PreCalculus Fun
Limit Analysis:
Through the Study of
Power, Polynomial and
Rational Functions
P( x )  x n
R( x ) 
P( x )
Q( x )
lim f ( x )
x c
C( t ) 
80,000p
C( p ) 
(100  p )

2011-2012
5t
0.01t 2  3.3

Contents
VIP Vocab: Limits, Extrema, Concavity and End Behavior……………………………………..…..….5
Lab 1: Power Functions……………………………………………………………………………….……………...8
Lab 2: Polynomial Long and Short Run Behavior.………………………………………………………...11
Lab 3: Polynomial Behavior Near Roots…….………………………………………………………………...14
Worksheet 1: Bringing it All Together – Polynomial Practice………………………………………...17
Lab 4: Limits………………..…………………………………………………………………………………………...21
Worksheet 2: Bringing it All Together – Limits Practice…………………………..…………………...26
Lab 5: Points of Discontinuity.…………………………………………………………………………………...28
Summarization: Limits, Holes and Asymptotes…………………………………………………………….34
Worksheet 3: Bringing it All Together – Rational Function Graphing Practice……….……...37
Worksheet 4: Bringing it All Together – Rational Function Graphing Practice……….……...42
Worksheet 5: Bringing it All Together – Polynomial and Rational Review……….…….……...44
Worksheet 6: Bringing it All Together – Polynomial and Rational Review……….…….……...47
VIP Vocabulary Introduction: Limits, Extrema and Concavity
From a Numerical and Graphical Perspective*
Vocabulary to describe how the height is
changing
A function that is “increasing” means:
A function that is “decreasing” means:
Vocabulary to describe how the slopes are
changing
A function that is “concave up” means:
A function that is “concave down” means:
(The Algebraic Perspective is what you will be studying down the road in calculus!)
Page 5
VIP Vocab: The place where the graph changes concavity is called an
.
VIP Vocab: The places where a curve would have a slope of zero are referred to as “local extrema.”
These are called local
and
values.
1. For the graph on the right, discuss the intervals pertaining to concavity and increasing/decreasing.
Note: The graph is “curvy throughout… i.e. no straight lines and the domain is all real numbers.
a. Write the intervals on which the graph has the following characteristics:
Concave Up & Increasing
Concave Up & Decreasing
Concave Down & Increasing
Concave Down & Decreasing
**Note:
You may write your interval notation in either of the following formats
a x b
or
( a, b )
a x b
or
( a, b )
a x b
or
[a, b ]
xb
or
( , b ] or
xa
or
( a,  )
etc.
b. What are the coordinates of any inflection points and local extrema?
Function End Behavior
Notation:
lim f ( x )  L
in words is “the limit of f(x) as x approaches infinity is L.”
x 
Interpretation: The symbols “ x   ” represents the idea “as gets really really big.” Similarly the
symbols “ x   ” represents the idea “as gets really really big negative.”
1
3. Consider the function f ( x )  .
x
a. Fill in the table of values
x
-and-
Graph f on x [0,100] and y  [1,1]
f ( x)
1
5
10
100
1000
10,000
b. Interpret your results from the table and graph to complete the following observation.
As the input gets bigger and bigger, the output gets closer and closer to _______.
c. Translate your statement above into the notation of limits:
lim
f ( x) 
x_
.
4. Evaluate each of the following limits.
a.
lim
x 
1

