Understanding Polynomial Functions in Practical Contexts

Polynomials are used in a variety of fields to model diverse phenomena. This lesson aims to show how polynomials can model real-life situations.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Binomial
Polynomial
Zeros of a Function
Degree of a Polynomial
Table of Values

Explore

Differences of Function Values

Consider the following table of values.

$x$	$f (x)$
$1$	$2$
$2$	$3$
$3$	$7$
$4$	$16$
$5$	$32$
$6$	$57$

Looking at the

x -

values, it can be noted that the values are equally spaced. It is possible to find the first finite differences by subtracting two consecutive values. Consider the following example.

f (2) - f (1) = 3 - 2 ⇓ f (2) - f (1) = 1

The first differences for each pair of consecutive terms can be found by doing the same process. Then, the second differences are found by subtracting consecutive first differences, third differences by subtracting consecutive second differences, and so on. Find every first, second, and third difference. Is there something that can be noted about the differences?

Example

Writing a Function From a Graph

As part of an entrepreneur unit in school, Zosia is learning some interesting math. During her lesson, Zosia's teacher explained that modeling data is useful to understanding business. He explained that there are various methods to write a polynomial function that models data.

As an example, the teacher told the students about a certain company that has experienced ups and downs in earnings due to economic downturns.

Before $1997,$ the company suffered losses due to the credits requested for the opening.
In the years $1997,$ $2002,$ and $2007,$ the company had neither profit nor losses.
From $1997$ to $2002,$ the company made a profit.
Between $2002$ and $2007,$ the company suffered losses.
From $2007$ onwards, the company's profits have been increasing.

The teacher showed the students the following graph, where $y$ represents the profit, in millions of dollars, and $x$ represents the years before and after $2000 .$ Here, negative values of $y$ represent losses.

Graph of a cubic polynomial passing through (-3,0), (0,7), (2,0), and (7,0)

Then the teacher asked the students to write the cubic function graphed. Help Zosia write the expression for the cubic function.

Hint

Notice that three of the points are $x -$ intercepts.

Solution

The first thing to notice in the given graph is that three of the given points are zeros of the function. The Factor Theorem indicates that the zeros of a polynomial function are factors of the function. Since the function is cubic, the three zeros indicate the three factors.

(7, 0), (- 3, 0), (2, 0) ⇓ f (x) = a (x - 7) (x + 3) (x - 2)

In this expression, the factors are given by the zeros and

a

is a constant. To find the value of

a,

the fourth point is substituted into the expression.

f (x) = a (x - 7) (x + 3) (x - 2)

SubstituteII

$x = 0$ , $f (x) = 7$

7 = a (0 - 7) (0 + 3) (0 - 2)

Solve for

a

IdPropAdd

\IdPropAdd

7 = a (- 7) (3) (- 2)

Multiply

7 = 42 a

DivEqn

$LHS / 42 = RHS / 42$

\frac{7}{4 2} = a

SimpQuot

Simplify quotient

\frac{1}{6} = a

RearrangeEqn

Rearrange equation

a = \frac{1}{6}

Now that the value of

a

has been found, it is possible to finish writing the function shown the graph, which represents the profits of the company.

f (x) = \frac{1}{6} (x - 7) (x + 3) (x - 2)

Discussion

Finite Differences and Their Properties

Sometimes it can be useful to examine the differences between function values when the input values are separated by a known quantity. These differences are known as finite differences.

Rule

Finite Difference

A finite difference is a mathematical expression that can be written in the following form.

$f (x + c_{1}) - f (x + c_{2})$

In this expression, $c_{1}$ and $c_{2}$ are constants. There are three basic finite differences that are especially useful.

Forward difference: $f (x + h) - f (x)$
Backward difference: $f (x) - f (x - h)$
Central difference: $f (x + h) - f (x - h)$

These differences are shown below.

Backward, central, and forward difference shown with a polynomial curve

Finite differences can be used to determine the degree of a polynomial function that models a data set. This is because finite differences have some really useful properties.

Rule

Properties of Finite Differences

The finite differences of polynomial functions satisfy the following two properties.

For a polynomial function of degree $n,$ the $n th$ differences of the function values corresponding to equally spaced values of the independent variable are constant and different from zero.
If the $n th$ differences of equally spaced values are constant and different from zero, then the values can be represented by a polynomial function of degree $n .$

Proof

Informal Justification

To justify these properties, the minor cases will be examined first.

