Coordinate Graphs

The most common type of graph using the Cartesian Coordinate System is one in which all values of both the x-coordinate and the y-coordinate are positive. This corresponds to Quadrant I of a Cartesian coordinate graph. The relationship between two physical quantities is often shown on this type of rectangular plot. The x-axis and the y-axis must first be labeled to correspond to one of the physical quantities. The units of measurement along each axis must also be established. For example, to show the relationship between reactor power level and time, the x-axis can be used for time in minutes and the y-axis for the reactor power level as a percentage of full power level. Data points are plotted using the associated values of the two physical quantities.

Example: The temperature of water flowing in a high pressure line was measured at regular intervals. Plot the following recorded data on a Cartesian coordinate graph.

The first step is to label the x-axis and the y-axis. Let the x-axis be time in minutes and the y-axis be temperature in F.

The next step is to establish the units of measurement along each axis. The x-axis must range from 0 to 120, the y-axis from 400 to 500.

The points are then plotted one by one. Figure 2 shows the resulting Cartesian coordinate graph.

Figure 2 Cartesian Coordinate Graph of Temperature vs. Time

Example: The density of water was measured over a range of temperatures. Plot the following recorded data on a Cartesian coordinate graph.

The first step is to label the x-axis and the y-axis. Let the x-axis be temperature in C and the y-axis be density in g/ml.

The next step is to establish the units of measurement along each axis. The x-axis must range from approximately 40 to 100, the y-axis from 0.95 to 1.00.

The points are then plotted one by one. Figure 3 shows the resulting Cartesian coordinate graph.

Figure 3 Cartesian Coordinate Graph of Density of Water vs. Temperature

Graphs are convenient because, at a single glance, the major features of the relationship between the two physical quantities plotted can be seen. In addition, if some previous knowledge of the physical system under consideration is available, the numerical value pairs of points can be connected by a straight line or a smooth curve. From these plots, the values at points not specifically measured or calculated can be obtained. In Figures 2 and 3, the data points have been connected by a straight line and a smooth curve, respectively. From these plots, the values at points not specifically plotted can be determined. For example, using Figure 3, the density of water at 65C can be determined to be 0.98 g/ml. Because 65C is within the scope of the available data, it is called an interpolated value. Also using Figure 3, the density of water at 101 C can be estimated to be 0.956 g/ml. Because 101 C is outside the scope of the available data, it is called an extrapolated value. Although the value of 0.956 g/ml appears reasonable, an important physical fact is absent and not predictable from the data given. Water boils at 100C at atmospheric pressure. At temperatures above 100C it is not a liquid, but a gas. Therefore, the value of 0.956 g/ml is of no significance except when the pressure is above atmospheric.

This illustrates the relative ease of interpolating and extrapolating using graphs. It also points out the precautions that must be taken, namely, interpolation and extrapolation should be done only if there is some prior knowledge of the system. This is particularly true for extrapolation where the available data is being extended into a region where unknown physical changes may take place.