x
b. lim( 2 / x ) 
x 
c. lim
x 
2x 1

1 x
Note the “−” sign!
5. For the graph in question 1:
a. lim f ( x ) 
x 
b. lim f ( x ) 
x 
6. Describe the function y  ln( x ) make sure you address the following areas:
What is the domain and range of? Are there any asymptotes? What is its x-intercept? Describe this function’s
behavior using the vocabulary you built in the sections above. What is the lim ln( x ) ? What is the lim ln( x ) ?
x 
x 0
Page 7
Lab 1: Exploring the Power Function
A power function is a function of the
form:
where k and p are constants.
From our prior knowledge, what effect does k have on f ( x ) ?
THE EFFECTS OF THE POWER "P":
Part 1: For each of the categories a-h below, give one example equation along with its
graph. Make a chart next to each graph make a table of data for when x=-1,0,1 and plot
those three points on the graph you just drew. 2) Describe the patterns that you see with
respect to: concavity, increasing/decreasing, inflection, domain/range, symmetry,
lim f ( x ) and any other characteristics that help distinguish it from other functions. The
x 
first two have been done for you as an example. Since we already understand the effect of
k, set it equal to 1 for all of the examples below.
a) P=0
b) P=1
0
Equation: f ( x )  x a.k.a. f ( x )  1
Equation: f ( x )  x1 a.k.a. f ( x )  x
x
y
x
y
1
1
1
1
0
1
0
0
-1
1
-1
-1
Concavity: Neither up nor down.
Neither increasing nor decreasing.
No inflection points.
Domain:
Range: f ( x )  1
Symmetry: y-axis
Other: Slope=0
Concavity: Neither up nor down.
Increasing
No inflection points.
Domain:
Range:
Symmetry: origin
Other: Slope=1
c) P  {Positive Even Integers}
Equation:
x
y
1
0
-1
d) P  {Positive Odd Integers}
Equation:
x
y
1
0
-1
e) P  {Negative Even Integers}
Equation:
x
y
1
0
-1
f) P  {Negative Odd Integers}
Equation:
x
y
1
0
-1
Page 9
1
2
Equation:
x
y
1
0
-1
g) P=
1
3
Equation:
x
y
1
0
-1
h) P=
Part 2:
1a) On one grid with a window of 0  x  1 and 0  y  1 , sketch:
y  x2 , y  x 4 , y  x 6
1b) On the interval 0  x  1 which of these graphs “dominates” (has a higher y- value)?
2a) On a grid with a window of -10 to 10 in both directions, sketch:
y  x2 , y  x 4 , y  x 6
2b) When x>1, which of these graphs “dominates” (has a higher y- value)?
3) Sketch a graph of f ( x )  x 4 and g( x )  2x on the same grid. For what interval is:
b) f ( x )  g( x )
c) f ( x )  g( x )
a) f ( x )  g( x )
4) Are f ( x )  x 4 and g( x )  4 x both power functions? Why or why not?
Lab 2: Exploring Polynomial Long & Short Run Behaviors
Examples of:
Polynomial….
Not a polynomial….
Based on the above examples can you come up with a definition for what is a polynomial:
The general formula for the family of polynomial functions can be written as:
Where “n” is the
called the
of P(x)
Term:
Coefficients:
Constant Term:
Leading Term:
Standard Form:
Page 11
EXPLORATION 1: LONG-RUN BEHAVIOR…. What happens to the y-values when x   .
Graph the equations in group “a” first and answer #1 below, then graph the equations in group “b” and
answer #2 below.
f ( x )  3x 2  2 x  7
f ( x )  4 x 3  2x2  5
a)
f ( x)  x  7
b)
f ( x )  4 x 6  5x 3  2 x
f ( x )  2 x 8  4 x2
f ( x)  x5  2x 4  3
Write down the long-run behavior (end behavior) you notice about the above functions:
1) For Odd-Degreed Polynomials
2) For Even-Degreed Polynomials
and leading coefficient >0
and leading coefficient >0
lim f ( x ) 
lim f ( x ) 
x 
x 
lim f ( x ) 
lim f ( x ) 
x 
x 
EXPLORATION 2: LONG-RUN BEHAVIOR…. What happens to the shape of the graph when
f ( x)  x  x
3
a) Graph at the same time:
Using a window of
X: –2 to 2
Y: -1 to 1
2
f ( x)  x3
X: -5 to 5
Y: -150 to 150
X: -10 to 10
Y: -1000 to 1000
What do you notice the further out you move?
b) Graph at the same time:
Using a window of
X: –2 to 2
Y: -1 to 1
f ( x)  x5  2x3
f ( x)  x5
X: -5 to 5
Y: -150 to 150
X: -10 to 10
Y: -1000 to 1000
What do you notice the further out you move?
c) Graph at the same time:
Using a window of
X: –2 to 2
Y: -1 to 1
f ( x )  2 x 4  3x 2  5
f ( x)  2x4
X: -5 to 5
Y: -150 to 150
What do you notice the further out you move?
Which term determines the shape of the graph in the long-run?
X: -10 to 10
Y: -1000 to 1000
x 
SUMMARY: When viewed on a large enough scale, the graph of the polynomial
p( x )  an x n  an 1 x n 1  ...  a1 x  a0 looks like the graph of
.
EXERCISE:
1) Suppose f ( x ) is an even-degreed polynomial function. For each of the statements, write true or false. If
the statement is false, explain why and give an example which illustrates this.
b)
f ( x ) is an “even function” (If you don’t know this word… look it up! Note: this is different than an
even-degreed polynomial)
f ( x ) cannot be an “odd function”
c)
If
a)
lim f ( x )   , then lim f ( x )  
x 
x 
EXPLORATION 3: SHORT RUN BEHAVIOR… (as x gets closer to zero)
a) Graph:
f ( x )  x 4  4 x 3  16 x  16
f ( x )  x 4  4 x 3  4 x 2  16 x
f ( x )  x 4  x 3  8 x2  12 x
On a large scale (ex: window from –8 to 8 on x and 0 to 4000 on y), all of these resemble:
On a smaller scale, however, they look quite different from each other.
Sketch each of the graphs using a window of
X: -5 to 5
Y: -35 to 15
b) Now graph:
f ( x )  x 5  3x 4  2 x  4
f ( x)  2x  4
Examine these two graphs using the following progression of windows:
X: -10 to 10
X: -2 to 2
X: -.5 to .5
Y: -10 to 10
Y: -10 to –1
Y: -5 to –3
X: -.25 to .25
Y: -5 to -3
What do you notice as you get closer to x=0? This behavior is called “short-run behavior” (behavior as
x  0 ).
SUMMARY: When viewed near x=0, the term with the
shape.
determines the
Page 13
Lab 3: Exploring the Behaviors Polynomials
Near Their Roots
REVIEW EXERCISES: Short and Long Run Behavior
1) If f ( x )  x11 lim f ( x ) 
and lim f ( x ) 
x 
2) If f ( x )  x 4 lim f ( x ) 
x 
x 
and lim f ( x ) 
x 
3) As x   , the graph of f ( x )  5x 3  2x6  3x  2 resembles:
4) As x  0 , the graph of f ( x )  5x 3  2x6  3x  2 resembles:
EXPLORATION: Behavior of Polynomials Near Their “Roots”
