Modeling with Polynomial Functions

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Modeling real-world phenomena with a function is an extremely useful tool to have at our disposal. This lesson will explain how to model a given set of data points with polynomial functions using the method of finite differences.

Modeling With Polynomial Functions

Suppose that the town of Algebra discovered a hammerhead shark near the town coastline in 2010. Here's an interesting fact! Hammerhead sharks are asexual, meaning that the female can reproduce all by herself - no male needed! Of course, this fact gave alarm to town officials, so they began tracking the number of hammerheads near the coastline each year, and the following chart shows how many hammerheads, H, were present each year, x, after 2010.

x 1 2 3 4 5 6 7
H 1 4 10 20 35 56 84

Some marine biologists were called in to analyze the population trends of the sharks in hopes to keep the population under control, and they found that the data in the chart can be modeled using the following function:

  • H = (1/6)x3 + (1/2)x2 + (1/3)x

This is an example of modeling with polynomial functions. A polynomial function is the sum of terms containing the same variable with different positive integer exponents. The highest exponent of a polynomial is called the degree of the polynomial, and the general form of a polynomial is when it is written with the term's exponents going in descending order. For example, some of the general forms of polynomials are shown in the image.


polymodfun1


We see that the polynomial representing the shark population has degree 3, and it is a cubic polynomial in general form. Speaking of which, how did the experts come up with this polynomial to model the given data? Let's explore!

Method of Finite Differences

When trying to find a polynomial function to model some given data, the first step is always to find the degree of the polynomial function. We can do this using the method of finite differences, which involves the following steps:

  1. Find the differences between the y-values, or outputs, of the data points. If they are constant, you can stop, and the polynomial function will have degree 1. If they are not constant, go on to step 2.
  2. Find the difference between the differences you just found. If they are constant, you can stop, and the polynomial will have degree 2. If they are not constant, repeat this step until they are constant. The number of times it takes for the differences to become constant is the degree of the polynomial.

For example, consider the set of data points:

x 0 1 2 3 4 5
y 1 4 9 16 25 36

To find the degree of a polynomial that could model the data, we first find the differences between the y values.


modpoly11


Since the differences aren't constant, we go onto step 2, and find the differences of the differences we just found.


polymod11


This time, the differences have a constant value of 2, so we stop here. Since it took two rounds to get a constant difference, the polynomial would have degree 2.

All together, we use the following steps to find a polynomial function to model a given set of data points.

  1. Find the degree of the polynomial using the method of finite differences.
  2. Write the general form of the polynomial function.
  3. Plug in your data points to create a system of equations. The number of equations in the system should be equal to the number of coefficients in the general form of the polynomial.
  4. Solve the resulting system of equations. Then plug these values back into the general form of the equation. This is your model.

There are many ways to solve systems of equations, and each one of those would constitute a lesson of its own, so this lesson will just concentrate on the process of finding the model, not on solving systems.

This may sound like a lot, but if we take it step by step, we can do it!

Applying the Steps

Let's consider our shark example again. We already know the polynomial function that can be used to model the data, but let's look at how this function was derived by taking the data points through our steps. First we would find the polynomials degree using the method of finite differences as shown in the image.


modpolyfun2


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