# Polynomial and Rational Functions

## Power Functions and Polynomial Functions


### Learning Objectives

1. Determine the degree of polynomials (IA 5.1.1).

2. 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). 

    

### Practice Makes Perfect


### 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  .

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  are polynomials.
There are three particular types of polynomials:
A monomial is a one term polynomial like  or 2.
A binomial is a two term polynomial like  .
A trinomial is a three term polynomial like  .
The degree of a polynomial in one variable is the highest exponent that appears on the variable in the polynomial. For example, the polynomial  has only one variable,  . The highest exponent on  is 2.


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


### Practice Makes Perfect


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



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


### Graph of Power Functions and End Behavior

ⓐ 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?
Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in .
The population can be estimated using the function 
     where 
       represents the bird population on the island 
       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 
      
      is
    
    and the function for the volume of a sphere with radius 
      
      is 
Both of these are examples of power functions because they consist of a coefficient, 
  or 
  multiplied by a variable 
  raised to a power.

### Identifying End Behavior of Power Functions

 shows the graphs of 
  and 
  which are all power functions with even, positive integer 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.
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 “ 
  approaches infinity,” which can be symbolically written as 
  we are describing a behavior; we are saying that 
  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 
  approaches positive or negative infinity, the 
  values increase without bound. In symbolic form, we could write 
 shows the graphs of 
  and 
  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.
These examples illustrate that functions of the form 
  reveal symmetry of one kind or another. First, in  we see that even functions of the form 
  even, are symmetric about the 
  axis. In  we see that odd functions of the form 
   odd, are symmetric about the origin.
For these odd power functions, as 
 
approaches negative infinity, 
 
decreases without bound. As 
 
approaches positive infinity, 
 
increases without bound. In symbolic form we write
The behavior of the graph of a function as the input values get very small ( 
  ) and get very large ( 
  ) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.
 shows the end behavior of power functions in the form 
  where 
  is a non-negative integer depending on the power and the constant.

### 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 
 
of the spill depends on the number of weeks 
 
that have passed. This relationship is linear.
We can combine this with the formula for the area 
 
of a circle.
Composing these functions gives a formula for the area in terms of weeks.
Multiplying gives the formula.
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.

### 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.

### 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 
 
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 .

### 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 
  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 .

### 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 
  degree is the product of 
  factors, so it will have at most 
  roots or zeros, or x-intercepts. The graph of the polynomial function of degree 
  must have at most 
  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.

### Key Equations


### Key Concepts

1. A power function is a variable base raised to a number power. See .
2. The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
3. The end behavior depends on whether the power is even or odd. See  and .
4. A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See .
5. 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 .
6. 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  and .
7. A polynomial of degree 
 
will have at most 
 
x-intercepts and at most 
 
turning points. See , , , , and .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.
For the following exercises, find the degree and leading coefficient for the given polynomial.
For the following exercises, determine the end behavior of the functions.
For the following exercises, find the intercepts of the functions.

### Graphical

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

### Numeric

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

### Technology

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

### 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.

### Real-World Applications

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