Function
y  ( x  3)( x  5)(2x  8)
Degree: 3 (odd)
Graph (Show x-intercepts)
End Behavior
x-intercepts
As
x  4, 3,5
x  ,
f ( x)  
As
x  ,
f ( x )  
y  ( x  3)( x  5)(2x  8)2
Function
y  ( x  3)2 ( x  5)(2x  8)
Graph (Show x-intercepts)
End Behavior
x-intercepts
y  ( x  3)( x  5)2 (2x  8)
y  ( x  3)( x  5)2 (2x  8)3
y  ( x  3)3 ( x  5)(2x  8)
Page 15
OBSERVATIONS:
1) Have you observed enough to perhaps come up with a hypothesis about how the
factors in the equation determine the behavior of polynomials near their roots? (In
other words, how the graphs look like near their x-intercepts?)
2) Using your “end behavior knowledge” and the hypothesis in #1, see if you can
sketch a quick graph of y   x  4   x  1  x  2  without a calculator. Explain
how you knew what the graph looked like:
2
3
Note: You can always double check your graph by choosing numbers between the roots and figuring
out whether the height values are positive or negative. That will tell you if graph is above or below the
x-axis.
“Going Backwards”
Given a graph you can figure out the equation by using the knowledge you have gained
about roots and end-behavior. “k” will be a stretch factor for the graph
f ( x )  k( x  a)( x  b )( x  c )
9
8
To write a polynomial’s equation from its graph, write the factors based on the xintercepts, plug in other coordinate you know and solve for k.
7
6
5
4
3
Example:
Apply the underlined statement above and see if you can solve for
a, b, and c so that you can write the equation of the following graph
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
(0,-3)-3
-4
-5
-6
-7
-8
-9
1
2
3
4
5
6
7
Worksheet #1: Bringing It All Together - Polynomials Practice
REVIEW:
1) If f ( x )  x 8 , lim f ( x ) 
and lim f ( x ) 
x 
x 
2) If f ( x )  x7 lim f ( x ) 
and lim f ( x ) 
x 
x 
3) As x   , the graph of f ( x )  5x 3  2x 6  3x  2 resembles:
4) As x  0 , the graph of f ( x )  5x 3  2x 6  3x  2 resembles:
5) What is the degree of f ( x )  ( x  2)( x  6)(2x  6) ?
6) For f ( x )  ( x  2)( x  6)(2x  6)
lim f ( x ) 
x 
and lim f ( x ) 
x 
Summary of Behavior Near X-intercepts
Let’s use the function from yesterday to summarize the patterns that we noticed:
f ( x )  ( x  4)2 ( x  1)3 ( x  2)
The degree of this polynomial is:
The leading coefficient will be positive/negative (circle one).
The lim f ( x ) 
x 
and lim f ( x ) 
x 
If a factor appears once (not squared, cubed, etc.) in the factored form of a polynomial, then the graph
will pass through that x-intercept/root
.
2
3
Ex: y  ( x  4) ( x  1) ( x  2)
If a factor appears twice (squared factor) in the factored form of a polynomial, then the graph will
that x-intercept/root.
2
3
Ex: y  ( x  4) ( x  1) ( x  2)
If a factor appears three times (cubed factor) in the factored form of a polynomial, then the graph will
pass through that x-intercept/root
.
2
3
Ex: . y  ( x  4) ( x  1) ( x  2)
Page 17
EXERCISES:
1) Sketch a graph of the following:
y  ( x  2)( x  6)(2x  6)
y  ( x  2)2 ( x  6)(2x  6)
y  ( x  2)2 ( x  6)(2x  6)
y  ( x  2)2 ( x  6)3 (2x  6)
2) Find possible equations of the following polynomials:
a)
b)
y  ( x  2)( x  6)3 (2x  6)
c)
3) Height Factor: f ( x )  k( x  a)( x  b )( x  c ) .
 Write Factors.
 Plug in non-x-intercept coordinate you know
 Solve for k.
Example: Find the equation of the graph to the right.
4) Find the equation of the following two polynomials (given the curve and a point on it):
Page 19
5) For the following, find possible formulas for the polynomials with the given properties.
a) f ( x ) has degree  2 , f (0)  f (2)  0 and f (3)  3
b) f ( x ) is a third-degree polynomial with f ( 3)  0 , f ( 1)  0 , f (4)  0 , and f (2)  5 .
c) f ( x ) has the least possible degree and goes through the points (-3, 0 ), (1, 0 ), and (0, -3 ).
In order to answer #6, remember there is a difference between the following two phrases
“Even Function” means:
“Even Degreed Polynomial” means:
6) The following statements about f(x) are true:
 f ( x ) is a polynomial function
 f ( x )  0 at exactly four different values of x.
 lim f ( x )  
x 
For each of the following statements, write true if the statement must be true, never true if the
statement is never true, or sometimes true if it is sometimes true and sometimes not true.
a) f ( x ) is an odd function
b) f ( x ) is an even function
c) f ( x ) is a fourth degree polynomial
d) f ( x ) is a fifth degree polynomial
Lab 4: The
lim and lim
x c
x 
This lab should take you about an hour; but, please think carefully about what you are doing
and write down complete solutions for each part. The importance of this lab rests on the fact that
ALL (!) of calculus rests on the single concept of the limit.
A couple of comments are in order. First, you need to be certain of the notation for limits,
and what it means in words. As you might expect, a limit simply means what happens to the
function as you get closer and closer to some number. To define this in formal terms is actually no
easy task, but to use the idea of a limit is really quite manageable. Now the notation:
lim f ( x )
x c
And this simply means, what happens to the y-values as x gets closer and closer to some number c.
BACKGROUND ON LIMIT NOTATION:
lim f ( x ) means what y value does the function f approach when x gets closer and closer to a values of “c.” We don’t
x c
care about what happens at x = c, but instead only in the neighborhood of x = c.
lim f ( x ) means the limit as we approach c coming from the right side of c.
x c 
lim f ( x ) means the limit as we approach c coming from the left side of c.
x c 
lim f ( x ) means the limit as we approach c coming both sides.
x c
For the limit to exist as x approaches “c”
lim f ( x ) = lim f ( x )
x c 
x c 
Part 1: Let’s examine this idea from a graphical perspective.
Using the graph of f(x) provided, evaluate the following limits:
a) lim f ( x )
b) lim f ( x )
d) lim f ( x )
e) lim f ( x )
x 
x 2 
x 

