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 pp and real variable xx, functions of the form f(x)=x^pf(x)=xp are called power functions of xx.

What does it mean that a power function x^pxp models a given set of ordered pairs? We know that, in general, the graph of a function ff is the set of ordered pairs (x, y)(x,y), such that y=f(x)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)}{(1,2),(2,5),(3,10)}. This corresponds to the points A(1, 2)A(1,2), B(2, 5)B(2,5) and C(3, 10)C(3,10) contained in the graph of the power function f(x)=x^pf(x)=xp.

So, we have the following equations:
(1)(1) 1^p=11p=1
(2)(2) 2^p =42p=4
(3)(3) 3^p=93p=9

We see that p=2p=2, therefore the power function is f(x)=x^2f(x)=x2

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