# Polynomial and Rational Functions

## Graphs of Polynomial Functions


### Learning Objectives

1. Recognize and use the appropriate method to factor a polynomial completely (IA 6.4.1)
2. Solve a quadratic equation by factoring (IA 6.5.2)

### Objective 1: Recognize and use the appropriate method to factor a polynomial completely (IA 6.4.1).

The following outline provides a good strategy for factoring polynomials.

### Practice Makes Perfect

Recognize and use the appropriate method to factor a polynomial completely.


### Objective 2: Solve a quadratic equation by factoring (IA 6.5.2)

If  , where  and  represent real numbers. What can you say about  and  ?

The Zero Product Property states that if  , then  ,  or  , or both.


We can use this property to solve equations.


### Practice Makes Perfect


Solve


### Practice Makes Perfect


Solve a quadratic equation by factoring.


Use the zero factor property to solve each of the following exercises.

The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in .
The revenue can be modeled by the polynomial function
where 
     represents the revenue in millions of dollars and 
       represents the year, with 
       corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.

### Recognizing Characteristics of Graphs of Polynomial Functions

Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous.  shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.

###  Using Factoring to Find Zeros of Polynomial Functions

Recall that if 
       is a polynomial function, the values of 
         for which 
         are called zeros of 
         If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.
We can use this method to find 
  intercepts because at the 
  intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases:
1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
2. The polynomial is given in factored form.
3. Technology is used to determine the intercepts.

### Identifying Zeros and Their Multiplicities

Graphs behave differently at various x-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and "bounce" off.


Suppose, for example, we graph the function shown.
Notice in  that the behavior of the function at each of the x-intercepts is different.
The x-intercept 
      
      is the solution of equation 
         The graph passes directly through the x-intercept at 
  The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.
The x-intercept 
       is the repeated solution of equation 
         The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.
The factor is repeated, that is, the factor 
       appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor, 
         has multiplicity 2 because the factor 
         occurs twice.
The x-intercept 
       is the repeated solution of factor 
         The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic—with the same S-shape near the intercept as the toolkit function 
         We call this a triple zero, or a zero with multiplicity 3.
For zeros with even multiplicities, the graphs touch or are tangent to the x-axis. For zeros with odd multiplicities, the graphs cross or intersect the x-axis. See  for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.
For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the x-axis.
For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x-axis.

### Determining End Behavior

As we have already learned, the behavior of a graph of a polynomial function of the form
will either ultimately rise or fall as 
      
      increases without bound and will either rise or fall as 
        
      decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.
Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, 
  is an even power function, as 
  increases or decreases without bound, 
  increases without bound. When the leading term is an odd power function, as 
  decreases without bound, 
  also decreases without bound; as 
  increases without bound, 
  also increases without bound. If the leading term is negative, it will change the direction of the end behavior.  summarizes all four cases.

###  Understanding the Relationship between Degree and Turning Points

In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function 
  in . The graph has three turning points.
This function 
 
is a 4th degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.

### Graphing Polynomial Functions

We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.

### Using the Intermediate Value Theorem

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x-axis, we can confirm that there is a zero between them. Consider a polynomial function 
  whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers 
  and 
  in the domain of 
  if 
  and 
  then the function 
  takes on every value between 
  and 
  (While the theorem is intuitive, the proof is actually quite complicated and requires higher mathematics.) We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function 
  at 
  lies above the 
  axis and another point at 
  lies below the 
  axis, there must exist a third point between 
  and 
  where the graph crosses the 
  axis. Call this point 
  This means that we are assured there is a solution 
  where 
  
In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the 
  axis.  shows that there is a zero between 
 
and 
 


###  Writing Formulas for Polynomial Functions

Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an x-intercept where each factor is equal to zero, we can form a function that will pass through a set of x-intercepts by introducing a corresponding set of factors.

### Using Local and Global Extrema

With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.
Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a global maximum or a global minimum. These are also referred to as the absolute maximum and absolute minimum values of the function.

### Key Concepts

1. Polynomial functions of degree 2 or more are smooth, continuous functions. See .
2. To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See , , and .
3. Another way to find the 
  intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the 
  axis. See .
4. The multiplicity of a zero determines how the graph behaves at the 
  intercepts. See .
5. The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
6. The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
7. The end behavior of a polynomial function depends on the leading term.
8. The graph of a polynomial function changes direction at its turning points.
9. A polynomial function of degree 
 
has at most 
 
turning points. See .
10. To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most 
 
turning points. See  and .
11. Graphing a polynomial function helps to estimate local and global extremas. See .
12. The Intermediate Value Theorem tells us that if 
 
have opposite signs, then there exists at least one value 
 
between 
 
and 
 
for which 
 
See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the 
  or t-intercepts of the polynomial functions.
For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.
For the following exercises, find the zeros and give the multiplicity of each.

### Graphical

For the following exercises, graph the polynomial functions. Note 
  and 
  intercepts, multiplicity, and end behavior.
For the following exercises, use the graphs to write the formula for a polynomial function of least degree.
For the following exercises, use the graph to identify zeros and multiplicity.
For the following exercises, use the given information about the polynomial graph to write the equation.

### Technology

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

### Extensions

For the following exercises, use the graphs to write a polynomial function of least degree.

### Real-World Applications

For the following exercises, write the polynomial function that models the given situation.