c) lim f ( x )

x 2 

x 2

f) f (2)
g) lim f ( x )
h) lim f ( x )
i) lim f ( x )
x 0 

x 0 
x 0











Page 21



Part 2: Let’s examine this idea from a numerical perspective.
Using the table of data for f(x) provided, evaluate the following limits:
1) Using the chart above:
lim g( x ) 
x 4 
2) Using the chart above:
lim f ( x ) 
x 4 
lim g( x ) 
lim g( x ) 
lim f ( x ) 
lim f ( x ) 
x 4 
x 4 
x 4
x 4
Part 3: Performing Numerical Analysis on Our Own
Let’s now use limits to study three of the different types of discontinuities that can occur.
Example 1: f ( x ) 
1
( x  3)2
What values of x can we not plug into the following function: f ( x ) 
1
( x  3)2
Using the TABLE feature of your calculator make a table of data in your notebook to estimate the
numerical values of the following limits.
a) Evaluate lim f ( x ) by choose successive values of f ( x ) at x = 2.8, 2.9, 2.99, 2.999, 2.9999,
x 3
what do you think happens to the values of f ( x ) as x increases toward 3?
b) Evaluate lim f ( x ) Choose successive values of f ( x ) at x = 3.1, 3.01, 3.001, 3.0001, 3.00001,
x 3
what do you think happens to the values of f ( x ) as x decreases toward 3?
c) Does the lim f ( x ) exist? If so, what is it? If not, why not?
x 3
d) Apply the same strategy for numbers near the specified x values below and numerically
compute each of the limits.
2x
( x  4)2
What values can we
not plug in?
f ( x) 
lim f ( x )
x 4
x 2
x4
What values can we
not plug in?
f ( x) 
lim f ( x )
x 4
( x  2)
( x  2)( x  3)
What values can we not
plug in?
f ( x) 
lim f ( x )
x 3
( x  1)( x  2)
( x  4)( x  1)
What values can we not
plug in?
f ( x) 
lim f ( x )
x 4
lim f ( x )
x 4 
lim f ( x )
x 4 
lim f ( x )
x 4
lim f ( x )
x 4 
lim f ( x )
x 4
lim f ( x )
x 3
lim f ( x )
x 3
lim f ( x )
x 4
lim f ( x )
x 1
lim f ( x )
x 1
lim f ( x )
x 0
lim f ( x )
x 0
lim f ( x )
x 2
lim f ( x )
x 2
lim f ( x )
x 2
lim f ( x )
x 2
lim f ( x )
x 1
lim f ( x )
x 2
lim f ( x )
x 2
lim f ( x )
x 0
lim f ( x )
x 2
lim f ( x )
x 2
lim f ( x )
x 2
Page 23
f ( x)
Part 4: End Behavior - The xlim

Using your calculator’s table feature, find the following limits:
x 2
x  x 2  x  6
lim
x 2
x  x 2  x  6
Degree Numerator=
Degree Denominator=
2) lim
x2  x  6
x 
x 2
x2  x  6
x 
x 2
Degree Numerator=
Degree Denominator=
3x 2  x  6
3) lim 2
x  2 x  3x  5
3x 2  x  6
lim
x  2 x 2  3x  5
1) lim
lim
Degree Numerator=
Degree Denominator=
Let’s summarize the patterns we see with lim f ( x )
x 
SUMMARY: When we have a polynomial divided by a polynomial the following things can
happen with the lim f ( x ) …
x 
CASE 1: If the degree of the denominator is larger than the degree of the numerator:
1
1
x2  8
EX: lim  lim 2 
lim 3

x  x
x  x
x  x  2
then the limit is
!
CASE 2: If the degree of the denominator is smaller than the degree of the numerator:
x3  8
EX: lim 2

