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AP Review – Linearizing a Graph
This review uses a technique called “linearizing the data.” It is nice to work with linear graphs. It is much
easier to calculate the slope of a linear graph. Calculating the area under the curve is easier too.
Unfortunately lots of functions are not linear and will not produce a straight line when graphed directly.
This is where linearizing the data comes in.
Example 1: The function
.A graph of y vs x is not linear. The reason it is not linear, is because x is
square rooted. To linearize it, square root all the x values and graph y vs
instead.
Example 2: Consider the function y = KA3. If I had data for y and A and wanted to determine K
graphically, what should be graphed? Graphing y vs A isn’t helpful. The graph is curved and I
can’t take the slope or area under the curve vary easily. But if I graphed y vs A3, that is linear and
the slope would give me K.
In the past two examples, only the x-axis had to be adjusted in order to linearize the graph. However,
sometimes both the x and the y axis need to be adjusted in order to produce a linear graph.
Practice Free Response Question:
A student is given the set of orbital data for some of the moons of Saturn shown below and is asked to
use the data to determine the mass MS of Saturn. Assume the orbits of these moons are circular.
a. Write an algebraic expression for the gravitational force between Saturn and one of its moons.
b. Use your expression from part (a) and the assumption of circular orbits to derive an equation for the
orbital period T of a moon as a function of its orbital radius R.
c. Which quantities should be graphed to yield a straight line whose slope could be used to determine
Saturn's mass?
d. Complete the data table by calculating the two quantities to be graphed. Label the top of each
column, including units.
e. Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers
to indicate the scale.
f.
Using the graph, calculate a value for the mass of Saturn.