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College Algebra with Corequisite Support

5.2 Power Functions and Polynomial Functions

College Algebra with Corequisite Support5.2 Power Functions and Polynomial Functions
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Chapter Review
      1. Key Terms
      2. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 7.1 Systems of Linear Equations: Two Variables
    3. 7.2 Systems of Linear Equations: Three Variables
    4. 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 7.4 Partial Fractions
    6. 7.5 Matrices and Matrix Operations
    7. 7.6 Solving Systems with Gaussian Elimination
    8. 7.7 Solving Systems with Inverses
    9. 7.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 8.1 The Ellipse
    3. 8.2 The Hyperbola
    4. 8.3 The Parabola
    5. 8.4 Rotation of Axes
    6. 8.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 9.1 Sequences and Their Notations
    3. 9.2 Arithmetic Sequences
    4. 9.3 Geometric Sequences
    5. 9.4 Series and Their Notations
    6. 9.5 Counting Principles
    7. 9.6 Binomial Theorem
    8. 9.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
  12. Index

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.

Corequisite Skills

Learning Objectives

  • Determine the degree of polynomials (IA 5.1.1).
  • Simplify expressions using properties of exponents (IA 5.2.1).

Objective 1: Simplify expressions using the properties of exponents (IA 5.2.1).

An exponential expression is an expression that has exponents (or powers).

A graphic pointing out the parts of an expression with a power of 3, where 2 is the coefficient, a is the base, and 3 is the power.

Properties of Exponents

Name of the Property Property Meaning
Product Property xm·xn=xm+nxm·xn=xm+n Keep the base, add the exponents
Power Property (xn)m=xn·m(xn)m=xn·m Keep the base, multiply the exponents
Quotient Property xmxn=xm-nxmxn=xm-n Keep the base, subtract the exponents (top minus bottom).
Power Property for products and quotients (xy)m=xmym(xy)m=xmym(xy)m=xmym(xy)m=xmym Raise each base to the power.
Zero Exponent x0=1x0=1 Any non-zero number or variable raised to the power 0 is equal to 1.
Negative exponent x-n=1xn1x-n=xnx-n=1xn1x-n=xn Move the base from numerator to denominator or vice versa to make an exponent positive.

Example 1

Use the property of exponents to simplify each of the following:

(3x2y)4(3x2y)4

Practice Makes Perfect
1.

Use properties of exponents to simplify.

(3xy2x2y-3)4(3xy2x2y-3)4

Objective 2: Determine the degree of polynomials (IA 5.1.1).

A term can be a number like -2, a variable like x, or a product of numbers and variables like 7x37x3 .

A polynomial is an expression with more than one term with no variables in the denominator and no negative exponents. Any exponent on the variables must be whole numbers.

For example 2x+1, 4x2-5x+32x+1, 4x2-5x+3 are polynomials.

There are three particular types of polynomials:

A monomial is a one term polynomial like 7x37x3 or 2.

A binomial is a two term polynomial like 7x3+27x3+2 .

A trinomial is a three term polynomial like 4x2-5x+34x2-5x+3 .

The degree of a polynomial in one variable is the highest exponent that appears on the variable in the polynomial. For example, the polynomial 5x2+7x-45x2+7x-4 has only one variable, xx . The highest exponent on xx is 2.

Example 2

Determine the degree of polynomials.

If the expression is a polynomial, identify it as a monomial, binomial, trinomial, or other polynomial, and then find the degree of each polynomial. If it is not a polynomial, state this and give a reason why.

Expression How many variables Classification
Polynomial? If not, why? If it is a polynomial, how many terms are there? Classify as a monomial, binomial or trinomial. Degree of polynomial?
5x2-2x+75x2-2x+7 1 yes 3, trinomial 2
y-2.5y-2.5 1 yes 2, binomial 1
-5x2y-5x2y 2 yes 1, monomial 3
4x2-2x34x2-2x3 1 No, there is a variable in the denominator
b3+13b2-12b+12b3+13b2-12b+12 1 yes 4, polynomial or quadrinomial 3
x-2-3xy-10y2x-2-3xy-10y2 1 No, there is a negative exponent on x
Power Functions

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.

A power function is a function that can be represented in the form f(x)=kxpf(x)=kxp where k  and p  are real numbers, and k  is known as the coefficient.

Example 3

  1. f(x)=x2f(x)=x2
  2. f(x)=5x4f(x)=5x4
  3. f(x)=x2/3f(x)=x2/3
  4. f(x)=3xf(x)=3x
  5. f(x)=-5x-2f(x)=-5x-2
Practice Makes Perfect

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial, then find the degree of each polynomial.