x  x  2
then the limit is
!
CASE 3: If the degree of the denominator is equal to the degree of the numerator:
x2  8
x3  4x  5
6 x2  8 x 3
EX: lim 2

lim 3

lim

x  x  4
x  2 x  8 x  4
x  5x 3  2
then the limit is
!
This works logically if we also have other functions too. Think big and small!
Translate the following into sentences and then evaluate the answer without using a calculator.
2 x2
3x  1
4) lim 2
5) lim
x  x  1
x  2 x  5
Translate the following into sentences and draw an example illustrate your sentence.
6) lim f ( x )  3
7) lim x2
8) lim g( x )  
x 2
x 5
x 2
Without using your calculator, observe and evaluate the following limits
5 x 2  3x  1
5 x 2  3x  1
5x  1
9) lim 2
10) lim
11) lim
x 
x 
x  7x  9
7x 2  9
7x  9
Page 25
Worksheet #2: Bringing It All Together - Limits Practice
1) Use the table feature of your calculator to find the following limits:
a) lim
x 4
x 2
x4
b) lim
x 2
ex 1
x 0
x
x 2
2
x  x 6
c) lim
sin x
(radian mode)
x 0
x
d) lim
2) Use the graph to answer the following
a) lim f ( x )
b ) lim f ( x )
c ) lim f ( x )
d ) lim f ( x )
e) f (5)
f ) lim f ( x )
x 1
x 5
x 1
x 1
x 
g ) lim f ( x )
x 
3) Use the graph to answer the following
a) lim g (t )
b ) lim g (t )
c ) lim g (t )
d ) lim g (t )
e) lim g (t )
f ) lim g (t )
g ) g (2)
h ) lim g (t )
t 0
t 2
t 0
t 2
t 4
t 0
t 2
4) Without a calculator, use the knowledge gained from the rules you generated about lim answer
x
the following:
x4
2
x  x  7x  12
a) lim
6 x2  4
x  3x 2  7x  12
b) lim
y

5) Sketch a graph of the piecewise
 x2
for x  2
function: f ( x )  
6  x for x  2
then evaluate the following limits:



x
a) lim f ( x )
x 2
b) lim f ( x )
x 2
c) lim f ( x )
x 2
d) lim f ( x )
x 













e) lim f ( x )
x 
Page 27
Lab 5: Points of Discontinuity
Part 1: Numerical Analysis
A “point of discontinuity” is a value of x that is not in the domain of a function. It is a value of x
that cannot be plugged into a function.
y
y
TYPES OF DISCONTINUITIES
y












x











x
x















 







REMOVABLE

DISCONTINUITIES
(A.K.A HOLES)
VERTICAL ASYMPTOTES

JUMP DISCONTINUITIES


Now the important question is…what in an equation causes the first
two types: holes and vertical asymptotes?
1. Using the data you collected in “Limits Lab 4” and your graphing calculator sketch a graph
for each of the following rational functions.
1
( x  3)2
Discontinuities at:
x3
f ( x) 
f ( x) 
Discontinuity at:
2x
( x  4)2
f ( x) 
Discontinuity at:
x 2
x4


f ( x) 
Discontinuities at:
( x  2)
( x  2)( x  3)
f ( x) 
( x  1)( x  2)
( x  4)( x  1)
Discontinuities at:
2. Examine your equations and graphs and come up with a hypothesis about characteristic in
an equation causes a discontinuity as well as what determines if the discontinuity is a hole
or a vertical asymptote.
Page 29
Part 2: Graphical Perspective – Vertical Asymptotes
3. Hypothesize in your group, what in the equation causes a graph to approach a vertical
asymptote going in the same direction or in opposite directions? (See diagrams below)
Approaching vertical asymptotes going in the same direction: In both of the pictures below
as x approaches 2 from both the left and the right, our y-values are approaching either:
both positive infinity
or
both negative infinity
lim f ( x )  
lim f ( x )  
x 2
x 2
Approaching vertical asymptotes going in the opposite directions: In both of the pictures
below as x approaches 2 from the left and the right, our y-values change from approaching
negative infinity to positive infinity or vice-versa:
lim f ( x )   lim f ( x )  
x 2
x 2
or
lim f ( x )   lim f ( x )  
x 2
x 2
Part 3: Practice
Based on your above hypotheses, what do you speculate would be the equation for these
functions?
y




x













y





x














Page 31
Part 4: The Conceptual Approach
Now that you have seen the numerical analysis supporting the behaviors of rational functions, let’s
explore the conceptual, more algebraic reasoning behind it.
1. Reciprocal Behavior – Answer the following and give an example of each:
a. What happens when you reciprocate a value of x  1 , what kind of value do you get? Also,
the bigger x is what happens to the value of its reciprocal?
b. What happens when you reciprocate a value of x  1 , what kind of value do you get?
c. What happens when you reciprocate a value of 0  x  1 , what kind of value do you get?
Also, the smaller a fraction x is what happens to the value of its reciprocal?
d. What happens when you reciprocate a value of 1  x  0 , what kind of value do you get?
e. What happens when you reciprocate a value of 0, what kind of value do you get?
2. The Reciprocal Function
a. Consider the cubic polynomial c( x)  ( x  1)2 ( x  2) whose graph is given below. Find the
y-coordinate for each lettered point A-E.

