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Activity: Analyzing local and global behavior of rational functions
A Review of Local Behavior and Introduction to Jump Discontinuity
When we observe the local behavior of a function f (x) around a specific
value x  p , we look to see if f (x) approaches a particular value as x
approaches p from two directions - the left and the right.
Consider the graph of the function f (x) shown below.
Say we wish to examine the local behavior of the function f (x) around the
value x  1 . We start by examining the behavior of the function as x
approaches 1 from the left (i.e. x  1 ).
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In the above example, we say that "as x approaches 1 from the left, f (x)
approaches 1" (i.e. if x  1 , then f (x )  1 ). Likewise "as x approaches 1
from the right, f (x) approaches 3" (i.e. if x  1 , then f (x)  3 ).
Since the function doesn't approach the same value from both sides of x  1 ,
we say that f (x) has a jump discontinuity at x  1 . If the ant wished to
switch positions from one side of x  1 to the other, we think she would find
this terminology quite appropriate!
Rational Functions Activity 1
In this activity, you will use your graphing calculator to explore rational
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functions - functions of the form y 
f (x)
g(x)
, with f (x) and g(x ) polynomials.
The table on the next page contains several rational functions. Complete
the missing values of the table by analyzing the local behavior of each
function at the location p specified in the table.
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Rational
Function
1
x 3
1
2
1
3
1
4
, n
1
x  a
Behavior of f (x) as
x  p (from right)
3
f (x )  
f (x)  
3
x  3
2n
Behavior of f (x) as
x  p (from left)
3
x  3
x  a
p
3
x  3
1
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2 n 1
, n
a
a
Solution Discussion. Below we illustrate how we completed the first row of
the table using a graphing calculator. Begin by typing the rational function
1
into the y= editor.
f (x) 
x 3
We'll examine local behavior around x  3 using two different methods - with
tables and with graphs.
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Using a Table. Set your calculator table to "ask mode", then type in values
for x that approach 3 from the left. Examine what happens to values of y as
x  3 . Similarly, type in values for x that approach 3 from the right.
Examine what happens to values of y as x  3 .
Tracing a graph. Alternately, graph the function. We chose window
settings that focused on values near x  3 .
Next, press 2 ND Trace 1: value . Manually enter x values that approach 3
from the left (see below).
Notice the circled y-values on the bottom of each screen. What appears to
happen to these values as x  3 ? Similarly, you can explore local behavior
as x approaches 3 from the right by manually entering values of x greater
than 3.
Questions for Further Thought
1) Based on the information you compiled in the table, what (if any)
generalizations can you make about rational functions of the form
1
, k .
k
x  a
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2) Using your knowledge of algebra, make predictions about the local
x 3
behavior of the rational function f (x ) 
near the value
(x  3)(x  2)
x  2 . Use your graphing calculator to check your predictions.
3) Consider the graph of a rational function of the form
1
where a1 , a2 , a3  . Use the graph to
f (x) 
(x  a1 )(x  a2 )(x  a3 )
predict the values of a1 , a2 and a3 .
Check your prediction using graphing features of your calculator.
4) Construct your own rational function f (x) such that f (x) satisfies the
following 4 properties:
(a) As x  3 , y  4
(b) As x  0 , y  
(c) As x  0 , y  
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(d) f (6)  1
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Introduction to Removable Discontinuity
In the previous activity, you focused much of your attention on rational
functions with a constant numerator. Rational functions of the form
1
have jump discontinuities at a1 , a2 , , an
f (x ) 
( x  a1 )( x  a2 ) ( x  an )
(i.e. the roots of the denominator).
In this section, we'll examine rational functions with linear factors in both
the numerator and denominator. As you will see, rational functions such as
these may or may not have jump discontinuities.
An Example
Think about it. Consider the rational function f (x ) 
you asked a student to "simplify" the algebraic expression
(x  2)(x  5)
(x  5)
. If
(x  2)(x  5)
, what
(x  5)
would they likely answer? What does this suggest about the graph of f (x) ?
Dig Deeper. With your graphing calculator, construct a graph of f (x) .
