Power Functions and Polynomial Functions

Learning Objectives

In this section, you will:

  • Identify power functions.
  • Identify end behavior of power functions.
  • Identify polynomial functions.
  • Identify the degree and leading coefficient of polynomial functions.
Three birds on a cliff with the sun rising in the background.

Figure 1. (credit: Jason Bay, Flickr)

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in (Figure).

Year 2009 2010 2011 2012 2013
Bird Population 800 897 992 1,083 1,169

The population can be estimated using the functionP(t)=0.3t3+97t+800,whereP(t)represents the bird population on the islandtyears after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.

Identifying Power Functions

Before we can understand the bird problem, it will be helpful to understand a different type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number.

As an example, consider functions for area or volume. The function for the area of a circle with radiusr
is

A(r)=πr2

and the function for the volume of a sphere with radiusr
is

V(r)=43πr3

Both of these are examples of power functions because they consist of a coefficient,πor43π,multiplied by a variablerraised to a power.

Power Function

A power function is a function that can be represented in the form

f(x)=kxp

wherek
andpare real numbers, andk
is known as the coefficient.

Isf(x)=2xa power function?

No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.

Identifying Power Functions

Which of the following functions are power functions?

f(x)=1Constant functionf(x)=xIdentify functionf(x)=x2Quadratic functionf(x)=x3Cubic functionf(x)=1xReciprocal functionf(x)=1x2Reciprocal squared functionf(x)=xSquare root functionf(x)=3xCube root function

Try It

Which functions are power functions?

f(x)=2x4x3g(x)=x5+5x3h(x)=2x513x2+4

Identifying End Behavior of Power Functions

(Figure) shows the graphs off(x)=x2,g(x)=x4andh(x)=x6,which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.

Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.

Figure 2. Even-power functions

To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbolfor positive infinity andfor negative infinity. When we say that “xapproaches infinity,” which can be symbolically written asx,we are describing a behavior; we are saying thatxis increasing without bound.

With the positive even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that asxapproaches positive or negative infinity, thef(x)values increase without bound. In symbolic form, we could write

as x±,f(x)

(Figure) shows the graphs off(x)=x3,g(x)=x5,andh(x)=x7,which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.

Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.

Figure 3. Odd-power functions

These examples illustrate that functions of the formf(x)=xnreveal symmetry of one kind or another. First, in (Figure) we see that even functions of the formf(x)=xnneven, are symmetric about they-axis. In (Figure) we see that odd functions of the formf(x)=xnn odd, are symmetric about the origin.

For these odd power functions, asx approaches negative infinity,f(x) decreases without bound. Asx approaches positive infinity,f(x) increases without bound. In symbolic form we write

asx,f(x)asx,f(x)

The behavior of the graph of a function as the input values get very small (x) and get very large (x) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.

(Figure) shows the end behavior of power functions in the formf(x)=kxnwherenis a non-negative integer depending on the power and the constant.

Graph of an even-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to positive infinity. Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an even-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an odd-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity.

Figure 4.

How To

Given a power functionf(x)=kxnwherenis a non-negative integer, identify the end behavior.

  1. Determine whether the power is even or odd.
  2. Determine whether the constant is positive or negative.
  3. Use (Figure) to identify the end behavior.

Identifying the End Behavior of a Power Function

Describe the end behavior of the graph off(x)=x8.

Identifying the End Behavior of a Power Function.

Describe the end behavior of the graph off(x)=x9.

Analysis

We can check our work by using the table feature on a graphing utility.

x f(x)
–10 1,000,000,000
–5 1,953,125
0 0
5 –1,953,125
10 –1,000,000,000

We can see from (Figure) that, when we substitute very small values forx,the output is very large, and when we substitute very large values forx,the output is very small (meaning that it is a very large negative value).

Try It

Describe in words and symbols the end behavior off(x)=5x4.

Identifying Polynomial Functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radiusr
of the spill depends on the number of weeksw
that have passed. This relationship is linear.

r(w)=24+8w

We can combine this with the formula for the areaA
of a circle.

A(r)=πr2

Composing these functions gives a formula for the area in terms of weeks.

A(w)=A(r(w))=A(24+8w)=π(24+8w)2

Multiplying gives the formula.

A(w)=576π+384πw+64πw2

This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

Polynomial Functions

Letn
be a non-negative integer. A polynomial function is a function that can be written in the form

f(x)=anxn+...+a2x2+a1x+a0

This is called the general form of a polynomial function. Eachai
is a coefficient and can be any real number, but
ancannot = 0. Each expressionaixi
is a term of a polynomial function.

