**Simple Graphs—Smooth Curves**

When the variables in a function are clearly defined, or when they can attain only specific values (called *discrete values*), graphs can be rendered simply. Here are some of the most common types.

**Smooth Curves**

Figure 1-6 is a graph showing two curves, each of which represents the fluctuations in the prices of a hypothetical stock during the better part of a business day. Let's call the stocks Stock X and Stock Y. Both of the curves represent functions of time. You can determine this using the vertical-line test. Neither of the curves intersects a movable, vertical line more than once.

Suppose, in the situation shown by Fig. 1-6, the stock price is considered the independent variable, and time is considered the dependent variable. To illustrate this, plot the graphs by ''standing the curves on their ears,'' as shown in Fig. 1-7. (The curves are rotated 90 degrees counterclockwise, and then mirrored horizontally.) Using the vertical-line test, it is apparent that time can be considered a function of the price of Stock X, but not a function of the price of Stock Y.

**Vertical Bar Graphs**

In a *vertical bar graph*, the independent variable is shown on the horizontal axis and the dependent variable is shown on the vertical axis. Function values are portrayed as the heights of bars having equal widths. Figure 1-8 is a vertical bar graph of the price of the hypothetical Stock Y at intervals of 1 hour.

**Horizontal Bar Graphs**

In a *horizontal bar graph*, the independent variable is shown on the vertical axis and the dependent variable is shown on the horizontal axis. Function values are portrayed as the widths of bars having equal heights. Figure 1-9 is a horizontal bar graph of the price of the hypothetical Stock Y at intervals of 1 hour.

**Histograms**

A *histogram* is a bar graph applied to a special situation called a *distribution*. An example is a portrayal of the grades a class receives on a test, such as is shown in Fig. 1-10. Here, each vertical bar represents a letter grade (A, B, C, D, or F). The height of the bar represents the percentage of students in the class receiving that grade.

In Fig. 1-10, the values of the dependent variable are written at the top of each bar. In this case, the percentages add up to 100%, based on the assumption that all of the people in the class are present, take the test, and turn in their papers. The values of the dependent variable are annotated this way in some bar graphs. It's a good idea to write in these numbers if there aren't too many bars in the graph, but it can make the graph look messy or confusing if there are a lot of bars.

Some histograms are more flexible than this, allowing for variable bar widths as well as variable bar heights. We'll see some examples of this in Chapter 4. Also, in some bar graphs showing percentages, the values do not add up to 100%. We'll see an example of this sort of situation a little later in this chapter.

**Point-to-Point Graphs**

In a *point-to-point graph*, the scales are similar to those used in continuous-curve graphs such as Figs. 1-6 and 1-7. But the values of the function in a point-to-point graph are shown only for a few selected points, which are connected by straight lines.

In the point-to-point graph of Fig. 1-11, the price of Stock Y (from Fig. 1-6) is plotted on the half-hour from 10:00 A.M. to 3:00 P.M. The resulting ''curve'' does not exactly show the stock prices at the in-between times. But overall, the graph is a fair representation of the fluctuation of the stock over time.

When plotting a point-to-point graph, a certain minimum number of points must be plotted, and they must all be sufficiently close together. If a point-to-point graph showed the price of Stock Y at hourly intervals, it would not come as close as Fig. 1-11 to representing the actual moment-to-moment stock-price function. If a point-to-point graph showed the price at 15-minute intervals, it would come closer than Fig. 1-11 to the moment-to-moment stock-price function.

**Choosing Scales**

When composing a graph, it's important to choose sensible scales for the dependent and independent variables. If either scale spans a range of values much greater than necessary, the *resolution* (detail) of the graph will be poor. If either scale does not have a large enough span, there won't be enough room to show the entire function; some of the values will be ''cut off.''

**Simple Graphs Practice Problems**

**Practice 1**

Figure 1-12 is a hypothetical bar graph showing the percentage of the work force in a certain city that calls in sick on each day during a particular work week. What, if anything, is wrong with this graph?

**Fig. 1-12. **Illustration for Practice 1 and 2.

**Solution 1**

The horizontal scale is much too large. It makes the values in the graph difficult to ascertain. It would be better if the horizontal scale showed values only in the range of 0 to 10%. The graph could also be improved by listing percentage numbers at the right-hand side of each bar.

**Practice 2**

What's going on with the percentage values depicted in Fig. 1-12? It is apparent that the values don't add up to 100%. Shouldn't they?

**Solution 2**

No. If they did, it would be a coincidence (and a bad reflection on the attitude of the work force in that city during that week). This is a situation in which the sum of the percentages in a bar graph does not have to be 100%. If everybody showed up for work every day for the whole week, the sum of the percentages would be 0, and Fig. 1-12 would be perfectly legitimate showing no bars at all.

Practice problems for these concepts can be found at:

Background Math Practice Test