D (-1, ____)
A (2, ____)




B (1.2, ____)
E (-2, ____)









C (1, ____)









b. Consider the function, f ( x) , that is the reciprocal of c( x) , namely the function
f ( x) 
x
c( x)
f ( x) 
A 2
B
1
c( x)
1
1
.

2
c( x) ( x  1) ( x  2)
Identify the location of points A-E under this transformation. Use
this information to make a sketch of f ( x) . Check your sketch using
your graphing calculator.
1.2
C 1
D −1
E
−2






























Summarize why the behavior near each of the asymptotes makes sense based on your
exploration of reciprocals.
Page 33
In a Nutshell: “Holes, Asymptotes, and lim
x  c ” Summarized
Vertical Asymptotes and Holes are points of discontinuity located at x-values which are
issues for the domain of a function. To figure out what kind of discontinuity we have at x = c,
we need to see what happens to the function as x is near value c. So we take the:
lim f ( x )
x c
How do we evaluate a limit of a rational function as it approaches a particular number? How do we tell if it is a
hole or an asymptote?
lim
x 4
1
x4
lim
x 4
x4
x2  16
If we look at these both from a data perspective, we will notice that although algebraically they reduce to the same
things, that they behave differently.
VERTICAL ASYMPTOTES
x
3.5
3.9
3.99
3.999
4
4.0001
4.001
4.01
4.5
1
x4
-2
-10
-100
-1000
UNDF
10000
1000
100
2
In this case: lim
x 4 
1
1
1
  and lim
  , hence lim
does not exist and there is a vertical asymptote
x

4

x

4
x4
x4
x4
located at x = 4.
A vertical asymptote comes
from the x-value that makes
the factor (located in the
DENOMINATOR ONLY) equal to zero.
THE OTHER CASE….HOLES
x
3.5
3.9
3.99
3.999
4
4.0001
4.001
4.01
4.5
x4
x 2  16
0.1333
0.12658
0.125156
.125016
UNDF
0.124998
0.1249
0.124884
0.117647
We can see from our data that the lim
x 4
x4
1
 .125  . What does it look like visually?
2
x  16
8
Visually there is a hole in the graph at x = 4. The height of this hole is at y 
1
. Essentially the graph
8
1
1
, but then just skips over the point (4, ). There is an open circle shown there.
8
8
You must be careful, since your calculator is not going to show you this information.
approaches a height of
A hole comes from x-value that makes the factor that is located in BOTH the denominator and the
numerator equal to zero.
This is a graph of
Notice that there is an asymptote and a hole in this graph! The asymptote
came from the (x+4) factor and the hole came from the (x-4) factor.
Where does this
f ( x) 
1
come from algebraically?
8
Note:
There is a hole at
x=4 and a vertical
asymptote at x = -4.
To figure out the height of the hole we first factor
x4
x4
. It can be re-written as
.
f ( x)  2
( x  4)( x  4)
x  16
We then eliminate the factor that is occurs in the
x4
top and the bottom
. Then we
( x  4) ( x  4)
evaluate the limit of the remaining function as x
approaches 4 (i.e. the location of the hole).
x4
1
1
1
lim
 lim

 .
x 4 ( x  4) ( x  4)
x 4 ( x  4)
44 8
Therefore the hole is located at the point (4,
x4
.
x2  16
-4
4
1
)!
8
With this same function, let’s analyze what happens numerically near x = -4. From the picture above, you can see
that there is a vertical asymptote…but let’s look at the table of data to see this also.
x
-4.5
-4.01
-4.001
4.0001
4
-3.999
-3.99
-3.9
-3.5
x4
x 2  16
-2
-100
-1000
-10000
UNDF
10000
100
10
2
Page 35
Horizontal Asymptotes are different “critters”… they are lines that in the long-run a graph
approaches. It just means “end-behavior”. So to do this we want to know what happens as x
gets bigger either positively or negatively:
lim f ( x )
x 
Some background, before we analyze the horizontal asymptotes of our two
functions…
When we have a polynomial divided by a polynomial the following things can happen…
CASE 1: If the degree of the denominator is larger than the degree of the numerator:
1
1
x2  8
EX: lim  0,
lim 2  0,
lim 3
 0 then the limit is zero!
x  x
x  x
x  x  2
CASE 2: If the degree of the denominator is smaller than the degree of the numerator:
x3  8
EX: lim 2
  then the fraction keeps getting larger (   ) and doesn’t approach a specific
x  x  2
number!
CASE 3: If the degree of the denominator is equal to the degree of the numerator:
x2  8
x3  4x  5 1
6 x2  8 x 3 8
EX: lim 2
 1,
lim 3
 , lim

