Graphs

According to the dictionary a graph is a diagram that represents the variation of a variable in comparison with that of one or more other variables. We are exposed to graphs almost daily. Figure 1 shows two graphs that appeared on my monthly utility bill, showing the amount of gas and electricity I used each month between February 1996 and February 1997.

Figure 1. Gas and electric usage between February 1996 and February 1997.

These graphs allow me to draw several conclusions:

My gas consumption varies significantly from month to month, and the gas consumption during the winter months is almost a factor of 10 higher than the gas consumption during the summer months.
My electric consumption is relatively constant during the entire year (no obvious correlation between winter and summer).

In general, a graphical representation of data makes it easier to observe trends. Graphs not only allow us to look for trends in the past, but also allow us to make predictions for future trends:

I expect that my usage of electricity remains pretty constant over the next 12 months.
I expect that I see a drastic reduction in my gas consumption during the coming summer.

Of course some of these trends are expected:

Since my house has a gas furnace, I expect a large increase in gas consumption during the heating season.

Other trends are not necessarily expected:

Besides gas my furnace also uses electricity for the blower. Obviously, this will lead to an increase in the use of electricity during the winter months. In addition, the significant reduction in the length of day light during the winter months, requires an increase in the usage of electricity for the lights in my house. However, the graph shown in Figure 1 does not suggest a significant change in the usage of electricity during the winter months, suggesting that the blower motor of my furnace and the energy usage of my lights is small compared to the power requirements for other household items, such as a vacuum cleaner, a dishwasher, a washing machine and dryer, etc., whose use is expected to have a negligible seasonal dependence.

Figure 2 shows several graphs that appeared on March 27, 1997, in the Business Section of the New York Times and show trends in the Standard&Poor's 500-stock index, in various Interest Rates, and in the values of the dollar against the yen. The time period covered by these graphs is 6 months, from October 1996 until March 1997. The first graph shows the value of the Standard&Poor's 500-stock index over this preriod. The bold line shows the value pf the stock index at the end of each week and shows significant variations on a week-to-week basis. The solid "bottom" line shows the average value of the stock index over the preceding 150-day period (a so-called running average). It shows significantly less week-to-week variations since it is based on a long-term average (and a significant variation for a given week will have a much smaller effect on the average overthe preceding 23 weeks). The running average is a much better indicator and predictor for long-term trends in the stock index.

Figure 2. Tracking the stock market.

Scientific Graphs

Physical concepts and the relationships between them can be represented in many ways. The most commonly used technique is the graphical representation. We will limit ourselves to two-dimensional graphs. Each point on a two-dimensional graph represents a pair of numbers. These two numbers are repreented by the location of a point with respect to two perpendicaulr lines called axes. On the graph shown in Figure 3, the vertical axis has a scale to represent mass, and the horizontal axis has a scale to represent volume. The single point shown in the graph in Figures 3 represents the mass and volume of a particular piece of aluminum with a mass of 5.4 g and a colume of 2.0 cm³. In the language of graphs, the number 2.0 is called the horizontal coordinate, or abscissa, and the number 5.4 is called the vertical coordinate, or ordinate.

Figure 3. Mass versus Volume graph.

Coordinates on a graph

Each point on a graph has two coordinates. Below you see two coordinate axes, labelled x(m) and y(m), that can be used to make a graph of the motion of a soccer player on a soccer field. A given position is uniquely defined by the projections of its location on the x and y axes. Use the mouse to click on this soccer field and the coordinates of that location will be shown next to the soccer field.

Trajectories

There are different ways to display motion. The motion of a soccer player on a soccer field can be displayed by plotting his/her position at various times on a two-dimensional "position" graph. This position can be uniquely defined by specifying two position coordinates (x, y). Use the mourse to indicate the position of the soccer player on the soccer field at different times. Your will see the trajectory of the motion emerge.

Instead of looking at a trajectory graph we can also describe the time dependence of the two position coordinates. The two smaller graphs on the right-hand side show the time dependence of the x and y cooridnates. These graphs are automatically updated when you plot the trajectory of the soccer player. Note that the shape of the graphs of the x and y coordinates as function of time are very different from the shape of the actual trajectory.

Experiments

Last updated on Monday, January 22, 2001 9:09