Identifying Polynomial Functions

Which of the following are polynomial functions?

f(x)=2x33x+4g(x)=x(x24)h(x)=5x+2

Identifying the Degree and Leading Coefficient of a Polynomial Function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

Terminology of Polynomial Functions

We often rearrange polynomials so that the powers are descending.

Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +…+a_2x^2+a_1x+a_0.

When a polynomial is written in this way, we say that it is in general form.

How To

Given a polynomial function, identify the degree and leading coefficient.

  1. Find the highest power ofx
    to determine the degree function.
  2. Identify the term containing the highest power ofx
    to find the leading term.
  3. Identify the coefficient of the leading term.

Identifying the Degree and Leading Coefficient of a Polynomial Function

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

f(x)=3+2x24x3g(t)=5t52t3+7th(p)=6pp32

Identify the degree, leading term, and leading coefficient of the polynomialf(x)=4x2x6+2x6.

Identifying End Behavior of Polynomial Functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms asx gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See (Figure).

Polynomial Function Leading Term Graph of Polynomial Function
f(x)=5x4+2x3x4 5x4 Graph of f(x)=5x^4+2x^3-x-4.
f(x)=2x6x5+3x4+x3 2x6 Graph of f(x)=-2x^6-x^5+3x^4+x^3.
f(x)=3x54x4+2x2+1 3x5 Graph of f(x)=3x^5-4x^4+2x^2+1.
f(x)=6x3+7x2+3x+1 6x3 Graph of f(x)=-6x^3+7x^2+3x+1.

Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in (Figure).

Graph of an odd-degree polynomial.

Figure 7.

Try It

Describe the end behavior, and determine a possible degree of the polynomial function in (Figure).

Graph of an even-degree polynomial.

Figure 8.

Identifying End Behavior and Degree of a Polynomial Function

Given the functionf(x)=3x2(x1)(x+4),express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Try It

Given the functionf(x)=0.2(x2)(x+1)(x5),express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

Identifying Local Behavior of Polynomial Functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

We are also interested in the intercepts. As with all functions, the y-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y-intercept(0,a0).The x-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x-intercept. See (Figure).

Figure 9.

Intercepts and Turning Points of Polynomial Functions

A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The y-intercept is the point at which the function has an input value of zero. The x-intercepts are the points at which the output value is zero.

How To

Given a polynomial function, determine the intercepts.

  1. Determine the y-intercept by setting x=0 and finding the corresponding output value.
  2. Determine the x-intercepts by solving for the input values that yield an output value of zero.

Determining the Intercepts of a Polynomial Function

Given the polynomial functionf(x)=(x2)(x+1)(x4),written in factored form for your convenience, determine the y– and x-intercepts.

Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial functionf(x)=x44x245,determine the y– and x-intercepts.

Try It

Given the polynomial functionf(x)=2x36x220x,determine the y– and x-intercepts.

Comparing Smooth and Continuous Graphs

The degree of a polynomial function helps us to determine the number of x-intercepts and the number of turning points. A polynomial function ofnthdegree is the product ofnfactors, so it will have at mostnroots or zeros, or x-intercepts. The graph of the polynomial function of degreenmust have at mostn1turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

Intercepts and Turning Points of Polynomials

A polynomial of degreenwill have, at most,nx-intercepts andn1turning points.

Determining the Number of Intercepts and Turning Points of a Polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points forf(x)=3x10+4x7x4+2x3.

Try It

Without graphing the function, determine the maximum number of x-intercepts and turning points forf(x)=10813x98x4+14x12+2x3.

Drawing Conclusions about a Polynomial Function from the Graph

What can we conclude about the polynomial represented by the graph shown in (Figure) based on its intercepts and turning points?

Graph of an even-degree polynomial.

Figure 12.

Try It

What can we conclude about the polynomial represented by the graph shown in (Figure) based on its intercepts and turning points?

Graph of an odd-degree polynomial.

Figure 14.

Drawing Conclusions about a Polynomial Function from the Factors

Given the functionf(x)=4x(x+3)(x4),
determine the local behavior.

Try It

Given the functionf(x)=0.2(x2)(x+1)(x5),determine the local behavior.