2.
Expression How many variables Classification
Polynomial? If not, why? If it is a polynomial, how many terms are there? Classify as a monomial, binomial or trinomial. Degree of polynomial?
5x4-25x4-2
2x3-6x2+12x3-6x2+1
-2x2/3-5x-2x2/3-5x
12x312x3
-3x6-5x4+4x3-3x6-5x4+4x3
3x-3+23x-3+2

Determine whether the following functions are power functions. If they are not, state it and the reason why.

3.

f(x)=x3f(x)=x3

4.

f(x)=2x7f(x)=2x7

5.

f(x)=4xf(x)=4x

6.

f(x)=7x-3f(x)=7x-3

7.

f(x)=-5x3/2f(x)=-5x3/2

Graph of Power Functions and End Behavior
Graphs of even power functions Graphs of odd power functions
Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.
Table 5 Figure 1

Graphs of f(x)=x2, f(x)=x4, f(x)=x6f(x)=x2, f(x)=x4, f(x)=x6

Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.
Figure 2

Graphs of f(x)=x3, f(x)=x5, f(x)=x7f(x)=x3, f(x)=x5, f(x)=x7

What are the similarities in the graphs of even power functions?
What are the similarities in the graphs of the odd power functions?
What are the differences between the graphs of the even power functions and the odd power 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 Table 1.

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

The population can be estimated using the function P(t)=0.3 t 3 +97t+800, P(t)=0.3 t 3 +97t+800, where P(t) P(t) represents the bird population on the island t t years 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 radius r r is

A(r)=π r 2 A(r)=π r 2

and the function for the volume of a sphere with radius r r is

V(r)= 4 3 π r 3 V(r)= 4 3 π r 3

Both of these are examples of power functions because they consist of a coefficient, π π or 4 3 π, 4 3 π, multiplied by a variable r r raised to a power.

Power Function

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

f(x)=k x p f(x)=k x p

where k k and p p are real numbers, and k k is known as the coefficient.

Q&A

Is f(x)= 2 x f(x)= 2 x a 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.

Example 1

Identifying Power Functions

Which of the following functions are power functions?

f(x) = 1 Constant function f(x) = x Identify function f(x) = x2 Quadratic function f(x) = x3 Cubic function f(x) = 1x Reciprocal function f(x) = 1x2 Reciprocal squared function f(x) = x Square root function f(x) = x3 Cube root function f(x) = 1 Constant function f(x) = x Identify function f(x) = x2 Quadratic function f(x) = x3 Cubic function f(x) = 1x Reciprocal function f(x) = 1x2 Reciprocal squared function f(x) = x Square root function f(x) = x3 Cube root function

Try It #1

Which functions are power functions?

f(x) = 2x4 x 3 g(x) = x 5 +5 x 3 h(x) = 2 x 5 1 3 x 2 +4 f(x) = 2x4 x 3 g(x) = x 5 +5 x 3 h(x) = 2 x 5 1 3 x 2 +4

Identifying End Behavior of Power Functions

Figure 2 shows the graphs of f(x)= x 2 ,g(x)= x 4 f(x)= x 2 ,g(x)= x 4 and h(x)= x 6 , h(x)= x 6 , 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 symbol for positive infinity and −∞ −∞ for negative infinity. When we say that “ x x approaches infinity,” which can be symbolically written as x, x, we are describing a behavior; we are saying that x x is 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 as x x approaches positive or negative infinity, the f( x ) f( x ) values increase without bound. In symbolic form, we could write

as x±, f(x) as x±, f(x)

Figure 3 shows the graphs of f(x)= x 3 ,g(x)= x 5 , f(x)= x 3 ,g(x)= x 5 , and h(x)= x 7 , h(x)= x 7 , 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 form f( x )= x n f( x )= x n reveal symmetry of one kind or another. First, in Figure 2 we see that even functions of the form f( x )= x n n f( x )= x n n even, are symmetric about the y- y- axis. In Figure 3 we see that odd functions of the form f( x )= x n n f( x )= x n n  odd, are symmetric about the origin.

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

as x, f(x)  as x, f(x) as x, f(x)  as x, f(x)

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

Figure 4 shows the end behavior of power functions in the form f(x)=k x n f(x)=k x n where n n is 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 function f(x)=k x n f(x)=k x n where n n is 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 4 to identify the end behavior.

Example 2

Identifying the End Behavior of a Power Function

Describe the end behavior of the graph of f(x)= x 8 . f(x)= x 8 .

Example 3

Identifying the End Behavior of a Power Function.

Describe the end behavior of the graph of f(x)= x 9 . f(x)= x 9 .

Analysis

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

x x f( x ) f( x )
–10 1,000,000,000
–5 1,953,125
0 0
5 –1,953,125
10 –1,000,000,000
Table 2

We can see from Table 2 that, when we substitute very small values for x, x, the output is very large, and when we substitute very large values for x, x, the output is very small (meaning that it is a very large negative value).

