How to establish a relationship from a graph

Draw a graph by assigning one variable to the horizontal axis and another variable to the vertical axis, and plotting points corresponding to pairs of data values. You plot the independent variable on the horizontal axis and the dependent variable on the vertical axis. For instance, if you were determining a relationship between force and acceleration, you might decide that force is the independent variable, and that the acceleration depends on it. Force would go on the horizontal axis and acceleration would go on the vertical axis. If you know the acceleration that results from a given force, you use that pair of values to plot a data point for your graph.

n The relationship between two variables graphed on an xy coordinate system is linear if you can draw a straight line through all the data points. The equation for such a line is of the form y = Cx + b, where x and y are the independent and dependent variables, C is a constant equal to the slope of the line, and b is the value of the point at which the line intersects the y axis.

y = Cx + b, where C and b are constant

n If the graph of the relationship appears to be a curve – that is, if the data points lie in positions which can have a smooth curve drawn through them – then deducing its equation becomes a little trickier. The relationship might be inverse. In this case, we say that y is inversely proportional to x. As x increases, y decreases. An inverse relationship has an equation of the form,

, where C is a constant

n The data might have an inverse squared relationship: y is inversely proportional to x squared. An inverse squared relationship has an equation of the form,

, where C is a constant

n Or it might be a direct squared relationship: y is directly proportional to x squared. A squared relationship has an equation of the form,

y = Cx², where C is a constant

n The two variables might be related by the square root function, either directly or inversely,

 or
, where C is a constant

n Finally, there might be no relationship at all. In this case, you may see data points scattered in an irregular cloud so that you cannot draw a straight line or a smooth curve through them.

Examine your graph and data points. If you can draw a straight line through them, try to write the equation for the line in the form y = Cx + b.

If your graph is a curve, consider the possibility that it represents one of the other types of relationship.

A way to judge whether your data fit one of the other equations is to substitute one pair of data values into an equation and calculate C. Then, using that value of C, see if other pairs of data values satisfy the equation. If they do, you can be fairly sure your whole data set fits the equation.

In the simulations in this textbook, much of the error associated with “real-world” labs is minimized. That may be a plus or a minus, depending on your perspective – learning how to deal with and minimize error is a crucial skill. The idealized nature of the simulations does, however, simplify the task of determining the mathematical relationships among variables.