Key Equations

general form of a polynomial function f(x)=anxn+...+a2x2+a1x+a0

Key Concepts

  • A power function is a variable base raised to a number power. See (Figure).
  • The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
  • The end behavior depends on whether the power is even or odd. See (Figure) and (Figure).
  • A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See (Figure).
  • The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See (Figure).
  • The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See (Figure) and (Figure).
  • A polynomial of degreen
    will have at mostn
    x-intercepts and at mostn1
    turning points. See (Figure), (Figure), (Figure), (Figure), and (Figure).

Section Exercises

Verbal

Explain the difference between the coefficient of a power function and its degree.

If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? Asx,f(x)
and asx,f(x).

Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.

f(x)=x5

f(x)=(x2)3

f(x)=xx4

f(x)=x2x21

f(x)=2x(x+2)(x1)2

f(x)=3x+1

For the following exercises, find the degree and leading coefficient for the given polynomial.

3x4

72x2

2x23x5+x6

x(4x2)(2x+1)

x2(2x3)2

For the following exercises, determine the end behavior of the functions.

f(x)=x4

f(x)=x3

f(x)=x4

f(x)=x9

f(x)=2x43x2+x1

f(x)=3x2+x2

f(x)=x2(2x3x+1)

f(x)=(2x)7

For the following exercises, find the intercepts of the functions.

f(t)=2(t1)(t+2)(t3)

g(n)=2(3n1)(2n+1)

f(x)=x416

f(x)=x3+27

f(x)=x(x22x8)

f(x)=(x+3)(4x21)

Graphical

For the following exercises, determine the least possible degree of the polynomial function shown.

Graph of an odd-degree polynomial.

Graph of an even-degree polynomial.
Graph of an odd-degree polynomial.

Graph of an odd-degree polynomial.
Graph of an odd-degree polynomial.

Graph of an even-degree polynomial.
Graph of an odd-degree polynomial.

Graph of an even-degree polynomial.

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

Graph of an odd-degree polynomial.

Graph of an equation.
Graph of an even-degree polynomial.

Graph of an odd-degree polynomial.
Graph of an odd-degree polynomial.

Graph of an equation.

Graph of an odd-degree polynomial.

Numeric

For the following exercises, make a table to confirm the end behavior of the function.

f(x)=x3

f(x)=x45x2

f(x)=x2(1x)2

f(x)=(x1)(x2)(3x)

f(x)=x510x4

Technology

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

f(x)=x3(x2)

f(x)=x(x3)(x+3)

f(x)=x(142x)(102x)

f(x)=x(142x)(102x)2

f(x)=x316x

f(x)=x327

f(x)=x481

f(x)=x3+x2+2x

f(x)=x32x215x

f(x)=x30.01x

Extensions

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

Theyintercept is(0,4).Thexintercepts are(2,0),(2,0).Degree is 2.

End behavior:asx,f(x),asx,f(x).

Theyintercept is(0,9).Thex-intercepts are(3,0),(3,0).Degree is 2.

End behavior:asx,f(x),asx,f(x).

Theyintercept is(0,0).Thexintercepts are(0,0),(2,0).Degree is 3.

End behavior:asx,f(x),asx,f(x).

Theyintercept is(0,1).Thexintercept is(1,0).Degree is 3.

End behavior:asx,f(x),asx,f(x).

Theyintercept is(0,1).There is noxintercept. Degree is 4.

End behavior:asx,f(x),asx,f(x).

Real-World Applications

For the following exercises, use the written statements to construct a polynomial function that represents the required information.

An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function ofd,the number of days elapsed.

A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function ofm,the number of minutes elapsed.

A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased byxinches and the width increased by twice that amount, express the area of the rectangle as a function ofx.

An open box is to be constructed by cutting out square corners of xinch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function ofx.

A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width (x).

Glossary

coefficient
a nonzero real number multiplied by a variable raised to an exponent
continuous function
a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph
degree
the highest power of the variable that occurs in a polynomial
end behavior
the behavior of the graph of a function as the input decreases without bound and increases without bound
leading coefficient
the coefficient of the leading term
leading term
the term containing the highest power of the variable
polynomial function
a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.
power function
a function that can be represented in the formf(x)=kxpwherekis a constant, the base is a variable, and the exponent,p, is a constant
smooth curve
a graph with no sharp corners
term of a polynomial function
anyaixiof a polynomial function in the formf(x)=anxn+...+a2x2+a1x+a0
turning point
the location at which the graph of a function changes direction