Rational Functions (3.10)
In-Class Exercises
These exercises should be completed and understood before attempting the homework exercises.
1. A Rational Function is a function formed by taking the ratio of two polynomials. The reciprocal functions we studied in 3.8 are simple examples of a rational functions.
Consider the rational function
 x

1
( x

2)( x

2)
.
Perform the analysis below without your calculator. As you learn more about the shape of this function, incorporate your findings by sketching the graph on the grid at the bottom of this page. a. Find the zeros ( x -intercepts) of R ( x ) and y -intercept of R ( x ). b. Why is the domain of R ( x ): ( , 2) ( 2, 2)

(2, ) ? c. Use the tables of values at right to analyze the behavior of R ( x ) near x = 2. d. Analyze the end-behavior of R ( x ) by mentally plugging in large positive and negative x -values into the function. Summarize your thinking using limit notation: lim ( ) x


______ and lim ( ) x


______ e. Figure out what is happening at x = −2 to conclude your analysis of this function.









 x R ( x )
         
2.1 2.68
2.01 25.187
2.001 250.19
2.00 Error










1.999 −249.81
1.99
1.90
−24.81
−2.38
After incorporating the information above to make an informed sketch, check yourself by graphing
R ( x ) on your calculator and/or Wolfram Alpha.

Rational Functions (3.10)
2. Based on similar reasoning to that used in Exercise 1, a student might conclude that the rational function y

( x

2)( x

3) x
2  
2
has xintercepts at x
 
2 and 3. Why is this wrong? Are there any vertical asymptotes?
3. Review/Summary of Rational Functions from Algebra 2 and from your work in Exercises 1-2.
When Graphing a Rational Function in the form R ( x )

N ( x )
D ( x )
, where N( x ) is the polynomial in the numerator and D( x ) is the polynomial in the denominator, consider the following ideas:
Note: Use the notation deg(N) to mean “degree of the numerator” and deg(D) to mean “degree of the denominator.” a.
There are vertical asymptotes on the graph where … b.
The end behavior asymptotes will be: i.
y = 0 when… deg(N) < deg(D) ii.
y = constant (non-zero) when… iii.
y = linear function (1 st
degree poly) when… iv.
y = quadratic function when…
…and so on. Thus, the degree of the end behavior asymptote is: c.
The zeros are found by… d.
The domain is… e.
There is a hole/deleted point when… i.
We find the y -value of the deleted point by…

Rational Functions (3.10)
Homework Exercises
Try these problems after you have completed the in-class exercises. Problems marked with an asterisk
(*) are written presentation problems which you may submit next class.
4. Identify the x -intercepts, end behavior, and the location of any vertical asymptotes for each of the following rational functions and use the process in exercise 1 to sketch a graph in the space below: a. y

( x

2)( x

3)
( x

1)( x

6) b.
y

( x

2)( x

3)
2 x
2 
11 x

15
5. Write a possible rational function formula for each of the graphs below:
a.
            







 y








      
b.










 x
      


 y













            x

6. Consider the rational function y
 x
3 
2 x
2 
7 x
2 
4
. a. Use long division to verify that x
3 
2 x
2 
7 x
2 
4 x

4 x

1 x
2 
4
.
Rational Functions (3.10) b. What happens to value of lim x

4 x

1 x
2 
4
4 x

1 x
2 
4
?
as values of x get bigger and bigger? In other words, what is the b.
The function y
 x 3 
2 x 2 
7 x
2 
4
has problems at

2 because at those values the denominator is
0, making the fraction undefined. Explain why the graph of complicated on the interval x [ 10,10] y
 x
3
, but simple on the interval

2 x
2 
7 x x
2 
4
looks
[ 100,100] . x [ 10,10] and y [ 10,10] x [ 100,100] and y [ 100,100] d. What is the equation of the line that dominates the graph to the right above? How is it related to the long division that started this Exercise?

7. A manufacturer wants to sell .5 liter (500 cm 3 ) cans of juice.
There are lots of different cans that can hold .5 liters – short wide cans, tall skinny cans, etc. – but the manufacturer wants to
Rational Functions (3.10) use the smallest amount of material possible to make this can.
In other words, the manufacturer wants to minimize the surface area needed to make the can.
You may need to remember/look up the surface area and volume of a cylinder to proceed with this problem. a.
Write an equation for the surface area, S , of the can in terms of its radius and height. b.
Using a formula for S ( r ), determine the dimensions of the optimal can. (Hint: explain why
  2
500 r h for the desired can of radius r and height h .)
8 * . Consider the rational function y
 x 4 x
2 
2 x
2 
1
.
For inputs sufficiently far away from its asymptotes, this function’s outputs get closer and closer to those of a certain quadratic polynomial. Identify the quadratic asymptote that this rational function tends toward and use the method highlighted in Exercises 1 -3 to sketch a graph of this function without a calculator. ( hint: draw in the quadratic asymptote as a graphing aid )