x  x  4
x  2 x  8 x  4
2 x  5x 3  2
5
then the limit is the ratio of the leading coefficients!
This works logically if we also have other functions too. Think big and small!
Using these ideas, what would be the horizontal asymptotes for our two functions:
f ( x) 
*a)
*b)
lim
1
x4
lim
x4
x 2  16
x  
x 
1
x4
and
f ( x) 
x4
x2  16
Looking at our graphs, do our results match what we expected?
*a) zero
*b) zero
Worksheet #3: Bringing It All Together – Graphing Rational Functions Practice
Please fill in the left column with the important characteristics (or state that a particular characteristic does not apply), then
sketch its corresponding graph.
1) Find discontinuities (values that make the _______________________ equal to ________.)
Holes: If factor appears_________________________________________________________
set it equal to ________ to find the x-coordinate of the hole. Then cross out those two factors,
plug the x-value you just got in the remaining factors and that tells you the _________________
of the hole.
Vertical Asymptotes: If factor appears_______________________________________________
set it equal to ________ to find the x-value of the vertical asymptotes.
2) Find the x-intercepts: If factor appears_______________________________________________
set it equal to ________ to find the x-intercept.
3) Find End-Behavior (a.k.a._______________________________). Take the limit of the function as x   .
4) Plot hole(s), vertical asymptote(s), x-intercept(s) and horizontal asymptote(s).
5) Sketch the remainder of the graph.
**Reminders:
Your graph may cross a ________________________.Your graph may NOT cross a _____________________________.
Near a vertical asymptote one of four things can happen:
Vertical Asypmtote factor is a squared factor


2
Vertical Asypmtote factor is just a single factor
Page 37
1)
1
f ( x) 
x4
y


Hole(s):


x





Vertical Asymptote(s):



















Horizontal Asymptote:
2)
f ( x) 

1
y

 x  4
2

Hole(s):


x





Vertical Asymptote(s):




Horizontal Asymptote:
3)

x 2
f ( x) 
x4
y


Hole(s):


x
Vertical Asymptote(s):







Horizontal Asymptote:



4)
x 2
f ( x) 
 x  4  x  2 
y



Hole(s):

x






















Vertical Asymptote(s):



Horizontal Asymptote:
5)
2x  6  x  4 
f ( x) 
 x  1 x  2 
y


Hole(s):


x





Vertical Asymptote(s):




Horizontal Asymptote::
6)
f ( x) 

1
x 2
y


Hole(s):


x
Vertical Asymptote(s):









Horizontal Asymptote:

Page 39
7)
f ( x) 
1
y

 x  2
2

Hole(s):


x





Vertical Asymptote(s):



















Horizontal Asymptote:
8)

x 1
f ( x) 
x 2
y


Hole(s):


x
Vertical Asymptote(s):









Horizontal Asymptote:
9)

x 1
f ( x) 
 x  3 x  1
y


Hole(s):


x

Vertical Asymptote(s):







Horizontal Asymptote:


10)
 x  1 x  2 
f ( x) 
 3x  6  x  5
y


Hole(s):


x





Vertical Asymptote(s):



















Horizontal Asymptote:
11)

 3x  6  x  4 
f ( x) 
 x  2  x  1
y


Hole(s):


x
Vertical Asymptote(s):









Horizontal Asymptote:

12)
 x  2
f (x 
2
 x  2  x  5
y


Hole(s):


x

Vertical Asymptote(s):








Horizontal Asymptote:

Page 41
Worksheet #4: Bringing It All Together – Rational Functions Practice
Sketch a graph of the following, labeling all important characteristics:
2 x2  10 x  12
x2  x  6
2 x2  2 x  4
2) r( x ) 
x2  x
x2  x  6
3) r( x )  2
x  3x
3x 2  6
4) r( x )  2
x  2x  3
1)
r( x ) 
5)
r( x ) 
x2  2 x  1
x 3  3x 2
Write the equations of the following:
























































From your book….Sketch a graph of the following:
9) r( x ) 
3
x 2
11) r( x ) 
x2
x2  x  6
13) r( x ) 
6
x 2
15) r( x ) 
6x  2
x  5x  6
2
2
Page 43
Worksheet #5:
Bringing It All Together - Polynomials and Rational Review
Worksheet 5: Polynomial and Rational Review
1) Guess as to what you think that the equation is for the following two sets of data.
x
2.7
2.9
2.95
3
3.05
3.1
3.3
y
12.1
101
401
UNDEF
401
101
12.1
x
1.5
1.9
1.95
2
2.05
2.1
2.5
y
-1.5
-9.5
-19.5
UNDEF
20.5
10.5
2.5
2) Sketch a graph of the following, labeling x- and y-intercepts, coordinates of holes and equations of any
vertical or horizontal asymptotes.
a)
f ( x) 
(2 x  6)( x  5)
( x  5)( x  4)
b)
4) Give a possible equation for the following graphs:
a)
f ( x) 
(2 x  6)( x  5)
( x  5)( x  4)2
b)
y




c) The polynomial contains the point (0,4)
x














5) Find a possible formula for the following rational or polynomial functions (might be helpful to sketch a
graph in order to guide you).
a)
The graph of y=h(x) has two vertical asymptotes: one at x=-2 and one at x=3. It has a horizontal
asymptote of y=1. The graph of h touches the x-axis once at x=5.
b)
This 5th degree polynomial has x-intercepts at x=-3, x=2, and x=5, and at x=6. It has a y-intercept of
7.
c)
This function has zeros (x-intercepts) at x=-3 and x=2, and vertical asymptotes at x=-5 and x=7. It has
a horizontal asymptote of y=1.
d)
This function has zeros at x=2 and x=3. It has a vertical asymptote x=5. It has a horizontal
asymptote of y=-3.
6) Without a calculator, use the functions described below to match i-vi with descriptors a-f. Some of the
descriptions may have no matching function or more than one function matching function. For the first
one, I’ve given you a hint to help you get started.
f ( x )   x  3
2
g( x )  x2  4
h( x )  x  1
j( x )  x 2  1
( i) p( x ) 
f ( x ) ( x  3)2

