Graphing and Math Modeling

The relationships between experimental variables are best visualized in the form of a graph. A graph consists of a rectangular grid imposed upon a set of coordinate axes. These axes are oriented perpendicularly to each other and are joined together at the origin O. The horizontal axis, pointing to the right, is called the abscissa or the x axis. The vertical axis which rises from the abscissa, at the origin O, is known as the ordinate or the y axis.

Each axis is divided into incremental units starting at the origin. Numbering of the axes starts at the origin with the lowest x and y values written at the origin. For example, examination of the data in the following table shows 1.0 as the minimum x value and 2.0 as the minimum y value.

The (x,y) pairs designate points on the graph. The point (5,10) is located by going to 5.0 on the x axis and then by moving up 10.0 units parallel to the y axis and placing a dot to mark the point. In a similar fashion, the other points are plotted on the graph. Experimental data contains error and the correlation between (x,y) pairs is not perfect. As seen in the graph of the error free data, the line passes through each point. In the error containing data, the correlation line does not pass through all the points – some of the points are located off the line.

To create a graph from a table of experimental data:

1.)    Select a piece of graph paper and place the origin at the lower left corner.

2.)    Draw a set of coordinate axes originating from the origin and label each axis.

3.)    Determine the range of the x data by subtracting the minimum x value from the maximum x value.

4.)    Determine the range of the y data by subtracting the minimum y value from the maximum y value.

5.)    Calculate the value of the major units along the x axis by dividing the x range by the number increments between the origin and the end of the x axis.

6.)    Calculate the value of the major units along the y axis by dividing the y range by the number increments between the origin and the end of the y axis.

7.)    Number each major unit along the x and y axis.

8.)    Plot the (x,y) data points on the set of coordinate axes.

Once the data had been plotted, different mathematical models can be fitted to the data. There are several different models that might potentially fit the data. The commonly applied models include linear, polynomial, exponential and logarithmic. The easiest way to fit a linear model (y= mx + b) to a data set is to plot the graph and to draw a best fit line among the points. Choose two points. (x₁, y₁) and (x₂, y₂) on the line and apply the two point form for the equation of a line.  Note: x1,y1 should be closer to the origin than x2, y2.

                                   =                 (1)

This equation can be rearranged to point slope form ( y=mx + b):

              y =  x   + ( y1 - (y2-y1))        (2)

where the slope m is the ratio (y2 – y1)/(x2-x1) and the intercept b is (y1-(x1/(x2-x1))(y2-y1)). The slope tells how rapidly one coordinates changes with respect to the other. The intercept is the coordinate where the line crosses the y or x axis.

Example

The resistance R in ohms of a wire coil as a function of temperature T in degrees Centigrade is:

EXPERIMENTAL PROCEDURE

The following data, which was taken from the Dallas Morning News, shows relative humidity as a function of temperature.

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1.      Label the x axis as “Temperature (˚F)” and the y axis as “Relative Humidity (%)”

2.      Divide the axes into enough units to cover the range of each variable.

3.      Plot the graph for each set of data and use the two points form to obtain the equation of the line that relates relative humidity and temperature.

4.      Clearly label each graph as to which data it refers to, and write the equation on the graph.

5.      From an examination of the graphs and equation, formulate a hypothesis about Upload graphs and the answer to #4 and #6 to 

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