Your graph should resemble that of the linear function f (x)  x  2 , with one
small difference. Complete the following statements to learn more about the
(x  2)(x  5)
rational function f (x ) 
.
(x  5)
(a) As x  5 , y  _____ (fill in the blank)
(b) As x  5 , y  _____ (fill in the blank)
(c) Does f (x) have a jump discontinuity at x  5 ? Why or why not?
(d) f (5)  _____ (fill in the blank)
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As you probably discovered, the graph of the rational function doesn't
"jump" at x  5 . As the following screenshots suggest, as x  5 , f (x)  3 .
Since the same y-value is approached from both sides, the function doesn't
"jump".
However, since evaluating the function at x  5 results in division by zero
the function itself is undefined at x  5 .
Because the graph breaks at x  5 , we still say that the function is
discontinuous at x  5 . The graph highlights an example of a removable
discontinuity (some find it helpful to think of "removable" discontinuities
as values of x where "holes" occur in the graph of the function).
Rational Functions Activity 2
In this activity, you will use your graphing calculator to explore rational
functions with linear factors in the numerator and denominator. Some of
the functions have removable discontinuities. Some have jump
discontinuities. Some have both types.
The table on the next page lists several rational functions. Complete the
missing values of the table by analyzing the local behavior of each function at
the locations specified in the table. Then determine whether the function has
a jump or removable discontinuity at the specified location.
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Rational
Function
p
x 3
3
x 3
(x  3)
Behavior of f (x)
as x  p (from
Behavior of f (x)
as x  p (from
left)
right)
f (x )  
f (x)  
Type of
discontinuity
at x  p
Removable
3
(x  3)2
(x  3)2
3
(x  3)
(x  3)
3
(x  3)3
(x  2)
2
(x  2)5
(x  2)5
2
(x  2)
(x  2)5
2
(x  2)5
Questions for Further Thought
5) Based on the information you compiled in the table, what (if any)
generalizations can you make about rational functions of the form
(a)
(b)
(x  a)m
x  a 
n
(x  a)m

xa

n
, m  n  0 and m  n an even integer.
, m  n  0.
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(x  a)m
x  a 
n
, m  n  0 and m  n an odd integer.
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6) Based on the information you compiled in the table, what (if any)
generalizations can you make about rational functions of the form
(a)
(b)
(x  a)m
x  a 
n
(x  a)m
x  a 
n
, m  n  0 and m  n an even integer.
, m  n  0 and m  n an odd integer.
7) Using your knowledge of algebra, make predictions about the local
( x 2  6x  9)
behavior of the rational function f ( x ) 
near the value x  3 .
( x  3)
Use your graphing calculator to check your predictions.
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8) Consider the rational function f ( x ) 
( x 2  2x  8)
.
( x 2  5x  6)
(a) Factor the numerator and denominator of the function. At what
values of x will the function have a removable discontinuity?
(b) Using factored form of numerator and denominator, simplify the
function by "cancelling out" like factors. The resulting function should
(x  a)
be in the form
with a  b .
(x  b)
Think about it. You can use the "long division algorithm" to write
the simplified form of the function in a way that clearly illustrates
locations of jump discontinuities. Here's an example using the rational
x 1
function g(x) 
.
x2
1
x  2 x 1
x2
-------1
so
x 1
1
1
x2
x2
there's a jump at x=2.
Note that graph of g(x ) is actually the graph of
1
x
translated to the
right 2 units and translated up 1 unit.
(b) Complete similar steps to find the jump discontinuities of the function
you wrote in part (b).
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9) Consider the graph below. An equation for the function that generates
(x  a)(x  b)(x  c)
the graph may be written in the form f (x ) 
where
(x  d)(x  e)
a,b,c,d,e ¢ (with values not necessarily distinct).
Find values of a,b,c,d, and e that generate the graph. Check your
prediction using graphing features of your calculator.
