Deriving Relationships From Graphs printable pdf download

Deriving Relationships from Graphs

Graphing is a powerful tool for representing and interpreting numerical data. In addition, graphs and

knowledge of algebra can be used to find the relationship between variables plotted on a scatter chart.

Consider the following examples and procedures. Become familiar with these approaches, and attempt to

thoroughly understand the processes illustrated here. You will be using these approaches regularly in

introductory physics laboratory activities. Please note that this is merely an introduction to deriving

relationships from graphs. There ultimately will be much more to learn.

Linear Relationships:

The graph to the left depicts a linear relationship. That is,

the data points all can be connected with a straight line.

Note that the best-fit line does not pass through the origin;

rather, it passes through the y-axis at -1m. How would one

define the relationship between the values of the x and y

coordinates in this relationship? It’s done with the equation

of a straight line, y = mx + b where m is the slope and b is

the value of the y-intercept – the value of y when x = 0. The

slope, m, is found from Δy/Δx = (y

- y

)/(x

- x

) = 2m/s in

this graph. The value of b can be found from inspection as

equal to -1m. The algebraic relationship is then represented

as y = mx + b where y is identified with the distance, x is

identified with time, m the slope, and b the y-intercept.

Thus, our relationship from the graph is:

distance = (2m/s)time – 1m.

Proportional Relationships:

The graph to the left approximates the proportional

relationship between voltage and current in a circuit. Note

that a relationship is directly proportional only when the

relationship is linear and passes through the origin. This is

not quite the case in the current graph. The value of b is

0.0180 amps. The linear fit derived by the computer used to

generate the graph performs an algebraic best fit to the data.

The equation generated is the best mathematical fit, but the

fit does not properly represent the physical reality of the

situation. If no voltage is applied, then no current flows. If

this is the case, the best-fit line must pass through the origin

in what is known as a physical fit of the data. A physical fit

demands y = Ax as the model for the fit. Under the new

analysis m = 2amps/volt. Then we find:

current = (2amps/volt)voltage

(continued)

Deriving Relationships From Graphs