or 1350) bacteria in the sample. Thus graphing can be helpful in making estimates and predictions.
Graphs for Predictions
Sometimes the purpose for drawing a graph may not be to view the data already known but to construct a mathematical model that will allow one to analyze data and make predictions. One of the simplest models that can be constructed from a set of data is called a bestfit line. Such a line is useful in situations in which the data are roughly linear—that is, they are increasing or decreasing at a roughly constant rate but do not fall precisely on a line. (See graph on the previous page, bottom right.)
A bestfit line can be a very useful tool for analyzing data because lines have very simple formulas describing their behavior. If, for instance, one has collected data up to time 5 and wishes to predict what the value will be at time 15, the value 15 can be inserted into the formula for the line to derive an estimation. One can also determine how good an estimate is likely to be by computing the correlation factor for the data. The correlation factor is a quantity that measures how close the set of data is to being linear; that is, how good a "fit" the bestfit line actually is.
Graphs for Functions
One of the most common uses of graphs is to display the information encoded in a function. A function, informally speaking, is an operation or rule that can be applied to numbers. Functions are usually graphed in the cartesian plane (that is, the x,y plane) with the horizontal or x axis representing the input variable and the vertical or y axis representing the output variable. The graph of a function differs from the other types of graphs described so far in that all the points on the graph represent actual information. A concrete relationship, usually given by a mathematical formula, connects the two objects being analyzed.
For example, the "squaring" function takes numbers and squares them. Thus an input of the number 1 corresponds to an output of 1; an input of 2 corresponds to an output of 4; an input of −7 corresponds to an output of 49; and so on. Therefore, the graph of this function contains the points (1, 1), (2, 4), (−7, 49), and infinitely many others.
Does the point (10, 78) lie on this graph? To determine the answer, examine which characteristics all the points on the graph have in common. Any point on the graph of a function represents an inputoutput pair, with the x coordinate representing input and the y coordinate representing output. With the squaring function, each output value is the square of the corresponding input value, so on the graph of the squaring function, each y coordinate must be the square of the corresponding x coordinate. Because 78 is not the square of 10, the point (10, 78) does not lie on the graph of the squaring function.
Graphing Notation
It is traditional to name graphs with an equation rather than with words. The equation of any graph, regardless of whether it is the graph of a function, is meant to be a perfect description of the graph—it should tell the viewer the relationship between the x  and y coordinates of the numbers being graphed.
For example, the equation of the graph of the squaring function is y = x ² because the y coordinate of any point on the graph is the square of the x coordinate. The line that passes through the point (0, 3) and slants upwards with slope 4 (that is, at a rate of four units up for every one unit to the right) has equation y = 4x + 3. This indicates that for every point on the graph, the y coordinate is 3 more than 4 times the x coordinate.
An equation of a graph has many uses: it is not only a description of the graph but also a mechanism for finding points on the graph and a test for determining whether a given point lies on the graph. For example, to find out whether the point (278, 3254) lies on the line y = 4x + 3, simply insert (278, 3254), resulting in the inequality 3254 ≠ 4(278) + 3. Because these numbers are not equal, the point does not lie on the line. However, the equation shows that the point (278, 1115) does lie on theline.
see also Data Collection and Interpretation; Graphs and Effects of Parameter Changes; Statistical Analysis.
Naomi Klarreich
Bibliography
Larson, Roland E., and Robert P. Hostetler. Precalculus, 3rd ed. Lexington, MA: D. C. Heath and Company, 1993.
Warner, Jack. Graph Attack! Understanding Charts and Graphs. New Jersey: Regents/Prentice Hall, 1993.
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