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

Question

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

1 Answer

Impact of this question

5110 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-06-01T22:45:14

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

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

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

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

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

Science

Math

Humanities

... and beyond

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

1 Answer

Related questions

Impact of this question