g( x ) x2  4
( ii) q( x ) 
h( x )
g( x )
( iii) r( x )  f ( x )  g ( x )
( iv) s( x ) 
g( x )
j( x )
( v ) t( x ) 
1
h( x )
( vi)
j( x )
f ( x)
(a) I have Two zeros, no vertical asymptotes, and a horizontal asymptote.
(b) I have Two zeros, no vertical asymptote, and no horizontal asymptote.
(c) I have One zero, one vertical asymptote, and a horizontal asymptote.
(d) I have One zero, two vertical asymptotes, and a horizontal asymptote.
(e) I have No zeros, one vertical asymptote, and a horizontal asymptote at y=1.
(f) I have No zeros, one vertical asymptote, and a horizontal asymptote at y=0.
Page 45
7) Polynomial fitting:
(a) Find a polynomial
p0 ( x) of degree zero
( f ( x)  7 is an example of such a polynomial) that
passes through the point (0, 4).
(b) Find a polynomial
p1 ( x) of degree one (a linear
function) that passes through both (2, 0) and (0, 4)
(c) Find a polynomial
p2 ( x) of degree two that passes
through (-1, 0), (2, 0), and (0, 4). Try creating a
polynomial in factored form f ( x)  a( x  b)( x  c) ,
then solve for a, b, and c. Do not forget that the
zeros of a polynomial correspond to x-intercepts.
(d) Find a polynomial of
p3 ( x) degree three that
passes through (-1,0), (1, 0), (2, 0) and (0, 4).
Worksheet #6:
Bringing It All Together - Polynomials and Rational Review
1) Graph: f ( x ) 
x2  5x  6
(2 x  6)( x  4)
2) f(x) is a 7th degree polynomial which has the following characteristics. Its only roots (xintercepts) are located -3, -1, 4. As x  , f ( x)   and as x  , f ( x )   . Also f(0) < 0.
Sketch a graph and write a possible equation for it.
x2  x
3) Sketch a graph of y  2
.
x 1
Note any x and y-intercepts, vertical and horizontal asymptotes, and holes.
4) Sketch a graph of:
p. 322: #31,30,33,38,48
5) Rational Word Problem: After a certain drug is injected into a patient, the concentration c of
the drug in the blood stream is monitored. At time t  0 (in minutes since the injection), the
concentration (in mg/L) is given by
30t
Hmm…. Weird….tricky tricky
c( t )  2
t 2
a) Draw a graph (no calculator)
b) What eventually happens to the concentration of the drug in the blood stream?
6) As a train moves toward an observer, the pitch of its whistle sounds higher to the observer than
it would if the train were at rest, because the crests of the sound waves are compressed closer
together. This phenomenon is called the Doppler Effect. The observed pitch P is a function of the
speed v of the train and is given by:
P s
P( v)  o o where Po is the actual pitch of the whistle at the source and so=332 m/s is the
so  v
speed of sound in air. Suppose that a train has a whistle pitched at Po=440 Hz.
a) Graph the function P(v).
b) How can the vertical asymptotes be interpreted physically?
( x  2)( x  3)
has x-intercepts at
x2  x  2
x  2 and 3. Why is this wrong? Are there any vertical asymptotes?
7) A student might conclude that the rational function y 
Write a process by which a student can always determine the location of any x-intercepts or
vertical asymptotes.
Page 47
8) Write a possible rational function formula for the graph below:
a.
y








x





































b. “A Thinker”
y








x





























9) A manufacturer wants to sell .5 liter (500 cm3 ) 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
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 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. Explain why 500   r 2 h for the desired can of radius r and height h.
c. Using a formula for S(r), determine the dimensions of the optimal can.
10) The Refrigerator: How much does it really cost? [NCSSM Distance Learning]
On the web site http://www.eren.doe.gov/consumerinfo/energy_savers/appliancesbody.html
there is a list of yearly cost for electricity for common household appliances.
Appliance
Average Cost/year in
electricity
Home Computer
$9
Television
$13
Microwave
$13
Dishwasher
$51
Clothes Dryer
$75
Washing Machine
$79
Refrigerator
$92
a) If we assume that a new refrigerator costs $550, determine the total annual cost for a
refrigerator that lasts for 15 years. Assume the only costs associated with the refrigerator are it
purchase cost and electricity.
b) Develop a function that gives the annual cost of a refrigerator as a function of the number of
years you own the refrigerator.
c) Sketch a graph of that function. What is an appropriate window?
d) Since this is a rational function, determine the asymptotes of this function?
e) Explain the meaning of the horizontal asymptote in terms of the refrigerator.
f) If a company offers a refrigerator that costs $1200, but says that it will last at least twenty years,
is the refrigerator worth the difference in cost?
Page 47