Download How to establish a relationship from a graph

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.
Note: The curve/line shapes that we studied in class are highlighted. BUT read the entire
document to fully understand graphing concepts.
 1) 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
 2) 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
 3) 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
NOTICE how the two curves above look very similar. That is why, anytime your data gives you a
curve, you cannot automatically assume that the curve is associated with any particular equation
until you test your hypothesis by trying to get a straight line. To test for an inverse relationship (as
in #2), you would re-plot your data. This time (instead of graphing y vs. x), plot y vs. 1/x. If this
graph results in a straight line, then you have found the relationship. You can now conclusively
say that y is directly proportional to 1/x, following the equation y = C/x. (By the way, another way
of saying this, which means the same thing, is: “y is inversely proportional to x”).
 4) 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
But, again, since this is a curve, you must test your hypothesis by plotting y vs. x2 and
testing for a straight line.
 5) The two variables might be related by the square root function, either directly or
inversely,
or
, where C is a constant
 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.