$n = 1$

Consider a linear function.

f_{1} (x) = a x + b

In this expression,

a

and

b

are constants different from zero. Suppose that the

x -

values are spaced by a constant

h .

The first differences between two arbitrary consecutive values of the function can be written as follows.

f_{1} (x + h) - f_{1} (x)

The expression for the linear function can be substituted here to see what happens.

f_{1} (x + h) - f_{1} (x)

SubstituteII

$f_{1} (x + h) = a (x + h) + b$ , $f_{1} (x) = a x + b$

a (x + h) + b - (a x + b)

Simplify

Distr

Distribute $- 1$

a (x + h) + b - a x - b

Distr

Distribute $a$

a x + a h + b - a x - b

SubTerms

Subtract terms

a h

Since both

a

and

h

are non-zero constants, their product is different from zero. Therefore, the first differences are always a non-zero constant, which confirms the property. It should be noted that, since the first differences are constant, all the second differences are zero.

$n = 2$

Now consider a quadratic function.

f_{2} (x) = a x^{2} + b x + c

In this expression,

a,

b,

and

c

are non-zero constants. The values of

x

are separated by a constant

h .

A first difference can be obtained by subtracting two consecutive function values.

f_{2} (x + h) - f_{2} (x)

Again, it is possible to substitute the expression for

f_{2} (x)

to find an expression for the first difference.

f_{2} (x + h) - f_{2} (x)

SubstituteExpressions

Substitute expressions

a (x + h)^{2} + b (x + h) + c - (a x^{2} + b x + c)

Simplify

Distr

Distribute $- 1$

a (x + h)^{2} + b (x + h) + c - a x^{2} - b x - c

ExpandPosPerfectSquare

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

a (x^{2} + 2 x h + h^{2}) + b (x + h) + c - a x^{2} - b x - c

Distr

Distribute $a$

a x^{2} + 2 a x h + a h^{2} + b (x + h) + c - a x^{2} - b x - c

Distr

Distribute $b$

a x^{2} + 2 a x h + a h^{2} + b x + b h + c - a x^{2} - b x - c

SubTerms

Subtract terms

2 a x h + a h^{2} + b h

Looking at the linear term

2 a x h,

it can be concluded that this difference is not constant. Now, try again with a second difference. Two consecutive first differences are needed to find a second difference.

f_{2} (x + h) - f_{2} (x) f_{2} (x + 2 h) - f_{2} (x + h)

A second difference is obtained by subtracting the first expression from the second one.

f_{2} (x + 2 h) - f_{2} (x + h) - (f_{2} (x + h) - f_{2} (x))

Distr

Distribute $- 1$

f_{2} (x + 2 h) - f_{2} (x + h) - f_{2} (x + h) + f_{2} (x)

SubTerms

Subtract terms

f_{2} (x + 2 h) - 2 f_{2} (x + h) + f_{2} (x)

SubstituteExpressions

Substitute expressions

a (x + 2 h)^{2} + b (x + 2 h) + c - 2 (a (x + h)^{2} + b (x + h) + c) + a x^{2} + b x + c

Simplify

ExpandPosPerfectSquare

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

a (x^{2} + 4 x h + 4 h^{2}) + b (x + 2 h) + c - 2 (a (x^{2} + 2 x h + h^{2}) + b (x + h) + c) + a x^{2} + b x + c

Distr

Distribute $a$

a x^{2} + 4 a x h + 4 a h^{2} + b (x + 2 h) + c - 2 (a x^{2} + 2 a x h + a h^{2} + b (x + h) + c) + a x^{2} + b x + c

Distr

Distribute $b$

a x^{2} + 4 a x h + 4 a h^{2} + b x + 2 b h + c - 2 (a x^{2} + 2 a x h + a h^{2} + b x + b h + c) + a x^{2} + b x + c

Distr

Distribute $- 2$

a x^{2} + 4 a x h + 4 a h^{2} + b x + 2 b h + c - 2 a x^{2} - 4 a x h - 2 a h^{2} - 2 b x - 2 b h - 2 c + a x^{2} + b x + c

SubTerms

Subtract terms

2 a h^{2}

Since

a

and

h

are non-zero constants, the second difference is a constant different from zero, confirming the property again. Similar to the previous case, since the second differences are constant, the third differences are zero.

Conclusion

It can be concluded from the previous cases that, for a polynomial of degree $n,$ its $n th$ differences are the last constant non-zero differences. Therefore, the $n th$ differences of a polynomial of degree $n$ are constant and nonzero. Note that this is an informal justification and should not be taken as a proof.