10) Construct your own rational function f (x) such that f (x) satisfies the
following 3 properties:
(a) As x  3 , y  
(b) As x  3 , y  
(c) A removable discontinuity exists at x  5
For those who like a challenge, modify your rational function to satisfy
the above three properties and the additional property listed below:
(d) f (2)  4
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Sketching Rational Functions with Multiple Factors in Numerator
and Denominator
Recall that the global behavior of a function f (x) describes values that
function approaches as x   (i.e. the end behavior of f (x) on the left)
and x   (i.e. the end behavior of f (x) on the right).
If we know the end behavior of a function and we know information about all
points of discontinuity, we can quickly draw a "rough sketch" of the function.
An Example
Suppose a rational function f (x) possesses the following characteristics:
o
o
Global (end) Behavior: f (x)  0 as x   , f (x)  0 as x  
A single discontinuity at x = 3 : f(x)   as x  3 ,
f (x)   as x  3
With this information, you should be able to sketch the "general idea" of the
graph of the function. Examine the above characteristics, then make a quick
sketch predicting the shape of the function's graph (don't use your calculator).
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Discussion: Many students find it helpful to draw a dotted line to indicate
the location of any jump discontinuities. The graph will "jump" at x  3 ,
so we sketch the dotted vertical line x  3 .
Next, we draw small "arcs" to indicate the local behavior of the function
"near" x  3 :
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Next, we draw additional "arcs" to indicate the global behavior of the
function - i.e. the left and right end behavior.
Lastly, we connect the arcs to complete our sketch.
From the sketch, you should be able to construct an equation that "roughly"
1
generates the graph. The sketch resembles the function f (x) 
flipped
x
1
over the x-axis and translated 3 units to the right (i.e.
).
(x  3)
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We can graph the function on a graphing calculator to test the
reasonableness of our prediction.
The graph looks quite similar to our sketch.
Rational Functions Activity 3
In this activity, you will use your your knowledge of local and global behavior
to sketch graphs of rational functions that satisfy particular
characteristics. The functions may have linear factors in the numerator,
denominator, or both. Some of the functions have removable
discontinuities. Some have jump discontinuities. Some have both types
of discontinuties.
In the following three exercises, we list characteristics of several rational
functions. Perform the following for each exercise: (a) Draw a rough sketch of
each function based on the characteristics; (b) Construct a function that you
believe will generate a curve similar to your sketch; and (c) Use your
graphing calculator to check your work. If your function doesn't match your
sketch, revise it and test again.
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Exercise 1
o
o
Global (end) Behavior: f (x)  1 as x   , f (x)  1 as x  
A single discontinuity at x = -2 : f (x)   as x  2 ,
f (x)   as x  2
(a) Sketch:
(b) Function that generates sketch:
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Exercise 2
o
o
Global (end) Behavior: f (x)  0 as x   , f (x)  0 as x  
Two discontinuities - one at x = -3 , one at x = -2 :
f (x)  1 as x  3 , f (x)  1 as x  3
f(x)   as x  2 , f(x)   as x  2
(a) Sketch:
(b) Function that generates sketch:
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Exercise 3
o
o
Global (end) Behavior: f (x)   as x   , f (x)   as x  
A single discontinuity at x = 1 : f(x)  2 as x  1 ,
f(x)  2 as x  1
(a) Sketch:
(b) Function that generates sketch:
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Questions for Further Thought
In problems 11-15, you are provided with a rational function. For each,
perform the following tasks: (a) identify location of all jump discontinuities;
(b) identify location of all removable discontinuities; (c) sketch a graph of the
function without your calculator; (d) check your work by graphing with your
calculator (if possible).
You may wish to factor or use the long division algorithm on one or more of
the items to help you construct an informed sketch.
11) f (x ) 
12) f (x ) 
13) f (x) 
14) f (x) 
15) f (x ) 
1
(x  2)2 (x  3)
x 2  5x  6
(x  2)2 (x  3)
x2  x  6
x2
x 2  6x  9
x 2  5x  6
(x  a)(x  b)(x  c)
(x  a)(x  c)
( x  a )2 ( x  b)
16) Graph the function y 
, where a  b  0  c  d .
( x  c)( x  d)2
17) Find a rational function whose graph is given
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