Question

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

Answer 1

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.