Try It #2

Describe in words and symbols the end behavior of f(x)=5 x 4 . f(x)=5 x 4 .

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 radius r r of the spill depends on the number of weeks w w that have passed. This relationship is linear.

r(w)=24+8w r(w)=24+8w

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

A(r)=π r 2 A(r)=π r 2

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 A(w) = A(r(w)) = A(24+8w) = π (24+8w) 2

Multiplying gives the formula.

A(w)=576π+384πw+64π w 2 A(w)=576π+384πw+64π w 2

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

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

f(x)= a n x n + ... + a 2 x 2 + a 1 x+ a 0 f(x)= a n x n + ... + a 2 x 2 + a 1 x+ a 0

This is called the general form of a polynomial function. Each a i a i is a coefficient and can be any real number, but a n a n cannot = 0. Each expression a i x i a i x i is a term of a polynomial function.

Example 4

Identifying Polynomial Functions

Which of the following are polynomial functions?

f(x) = 2 x 3 3x+4 g(x) = x( x 2 4) h(x) = 5 x+2 f(x) = 2 x 3 3x+4 g(x) = x( x 2 4) h(x) = 5 x+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 of x x to determine the degree function.
  2. Identify the term containing the highest power of x x to find the leading term.
  3. Identify the coefficient of the leading term.

Example 5

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+2 x 2 4 x 3 g(t) = 5 t 5 2 t 3 +7t h(p) = 6p p 3 2 f(x) = 3+2 x 2 4 x 3 g(t) = 5 t 5 2 t 3 +7t h(p) = 6p p 3 2
Try It #3

Identify the degree, leading term, and leading coefficient of the polynomial f(x)=4 x 2 x 6 +2x6. f(x)=4 x 2 x 6 +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 as x x 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 Table 3.

Polynomial Function Leading Term Graph of Polynomial Function
f(x)=5 x 4 +2 x 3 x4 f(x)=5 x 4 +2 x 3 x4 5 x 4 5 x 4 Graph of f(x)=5x^4+2x^3-x-4.
f(x)=2 x 6 x 5 +3 x 4 + x 3 f(x)=2 x 6 x 5 +3 x 4 + x 3 2 x 6 2 x 6 Graph of f(x)=-2x^6-x^5+3x^4+x^3.
f(x)=3 x 5 4 x 4 +2 x 2 +1 f(x)=3 x 5 4 x 4 +2 x 2 +1 3 x 5 3 x 5 Graph of f(x)=3x^5-4x^4+2x^2+1.
f(x)=6 x 3 +7 x 2 +3x+1 f(x)=6 x 3 +7 x 2 +3x+1 6 x 3 6 x 3 Graph of f(x)=-6x^3+7x^2+3x+1.
Table 3

Example 6

Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in Figure 7.

Graph of an odd-degree polynomial.
Figure 7

Try It #4

Describe the end behavior, and determine a possible degree of the polynomial function in Figure 8.

Graph of an even-degree polynomial.
Figure 8

Example 7

Identifying End Behavior and Degree of a Polynomial Function

Given the function f(x)=3 x 2 (x1)(x+4), f(x)=3 x 2 (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 #5

Given the function f(x)=0.2(x2)(x+1)(x5), f(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, a 0 ). (0, a 0 ). 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 9.

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 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.

Example 8

Determining the Intercepts of a Polynomial Function

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

Example 9

Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function f(x)= x 4 4 x 2 45, f(x)= x 4 4 x 2 45, determine the y- and x-intercepts.

Try It #6

Given the polynomial function f(x)=2 x 3 6 x 2 20x, f(x)=2 x 3 6 x 2 20x, 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 of nth nth degree is the product of n n factors, so it will have at most n n roots or zeros, or x-intercepts. The graph of the polynomial function of degree n n must have at most n1 n1 turning 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 degree n n will have, at most, n n x-intercepts and n1 n1 turning points.

Example 10

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 for f(x)=3 x 10 +4 x 7 x 4 +2 x 3 . f(x)=3 x 10 +4 x 7 x 4 +2 x 3 .

Try It #7

Without graphing the function, determine the maximum number of x-intercepts and turning points for f(x)=10813 x 9 8 x 4 +14 x 12 +2 x 3 . f(x)=10813 x 9 8 x 4 +14 x 12 +2 x 3 .

Example 11

Drawing Conclusions about a Polynomial Function from the Graph

What can we conclude about the polynomial represented by the graph shown in Figure 12 based on its intercepts and turning points?

Graph of an even-degree polynomial.
Figure 12

Try It #8

What can we conclude about the polynomial represented by the graph shown in Figure 14 based on its intercepts and turning points?

