Module 5: RATIONAL FUNCTIONS
 Exploring Calculus

Anyone working in mathematics or in the sciences needs to understand and to be able to work with rational functions. We define a rational function as where P (x) and Q (x) are polynomial functions and Q (x)  0.

Part I
 

Unlike polynomial functions, rational functions have breaks in them and they may  flatten out  (i.e., level off to a certain y-value) as x approaches a very large positive or a very small negative value. You will investigate these situations and discover how to predict when these kinds of behavior will occur in a rational function.
 
 

Critical Thinking Questions
 
 

Display the graphs for  Qs. 1 – 3 below.
 

1. What are the x-values where there are breaks in the graph of the following rational functions?

    a.  y =

    b. y = 
 

2. Why do you think the breaks occur at these particular x-values?
 

3. Describe the algebraic method you would use to determine where the breaks occur in a rational function.
 

4. Given a rational function Click here to display the hints for this problem.Hint

    a. If it has a quadratic denominator, will it always have a break in its graph?

    if not, Give an example to illustrate your answer.

    b. Will it always have the same number of breaks in its graph if it has a cubic denominator?

Give examples to illustrate each possible case, and show the number of breaks in each case.

    c. What are the minimum and the maximum number of breaks it will have if its denominator is an odd degree polynomial?

Explain your answer.
 

5. What is the relationship between the degree of the function in the denominator of a rational function and the number of breaks in its graph?
 

6. This problem will help you discover how a rational function behaves on either side of the breaks in its graph.

The breaks are called vertical asymptotes.

   a1. Another way to say that a number gets larger and larger is to say that the number approaches .

   a2. Another way to say that a number gets more and more negative is to say that the number approaches .

    b. Consider the rational function f (x) =  .

Let x take the values 1.9, 1.99, 1.999 and 1.9999 as well as the values 2.1, 2.01, 2.001 and 2.0001. Record the resulting y-values in the tables below.
 

.

x

f (x)

.

x

f (x)

.

1.9

.

.

2.1

.

.

1.99

.

.

2.01

.

.

1.999

.

.

2.001

.

.

1.9999

.

.

2.0001

.

    c1. As x approaches_____  from the left, y approaches _____ .

    c2. As x approaches _____ from the right, y approaches _____.
 

7. Consider the rational function f (x) =  .

    a. As x approaches -2 from the left, what does the y-value approach?

    b. As x approaches -2 from the right, what does the y-value approach?

    c. As x approaches 2 from the left, what does the y-value approach?

    d. As x approaches 2 from the right, what does the y-value approach?
 

8 a. Do you think a function's graph always behaves the same way on either side of its vertical asymptote?

    b. For a function with one vertical asymptote, sketch freehand graphs representing each of the possible ways the function may behave on either side of this vertical asymptote.
 
 

Skill Exercises
 
 

Answer the following question without using the computer.

1. Let y =  .

    a. At what x-value does the break occur?

    b. How does the graph behave as you get closer and closer to the left side of the break?

    c. How does the graph behave as you get closer and closer to the right side of the break?
 

     Display the graph for  Qs. 2--4 .
 

2. Let y =  .

    a. Are there any vertical asymptotes?

    b. Why or why not?
 

3. Let y =  .

    a. Write the equations of the vertical asymptotes.

    b. Describe the behavior of the graph about each vertical asymptote.
 

4. Let y =  .

    a. Determine the equations of the vertical asymptotes.

Show your work.

    b. Describe the behavior of the graph about each vertical asymptote.
 

 

Three parts to this module ...





 

Part II
 
 

Rational functions differ from polynomial functions in another significant way. Whereas it is always true that a polynomial function grows very large or very small as x increases (or decreases), a rational function may flatten out as x becomes very large or very small.
 
 

Critical Thinking Questions
 
 

1.  Display the graphs to answer the following questions.

    Hint: Use the zoom feature in the graph to look at the expressions for large (>100) values of x in questions 1a.--c. below.

   a. For large values of x , is there a significant difference between the expression  "3x + 4"  and the expression "3x" ?

   b. For large values of x , is there a significant difference between the expression  "2x2 + 3x + 4" and the expression "2x2 " ?

   c. For large values of x , is there a significant difference between the expression  "2x3 + 6x2 + 5x + 3" and the expression "2x3 " ?

   d. Which term in a polynomial defines its behavior as x approaches (denoted)?

   e. Is your answer above the same as x approaches  (denoted)?
 

2.