How do I find the power function that models a given set of ordered pairs?

1 Answer
Jun 1, 2015

First, let's remember what a power function is. For any constant real number #p# and real variable #x#, functions of the form #f(x)=x^p# are called power functions of #x#.

What does it mean that a power function #x^p# models a given set of ordered pairs? We know that, in general, the graph of a function #f# is the set of ordered pairs #(x, y)#, such that #y=f(x)#.

So, a given set of ordered pairs modeled by a power function corresponds to a set of points contained in the graph of the power function.

The problem becomes that of finding the equation of the power function given that we know the coordinates of a number of points of its graph. This is solved by solving the resulting system of equations.

Example:

Consider a given set of ordered pairs: #{(1, 2), (2, 5), (3, 10)}#. This corresponds to the points #A(1, 2)#, #B(2, 5)# and #C(3, 10)# contained in the graph of the power function #f(x)=x^p#.

So, we have the following equations:
#(1)# #1^p=1#
#(2)# #2^p =4#
#(3)# #3^p=9#

We see that #p=2#, therefore the power function is #f(x)=x^2#

This is a simple illustration of the general idea. In practice, the problems can be more complex.