Graph of an odd-degree polynomial.
Figure 14

Example 12

Drawing Conclusions about a Polynomial Function from the Factors

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

Try It #9

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

5.2 Section Exercises

Verbal

1.

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

2.

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

3.

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

4.

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

5.

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

Algebraic

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

6.

f(x)= x 5 f(x)= x 5

7.

f(x)= ( x 2 ) 3 f(x)= ( x 2 ) 3

8.

f(x)=x x 4 f(x)=x x 4

9.

f(x)= x 2 x 2 1 f(x)= x 2 x 2 1

10.

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

11.

f(x)= 3 x+1 f(x)= 3 x+1

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

12.

3x 4 3x 4

13.

72 x 2 72 x 2

14.

2 x 2 3 x 5 +x6  2 x 2 3 x 5 +x6 

15.

x( 4 x 2 )(2x+1) x( 4 x 2 )(2x+1)

16.

x 2 ( 2x3 ) 2 x 2 ( 2x3 ) 2

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

17.

f( x )= x 4 f( x )= x 4

18.

f( x )= x 3 f( x )= x 3

19.

f( x )= x 4 f( x )= x 4

20.

f( x )= x 9 f( x )= x 9

21.

f(x)=2 x 4 3 x 2 +x1  f(x)=2 x 4 3 x 2 +x1 

22.

f(x)=3 x 2 +x2 f(x)=3 x 2 +x2

23.

f(x)= x 2 ( 2 x 3 x+1 ) f(x)= x 2 ( 2 x 3 x+1 )

24.

f(x)= (2x) 7 f(x)= (2x) 7

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

25.

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

26.

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

27.

f(x)= x 4 16 f(x)= x 4 16

28.

f(x)= x 3 +27 f(x)= x 3 +27

29.

f(x)=x( x 2 2x8 ) f(x)=x( x 2 2x8 )

30.

f(x)=(x+3)( 4 x 2 1 ) f(x)=(x+3)( 4 x 2 1 )

Graphical

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

31.
Graph of an odd-degree polynomial.
32.
Graph of an even-degree polynomial.
33.
Graph of an odd-degree polynomial.
34.
Graph of an odd-degree polynomial.
35.
Graph of an odd-degree polynomial.
36.
Graph of an even-degree polynomial.
37.
Graph of an odd-degree polynomial.
38.
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.

39.
Graph of an odd-degree polynomial.
40.
Graph of an equation.
41.
Graph of an even-degree polynomial.
42.
Graph of an odd-degree polynomial.
43.
Graph of an odd-degree polynomial.
44.
Graph of an equation.
45.
Graph of an odd-degree polynomial.

Numeric

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

46.

f(x)= x 3 f(x)= x 3

47.

f(x)= x 4 5 x 2 f(x)= x 4 5 x 2

48.

f(x)= x 2 ( 1x ) 2 f(x)= x 2 ( 1x ) 2

49.

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

50.

f(x)= x 5 10 x 4 f(x)= x 5 10 x 4

Technology

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

51.

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

52.

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

53.

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

54.

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

55.

f(x)= x 3 16x f(x)= x 3 16x

56.

f(x)= x 3 27 f(x)= x 3 27

57.

f(x)= x 4 81 f(x)= x 4 81

58.

f(x)= x 3 + x 2 +2x f(x)= x 3 + x 2 +2x

59.

f(x)= x 3 2 x 2 15x f(x)= x 3 2 x 2 15x

60.

f(x)= x 3 0.01x f(x)= x 3 0.01x

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.

61.

The y- y- intercept is (0,4). (0,4). The x- x- intercepts are (2,0)(2,0), (2,0). (2,0). Degree is 2.

End behavior: as xx, f(x)f(x); as xx, f(x). f(x).

62.

The y- y- intercept is (0,9). (0,9). The x- x- intercepts are (3,0)(3,0), (3,0). (3,0). Degree is 2.

End behavior: as xx, f(x)f(x), as xx, f(x). f(x).

63.

The y- y- intercept is (0,0). (0,0). The x- x- intercepts are (0,0)(0,0), (2,0). (2,0). Degree is 3.

End behavior: as xx, f(x)f(x), as xx, f(x). f(x).

64.

The y- y- intercept is (0,1). (0,1). The x- x- intercept is (1,0). (1,0). Degree is 3.

End behavior: as xx, f(x)f(x), as xx, f(x). f(x).

65.

The y- y- intercept is (0,1). (0,1). There is no x- x- intercept. Degree is 4.

End behavior: as xx, f(x)f(x), as xx, f(x). f(x).

Real-World Applications

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

66.

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 of d, d, the number of days elapsed.

67.

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 of m, m, the number of minutes elapsed.

68.

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

69.

An open box is to be constructed by cutting out square corners of x- x- inch 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 of x. x.

70.

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 x ).

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