Linearization

Power Functions:

A common possibility for the functionality of your data is that it can be represented by a power function in the form:

y = Axn

Where A (the coefficient) is some number, and n is some power like 1, 2, -1, -2,  1/2, (typically)

• For example, Newton's second law F = ma, if F is y and a is x, then the coefficient A would be the mass, and the power n would be 1

• For example, the energy stored in a spring PE = 1/2kx2, if PE is y and x is x, then the coefficient A would be  1/2k, and the power n would be 2

• For example, Coulomb's law F = kq1q2/r2, if F is y and r is x,  the the coefficient would be kq1q2, and the power n would be -2.

For a data set, you can make a log log graph, and from the equation of the line, deduce the value of the coefficient A, and the power n:

If   y = Axn    then 

log(y) = nlog(x) + log(A)

which is a straight line with a slope equal to the exponent "n", and a y-intercept of "log(A)".

For the three sets of data below, determine the value of the coefficient A, and the power n, and make a linearized graph 

1. Calculate the log(y) and the log(x)

2. Make a graph of log(y) vs log(x) (If it is not linear at this point, it is not a power function BTW)

3. Put a trendline on it and display the equation.  The slope is your exponent n, and the coefficient, A is the anti log of the y-intercept.

4. You can also then linearize the data by graphing y vs xn.

Sample Data sets: (These are in tabs on the bottom of the spreadsheet - if you are on Canvas use the one linked to the assignment)

Sample Data

Here is how to do the log log graph:  (Steps 1-3 above)

How to make a Linearized graph from your log log graph:

Exponyential Functions:

Another possibility is that you have an exponential function:

y = Aekx

Radioactive decay is like this:

N = Noe-λt

If your data does not become linear when you do a log log graph, make a graph of ln(y) vs x :

ln(y) = ln(Aekx) = ln(A) + kx

so the y - intercept of that graph is the ln of the coefficient A, and the slope is the coefficient of the exponent k.

For the data set below, use this technique to find the value of A, and the coefficient of the exponent k.

Sample Data Set: (This is on a tab on the bottom of the sheet.  If you are using Canvas use the spreadsheet linked to the assignment)

Sample Data

Here is an example of how to do this: