Bibb 1
Table of Contents
Overall Project Topic ---------------------------------------------------------------------------------------- 3
Students Understandings and Obstacles -------------------------------------------------------------- 3 – 4
Investigations
Vertical Asymptotes of Rational Functions ------------------------------------------------ 5 – 10
Horizontal Asymptotes of Rational Functions -------------------------------------------- 11 – 15
Homework---------------------------------------------------------------------------------------------- 16 – 19
Review ------------------------------------------------------------------------------------------------------- 20
Quiz ----------------------------------------------------------------------------------------------------- 21 – 24
Bibb 2
Overall Project Topic
For this project I am going to focus on vertical and horizontal asymptotes of rational
functions. I know for myself I struggled with this topic because I was so dependent on
memorizing the rules that I when I forgot the rules I could not find the asymptotes. Thus this
project helped me further my understandings as to what is occurring in the rational function at
the asymptotes instead of relying on the rules.
Students Understandings and Obstacles
Before a student can learn about asymptotes, they must have the understandings behind
why rational functions exist. Rational functions are used to model many real world events such
as representing the concentration of a medication in a person’s bloodstream with respect to time,
cleaning up an oil spill, and many other situations. Students have the previous knowledge of
polynomial functions in the form of:
anxn + an-1xn-1 + … + a1x + a0 where n is a nonnegative integer
When we divide one polynomial function by another polynomial function, a rational
function is created. Generally rational functions are represented by:
𝑝(𝑥)
f(x) = 𝑞(𝑥) when q(x) ≠ 0
Rational functions contain zeros, vertical and horizontal asymptotes. Through my
investigation students are going to uncover how rational functions behave at vertical and
horizontal asymptotes. But what causes an asymptote anyways? An asymptote is a line that the
rational function value approaches as the input value approaches some value whether it be the
root of the denominator or grows without bound. This “line” is not actually part of the graph, but
rather a boundary for the function.
Note: I find that students are going to have difficulty understanding when a rational function
crosses a horizontal asymptote. Students are often going to try to generalize that the rational
function cannot cross the asymptote but that is not true in all cases. Teachers need to be aware of
this misconception and reinforce that a rational function only approaches its vertical asymptote
because the vertical asymptote is where the function is not defined, thus it can never take on that
value (or touch the vertical asymptote). Whereas, a rational function is approaching the
horizontal asymptote that is representing the limit of the function but there is no reason why the
rational function cannot take on some values for a finite set of input values. The horizontal
asymptote is not where the function is undefined, rather just placing a limit to where the function
is approaching.
Bibb 3
Before learning the general reasoning’s behind the vertical and horizontal asymptotes,
students are going to investigate for themselves from a table of values to uncover the patterns in
the output values of the rational function.
First students are going to need to develop the vertical asymptote(s) of a rational
function. The vertical asymptote is a vertical line (x = __) where the function is not defined.
Thus a function is not defined when the denominator of a rational function is equal to zero. So,
by setting the denominator equal to zero and finding the x values of the denominator, we will
find the values of the rational function where the function does not exist. Now that we know
where the function is not defined, we can say that there is a vertical asymptote at those values
since the function will never be able to equal those values but become closer and closer to this
value, without ever reaching the value, as the input values gets closer and closer to the root of the
denominator. Students also need to realize that there can be more than one vertical asymptote, or
none at all.
Next the students are going to derive horizontal asymptotes. Similar to a vertical
asymptote, a horizontal asymptote is a horizontal line (y = __) where the function either
approaches ∞ or -∞. Unlike vertical asymptotes, a rational function has 3 ways to determine the
horizontal asymptote.
1. When the power of the leading coefficient of the numerator is less than the power of the
leading coefficient of the denominator the denominator is getting infinitely small, thus
the whole function is getting infinitely small so the function is going to approach 0. Thus
there is a horizontal asymptote at y = 0.
2. When the power of the leading coefficient of the numerator is equal to the power of the
leading coefficient of the denominator, so the function “pulls” evenly. The numerator
does not outgrow the denominator and vice versa, thus the horizontal asymptote is found
by dividing the ratio of the leading coefficient of the numerator by the leading coefficient
of the denominator.
3. When the power of the leading coefficient of the numerator is greater than the power of
the leading coefficient of the denominator the numerator is getting infinitely larger, thus
the whole function is getting infinitely larger so the function is going to approach ∞. Thus
there is no horizontal asymptote.
If students have the reasoning’s behind why these asymptotes exist, there is no reason that they
need to memorize any “rules” but rather have a deeper understanding of the mathematics.
Bibb 4
Investigations
Vertical Asymptotes of Rational Functions
Name: ____________________
1. The cost of cleaning up p percent of the pollution in the air in the United States may be
0.06𝑝
modeled by 𝐶 = ℎ(𝑝) = 100−𝑝 , where C is the cost in billions of dollars.
Question Rational: This question is designed for the students to work through a
context situations with a table of values and finding the rate of change to see the
behavior of the rational function. They will then discover that it is impossible to
calculate the rational function at a certain value. Leading them to the idea of
vertical asymptotes.
a) Create a table of values below for C and ΔC in terms of p, as p increases from 0 to
100 in increments of 25 percent.
Percent of
pollution
removed (p)
Cost in
billions of $
(C)
0
0
Change in the
Cleanup Cost
in billions of
$ (ΔC)
0.02
25
0.02
0.04
50
0.06
0.12
75
0.18
unknown
100
undefined
b) What is happening to the cleanup cost as each additional 25 percent of pollution is
removed? (Refer to the values of ΔC from part (a) above)
We see that the cost to remove the first 25% of pollution is 0.02 billion dollars.
The cost to remove the second 25% of pollution is 0.06 billion dollars. The cost to
remove the third 25% of pollution is 0.18 billion dollars. The cost to remove the
last 25% of pollution is undeterminable. The rate of change show us that the cost
to remove pollution is continually increasing at an increasing rate.
Bibb 5
c) Explain why it is impossible to use the model to calculate the cost of removing 100
percent of the pollution.
It is impossible to calculate the cost of removing 100% of the pollution because
C(100) is undefined. Thus through the use of this model, we cannot calculate the
cost to removing 100% of the pollution.
𝑥
2. Given that 𝑓(𝑥) = 2𝑥−8 , answer the questions below.
Question Rational: Through another table of values the students are going to work
from the left and from the right side of the undefined value in separate table to
determine the behavior of the rational function. To begin the students it is best to
break the table apart into one table that is approaching from the left and the other
table that is approaching from the right. The students are going to derive that the
rational function is approaching this undefined value from both the left and right
side.
a) Complete the following table of values. Show all of your calculations.
x
3
f(x)
𝟑
𝒇(𝟑) =
= −𝟏. 𝟓
𝟐(𝟑) − 𝟖
3.5
𝒇(𝟑. 𝟓) =
3.9
𝒇(𝟑. 𝟗) =
3.99
𝒇(𝟑. 𝟗𝟗) =
3.999
4
𝒇(𝟑. 𝟗𝟗𝟗) =
𝒇(𝟒) =
𝟑. 𝟓
= −𝟑. 𝟓
𝟐(𝟑. 𝟓) − 𝟖
𝟑. 𝟗
= −𝟏𝟗. 𝟓
𝟐(𝟑. 𝟗) − 𝟖
𝟑. 𝟗𝟗
= −𝟏𝟗𝟗. 𝟓
𝟐(𝟑. 𝟗𝟗) − 𝟖
𝟑. 𝟗𝟗𝟗
= −𝟏𝟗𝟗𝟗. 𝟓
𝟐(𝟑. 𝟗𝟗𝟗) − 𝟖
𝟒
= 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
𝟐(𝟒) − 𝟖
b) What happens to the values of f(x) as x approaches 4 from the left (from values that
are less than 4)? Why does this occur?
The values of f(x) get smaller (more negative) at a very fast rate. This occurs
because the rational function inputs are getting very close to 4 where the rational
function is undefined, thus the rational function will only get more negative the
closer the input values get to the left of 4 but the rational function will never
exist at 4.
Bibb 6
c) Complete the following table of values. Show all of your calculations.
x
f(x)
𝟒
= 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
𝟐(𝟒) − 𝟖
4
𝒇(𝟒) =
4.001
𝒇(𝟒. 𝟎𝟎𝟏) =
4.01
𝒇(𝒙) =
𝟒. 𝟎𝟎𝟏
= 𝟐𝟎𝟎𝟎. 𝟓
𝟐(𝟒. 𝟎𝟎𝟏) − 𝟖
𝟒. 𝟎𝟏
= 𝟐𝟎𝟎. 𝟓
𝟐(𝟒. 𝟎𝟏) − 𝟖
𝟒. 𝟏
= 𝟐𝟎. 𝟓
𝟐(𝟒. 𝟏) − 𝟖
4.1
𝒇(𝒙) =
4.5
𝒇(𝒙) =
5
𝟒. 𝟓
= 𝟒. 𝟓
𝟐(𝟒. 𝟓) − 𝟖
𝒇(𝒙) =
𝟓
= 𝟐. 𝟓
𝟐(𝟓) − 𝟖
d) What happens to the values of f(x) as x approaches 4 from the right (from values that
are greater than 4)? Why does this occur?
The values of f(x) get smaller (less positive) at a very fast rate. This occurs
because the rational function inputs are getting very close to 4 where the rational
function is undefined, thus the rational function will only get less positive the
closer the input values get to the right of 4 but the rational function will never
exist at 4.
e) Using a calculator graph f. Explain how the graph supports your conclusions above.
The graph supports my conclusions above because it does show the rational
function to the left of 4 becoming more negative at an increasing rate, as well as,
to the right of 4 where the rational function is becoming less positive at a
decreasing rate.
Bibb 7
1
3. Given that ℎ(𝑥) = 𝑥−2 , answer the questions below.
Question Rational: Similar to the question above, the students are going to use a
table of values to approach the undefined value from the left and from the right side
of the rational function. The students previously used two table to do this, but now
they are going to be asked to compute it in one table. This makes it a little more
challenging and for the student to really be able to identify what they are looking for
to answer the question correctly.
a) Complete the following table for x values close to 2.
x
1.00
h(x)
𝟏
𝒉(𝟏. 𝟎𝟎) =
= −𝟏
(𝟏. 𝟎𝟎) − 𝟐
1.90
𝒉(𝟏. 𝟗𝟎) =
1.99
𝒉(𝟏. 𝟗𝟗) =
2.00
𝒉(𝟐. 𝟎𝟎) =
𝟏
= −𝟏𝟎
(𝟏. 𝟗𝟎) − 𝟐
𝟏
= −𝟏𝟎𝟎
(𝟏. 𝟗𝟗) − 𝟐
𝟏
= 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
(𝟐. 𝟎𝟎) − 𝟐
𝟏
= 𝟏𝟎𝟎
(𝟐. 𝟎𝟏) − 𝟐
2.01
𝒉(𝟐. 𝟎𝟏) =
2.10
𝒉(𝟐. 𝟏𝟎) =
3.00
𝟏
= 𝟏𝟎
(𝟐. 𝟏𝟎) − 𝟐
𝒉(𝟑. 𝟎𝟎) =
𝟏
=𝟏
(𝟑. 𝟎𝟎) − 𝟐
b) What happens to the values of h(x) as x approaches 2 from the left?
The values of h(x) get smaller (more negative) at a very fast rate. This occurs
because the rational function inputs are getting very close to 2 where the rational
function is undefined, thus the rational function will only get more negative the
closer the input values get to the left of 2 but the rational function will never
exist at 2.
c) What happens to the values of h(x) as x approaches 2 from the right?
The values of h(x) get smaller (less positive) at a very fast rate. This occurs
because the rational function inputs are getting very close to 2 where the rational
function is undefined, thus the rational function will only get less positive the
Bibb 8
closer the input values get to the right of 2 but the rational function will never
exist at 2.
d) Using a calculator graph h. Explain how the graph supports your conclusions above.
The graph supports my conclusions above because it does show the rational
function to the left of 2 becoming more negative at an increasing rate, as well as,
to the right of 2 where the function is becoming less positive at a decreasing rate.
A vertical asymptote is a vertical line (x = __) where the rational function is not defined. If f(x)
increases without bound or decreases without bound, as x approaches a real number b from the
left and from the right, then f has a vertical asymptote at x = b. We can say that there is a vertical
asymptote at b since the rational function will never be able to equal b but will become infinitely
close to b, without ever reaching b, as the input values gets closer and closer to the root of the
denominator. In Problem #1 above, a vertical asymptote occurred at p = 100.
Note: A rational function can have more than one vertical asymptote, or none at all.
4. Where did the vertical asymptotes occur in problems 2 and 3? What is significant about
these input values to the function?
Question Rational: Now that the students know that the rational function is
undefined at certain values, and that the undefined value is the vertical asymptote,
this question will solidify to the instructor that the student knows where the vertical
asymptote came from and what the value actually is.
In Problem #2, the vertical asymptote occurs at x=4 and in Problem #3, the vertical
asymptote occurs at x=2. These input values are significant to their rational
functions because this is the input value that makes the rational function undefined.
The rational function is undefined because the denominator cannot equal 0. These
input values make the denominator of their rational functions 0, thus these input
values create the vertical asymptote for their rational function.
5. For each of the functions below, estimate where a vertical asymptote will exist. Then test
your prediction with the graphing calculator.
Question Rational: These questions are no longer guiding the students through
finding the asymptote through a table of values. Although a student can still create
the table of values if they need, but these questions are designed to push them
towards generalizing how to find the vertical asymptote by looking at the rational
function because they have the full understanding as to why and where the
asymptote occurs.
a) 𝑔(𝑥) =
𝑥 4 +2
𝑥+1
Vertical Asymptote: x = 1
Bibb 9
b) 𝑚(𝑥) =
4(𝑥+3)
𝑥+6
Vertical Asymptote: x = -6
c) 𝑑(𝑥) =
𝑥+17
𝑥 2 +4𝑥−5
Factor the Denominator to: (x+5)(x-1) to find the zeros of the denominator
easier.
Vertical Asymptote: x = -5 & x = 1
6. Write an explanation describing how to determine the vertical asymptote of a rational
function and why this approach works.
Question Rational: This final question of the investigation is pushing the student to
generalize their understanding of how to determine the vertical asymptote and why
it exists in words, instead of through a context situation so the students can apply
their understanding for any rational function.
In order to determine the vertical asymptotes of a ration function we need to find
the input value(s) that make the rational function undefined. From previous
knowledge, we have learned that dividing by zero will make a function undefined.
Thus, to find the vertical asymptotes we must find the input value(s) that make the
denominator equal to zero. Once the denominator is equal to zero, the rational
function will become undefined. Those input value(s) are the value(s) of our vertical
asymptotes because the rational function will never equal those values.
Bibb 10
Horizontal Asymptotes of Rational Functions
Name: ____________________
1. Suppose the average cost (in dollars) per gram of a new pharmaceutical medication, A, of
800,000,000+10𝑔
producing g grams of the new medication is defined by𝐴 = 𝑓(𝑔) =
𝑔
Question Rational: Through this context situation the students are first going to
evaluate a table of values and find the rate of change to determine the behavior of
the rational function. After determining how the rational function is behaving, the
students are going to complete another table to values to lead them to the value that
the outputs of rational function is approaching. Thus the horizontal asymptote.
a) Evaluate A(8) and explain what the numerical answer means in its real-world context.
A(8) = 100,000,010
A pharmaceutical company sells 8 grams of a new medication, at an average cost
of $100,000,010 per gram.
b) Create a table of values below for A and the rate of change in terms of g, as g
increases from 0 to 50 in increments of 10 grams.
Grams of new
medication, g
Average cost
in dollars per
gram, A
0
undefined
Rate of Change
∆𝑨
∆𝒈
unknown
10
800,000,010
-$4,000,000 / gram
20
400,000,010
-$1,333,333.33 / gram
30
26,666,676.67
-$666,666.67 / gram
40
20,000,010
-$400,00 / gram
50
16,000,010
Bibb 11
c) What is happening to the average cost as each additional 10 grams is produced?
(Refer to the values of the rate of change from part (b) above)
As the number of grams produced increase the average cost is decreasing. From
the rate of change, we can see that the average cost is decreasing at a slower and
slower rate.
d) Complete the table of values below
Grams of new
medication, g
0
100
1
1,000
10,000
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000
10,000,000,000
100,000,000,000
Average cost in
dollars per gram,
A
undefined
$8,000,010
$800,010
$80,010
$8,010
$810
$90
$18
$10.80
$10.08
$10.008
e) What happens to the values of A as g approaches ∞?
As g approaches ∞, the values of A decrease but are approaching 10. Thus, as
the number of grams produces increases without bound, the average cost per
gram approaches $10.00.
A horizontal asymptote is a horizontal line (y = ___) where the rational function approaches. If
f(x) approaches a real number b as x increases without bound and/or as x decreases without
bound, then f has a horizontal asymptote at y = b. In Problem #1 above, the horizontal asymptote
occurs at A= 10.
Note: A rational function is able to cross its horizontal asymptote for a finite set of values.
Bibb 12
𝑥+1
2. Given that 𝑓(𝑥) = 2𝑥−8 , answers the questions below.
Question Rational: This question is designed to lead the students through how the
numerator is behaving and how the denominator is behaving separately when the
power of the lead coefficient is equal. Next, they will see how the rational function is
behaving as a whole, when the input values increase and decrease without bound.
Finally, developing the horizontal asymptote.
a) Complete the following table of values
x
-1,000,000
-10,000
-100
100
10,000
1,000,000
x+1
-999,999
-9,999
-99
101
10,001
1,000,001
2x-8
-2,000,008
-20,008
-208
192
19,992
1,999,992
f(x)
0.4999975
0.4997501
0.4759615
0.5260417
0.5002501
0.5000025
b) What happens to the values of f(x) as x increases without bound?
As x increases without bound, the values of f(x) approach 0.5.
c) What happens to the values of f(x) as x decreases without bound?
As x decreases without bound, the values of f(x) approach 0.5.
d) Where did the horizontal asymptote of f(x) occur?
The horizontal asymptote of f(x) will occur at y = 0.5.
8
3. Given that 𝑔(𝑥) = 𝑥+1 , answer the questions below.
Question Rational: This question is designed to lead the students through how the
denominator is behaving when the power of the lead coefficient is larger in the
denominator. Next, they will see how the rational function is behaving as a whole
when the input values increase and decrease without bound. Finally, developing the
horizontal asymptote.
Bibb 13
a) Complete the following table of values
x
-1,000,000
-10,000
-100
100
10,000
1,000,000
x+1
-999,999
-9,999
-99
101
10,001
1,000,001
g(x)
-0.000008000008
-0.0008008
-0.08080808
0.079208
0.0007992
0.000007999992
b) What happens to the values of g(x) as x increases without bound?
As x increases without bound, the values of g(x) approach 0.
c) What happens to the values of g(x) as x decreases without bound?
As x decreases without bound, the values of g(x) approach 0.
d) Where did the horizontal asymptote of g(x) occur?
The horizontal asymptote of g(x) will occur at y = 0.
𝑥2
4. Given that ℎ(𝑥) = 𝑥+4 , answer the questions below.
Question Rational: This question is designed to lead the students through how the
numerator is behaving and how the denominator is behaving separately when the
power of the lead coefficient is larger in the numerator. Next, they will see how the
rational function is behaving together as the input values increase and decrease
without bound. Finally, developing the horizontal asymptote.
a) Complete the following table of values
x
-1,000,000
-10,000
-100
100
10,000
1,000,000
x2
1,000,000,000,000
100,000,000
10,000
10,000
100,000,000
1,000,000,000,000
x+4
--999,996
-9,996
-96
104
10,004
1,000,004
h(x)
-1,000,004
-10,004.0016
-104.16667
96.1538
9996.001599
999,996
b) What happens to the values of h(x) as x increases without bound?
As x increases without bound, the values of h(x) increase without bound.
Bibb 14
c) What happens to the values of h(x) as x decreases without bound?
As x decreases without bound, the values of h(x) decrease without bound.
d) Where did the horizontal asymptote of h(x) occur?
The horizontal asymptote of h(x) does not exist.
5. Generalize your reasoning’s from Problems 2, 3, and 4 above:
Question Rational: This question is designed for the students to be able to generalize
their reasoning’s from the questions about. There are three different ways to
determine a horizontal asymptote, thus the questions previously and this question is
essential for the students to have the complete understanding behind horizontal
asymptotes.
a) What conditions are necessary for a rational function to have a horizontal asymptote
at y = 0?
When the power of the leading coefficient of the numerator is less than the
power of the leading coefficient of the denominator the denominator is getting
infinitely small, thus the whole function is getting infinitely small so the function
is going to approach 0. Thus there is a horizontal asymptote at y = 0.
b) What conditions are necessary for a rational function not to have a horizontal
asymptote?
When the power of the leading coefficient of the numerator is greater than the
power of the leading coefficient of the denominator the numerator is getting
infinitely larger, thus the whole function is getting infinitely larger so the
function is going to approach ∞. Thus there is no horizontal asymptote.
c) What conditions are necessary for a rational functions to have a horizontal asymptote
at y = b where b ≠ 0? How can you determine the exact location of the horizontal
asymptote?
When the power of the leading coefficient of the numerator is equal to the power
of the leading coefficient of the denominator, so the function “pulls” evenly. The
numerator does not outgrow the denominator and vice versa, thus the horizontal
asymptote is found by dividing the ratio of the leading coefficient of the
numerator by the leading coefficient of the denominator.
Bibb 15
Homework
I chose this particular questioning sequencing because the students now have been
accustomed to both vertical and horizontal asymptotes thus I began their homework with
individual questions focusing on just vertical or horizontal asymptotes. The rational functions
that they are given are sequenced from basic to more complex. After they were given questions
that asked to find both the vertical and horizontal of one function.
Vertical and Horizontal Asymptotes Homework
Name: _____________________
For Exercises 1 – 3, find the vertical asymptotes, if any, for each rational function.
Question Rational: The first three questions are asked for the students to feel concrete in
their understandings of vertical asymptotes separately before asking them to find the
vertical and horizontal asymptotes of one rational function.
1. 𝑚(𝑥) =
𝑥−3
𝑥−7
Vertical Asymptote: x = 7
2. 𝑛(𝑥) =
𝑥 2 −9
𝑥 3 +9𝑥 2
Need to factor the denominator first to x2(x+9) in order to find the zeros of the
denominator
Vertical Asymptote: x = -9 and x = 0
3. 𝑘(𝑥) =
𝑥(2−𝑥)
(𝑥−6)
Vertical Asymptote: x = 6
For Exercises 4 – 6, find the horizontal asymptotes, if any, for each rational function.
Question Rational: The next three question are asked for the students to feel concrete in
their understandings of horizontal asymptotes separately before asking them to find the
vertical and horizontal asymptotes of one rational function.
3
4. 𝑓(𝑥) = 2𝑥−6
Horizontal Asymptote: y = 0
5. 𝑔(𝑥) =
2𝑥 2 +𝑥−3
2𝑥−2
Horizontal Asymptote: Does Not Exist
Bibb 16
6. ℎ(𝑥) =
(1−𝑥)(2+3𝑥)
2𝑥 2 +1
𝟑
Horizontal Asymptote: y = − 𝟐
7.
Let 𝑓(𝑥) =
1
(𝑥+3)2
Question Rational: The final question takes the students back to similar tables that
they saw in their investigation but now they have been combined together for the
students to find both the vertical and horizontal asymptote of the rational function.
a) Complete the table below for x values close to -3
x
f(x)
-4.00
-3.10
1
-3.01
-3.00
-2.99
-2.90
-2.00
1
100
10,000
undefined
10,000
100
1
b) What happens to the values of f(x) as x approaches -3 from the left and from the
right?
The values of f(x) will increase without bound as the x values approach -3 from
the left and from the right.
c) Complete the tables below
x
f(x)
x
f(x)
5
50
1
500
5,000
50,000
0.015625
0.000356
0.0000039524
0.00000000399952
0.0000000004
-5
-50
1
-500
-5,000
-50,000
0.25
0.00045269
0.0000040484
0.000000040048
0.0000000004
Bibb 17
d) Use the tables above to determine what happens to the values of f(x) as the x values
increase and decreases without bound.
The values of f(x) will approach 0 as the x values increase and decrease without
bound.
e) Using the above information, give the equations for the horizontal and vertical
asymptotes.
Vertical Asymptote: x = -3
Horizontal Asymptote: y = 0
8. The speed limit on most Interstate Highways in Arizona through rural areas is 75mph.
Question Rational: This question is asking the students to find the horizontal and
vertical asymptotes through a context situation. This problem is design to pull back
on the student’s previous knowledge and integrating in the new knowledge that they
have learned. The question might be difficult for students because the asymptotes
are at y = 0 and x = 0, thus they really need to understand asymptotes in order to
find these for this rational function.
a) If you maintain an average speed of 75 mph, how long will it take to make a 150
mile journey?
𝟏 𝒉𝒐𝒖𝒓
𝟏𝟓𝟎 𝒎𝒊𝒍𝒆𝒔
×
= 𝟐 𝒉𝒐𝒖𝒓𝒔
𝟕𝟓 𝒎𝒊𝒍𝒆𝒔
𝟏
If you maintain an average speed of 75 mph, it will take you 2 hours to make a
150 mile journey.
b) Write a function, f, which defines the average speed (in mph), s, as a function of
travel time (in hours), t for a 150 mile journey.
𝒕 = 𝒇(𝒔)
𝟏𝟓𝟎
𝒇(𝒔) =
𝒔
Bibb 18
c) When there are fatal accidents, the highway patrol officers slow or stop the traffic on
the highway. Create a table for the time it takes to travel 150 miles when traveling at
s = 20, 15, 10, 5, 4, 3, 2, 1 miles per hour.
Average speed (in miles per hour), s Travel time (in hours), t
20
7.5
15
10
10
15
5
30
4
37.5
3
50
2
75
1
150
d) As the average speed, s, increases, what happens to the travel time, t? Explain what
this means in this real-world context.
As the average speed of a person traveling a 150 mile journey increases, the
travel time in hours will decrease. Thus, the higher the average speed is, the less
time it will take to complete your 150 mile journey.
e) Give the equations for the horizontal and vertical asymptotes and explain what these
asymptotes mean in this context.
Horizontal Asymptote: y = 0
Vertical Asymptote: x = 0
This means that a person can never have an average speed of 0 miles per hour
and a person’s travel time can never be 0 hours for this journey.
Bibb 19
Review
Vertical and Horizontal Asymptotes Review
Name: _____________________
Date of Quiz: ___________________________________
Outlined below are the key concepts and ideas that you will be tested on. You are
responsible for having a solid understanding of all concepts and being able to apply them to
problems. As always, review your investigations, homework and classroom note in order to fully
prepare for the quiz. All the problems can be found in the student workbook. The problems with
an * are due on the day of the quiz, but I would suggest practicing all of them.
(I am assuming that the students will have access to the Pathways workbook for extra
review problems or I would photocopy these problem sets for the students and attach it to
their review)
Vertical Asymptotes
 Define why a vertical asymptote exists
 Can you identify the vertical asymptote of a rational function from an equation or a
graph?
 Determine how the rational function is behaving as the input values approach the
asymptote from the left and from the right
 Practice Problems: (p. 251 – 253) #2*, 6, 8*, 9* (b, e, and f)
Horizontal Asymptotes
 Can you identify the horizontal asymptote of a rational function from an equation or a
graph?
 Define why a horizontal asymptote exists at:
o y=0
𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑝𝑜𝑤𝑒𝑟
o y = 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 ℎ𝑖𝑔ℎ𝑒𝑟 𝑝𝑜𝑤𝑒𝑟

o No Horizontal Asymptote
Determine how the rational function is behaving as the input values are increasing and
decreasing without bound.
 Practice Problems: (p. 254 – 265) #10*, 11, 14*, 16* (c, d, f, and h)
** Do not solely rely on this review sheet. Review your notes, chapter readings, worksheets,
and previous homework problems.
Any Questions? Please come to my room, tutoring center, or send emails. I am here to help.
Email: lbibb@asu.edu
Bibb 20
Quiz
Vertical and Horizontal Asymptotes Quiz
1. Let 𝑓(𝑥) =
Name: _____________________
(1−𝑥)(2+3𝑥)
𝑥−9
f) Complete the table below for x values close to 9
x
f(x)
10.00
9.10
1
9.01
9.00
8.99
8.90
8.00
-288
-2373.3
-23253.03
undefined
23147.03
2267.3
182
g) What happens to the values of f(x) as x approaches 9 from the left and from the right
(use your table and graphing calculator)?
As the x values approach 9 from the left, f(x) is decreasing until reaching the
minimum where f(x) will begin to increase without bound.
As the x values approach 9 from the right, f(x) is increasing until reaching the
maximum where f(x) will begin to decrease without bound.
h) Complete the tables below
x
f(x)
x
f(x)
5
50
1
500
5,000
17
-181.6585
-1526.473
-15026.05
50,000
-150026
-5
-50
1
-500
-5,000
-50,000
5.5714
127.93
1474.4558
14974.046
149974
i) Use the tables above to determine what happens to the values of f(x) as the x values
increase and decreases without bound.
As the x values increase, f(x) decreases without bound.
As the x values decrease, f(x) increases without bound.
Bibb 21
j) Using the above information, give the equations for the horizontal and vertical
asymptotes.
Vertical Asymptote: x = 9
Horizontal Asymptote: Does Not Exist
2. Which of the statements is false about horizontal asymptotes?
(a) A rational function has a horizontal asymptote if the power of the leading
coefficient of the numerator is less than or equal to the power of the leading
coefficient of the denominator.
(b) The x-axis is the horizontal asymptote of a rational function if the power of the
leading coefficient is smaller than the power of the leading coefficient of the
denominator.
(c) For a rational function to have no horizontal asymptote, the power of the leading
coefficient of the numerator is greater than the power of the leading coefficient of
the denominator.
(d) The end behavior of a rational function always resembles a horizontal line.
(e) The line y = 6 will be the horizontal asymptote for the ration function:
𝑦=
12𝑥 2 +6𝑥+1
2𝑥 2 +4𝑥+2
3. For the table below, give the equation of the vertical asymptote.
x
f(x)
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
3.0
2.8
2.5
2.1
1.7
1.0
0
-1.7
-5.0
-15.0
undefined
25
15
Vertical Asymptote: x = 4.0
Bibb 22
4. For the table below, circle the correct equation of the horizontal asymptote.
x
f(x)
-6.0
0.0
6.0
12.0
16.0
21.0
36.0
46.0
96.0
196.0
396.0
1096.0
4096.0
-3.000
0.000
0.600
0.750
0.800
0.840
0.900
0.920
0.960
0.980
0.990
0.996
0.999
(a) x = 1
(b) y = 0
(c) y = 1
(d) x = -1
(d) Does Not Exist
(e) None of The Above
5. The value of a new car begins to depreciate the minute that the car is drove off of the lot.
Assume that the new car costs C dollars, and the value of the car as scrap metal is T
dollars. The life of the car is N years, then the average amount in dollars, D, by which the
𝐶−𝑇
car loses value each year is given by the multivariable function, D(C,T,N) = 𝑁
a) If the purchase price of the car is $26,000 and the value of the scrap metal is $1,500,
write a formula for D as a function of N
𝟐𝟔𝟎𝟎𝟎 − 𝟏𝟓𝟎𝟎
𝐃(𝟐𝟔𝟎𝟎𝟎, 𝟏𝟓𝟎𝟎, 𝐍) =
𝑵
b) Complete the table of values below for D
N
D(N)
1
24500
3
8166.67
5
4900
7
3500
9
2722.22
11
2227.27
13
1884.62
15
1633.33
Bibb 23
c) Using the above information, give the equations for the horizontal and vertical
asymptotes and explain what they mean in terms of the context situation.
Vertical Asymptote:
This represents that the average amount that the car loses value each year
can never be $0.
Horizontal Asymptote: x = 0
This represents that the life of the car can never be 0 years.
6. A rational function has two vertical asymptotes at x = 7 and x = -3, and a horizontal
asymptote at y = 3. Which of the following could be the equation of this rational
function?
(a) 𝒚 =
(b) 𝒚 =
(c) 𝒚 =
(d) 𝒚 =
(e) 𝒚 =
𝟑𝒙𝟐 +𝟏
𝟗𝒙𝟐 +𝟒𝒙−𝟐𝟏
𝒙𝟐 +𝟏𝟎𝒙+𝟐𝟏
𝟑𝒙𝟐
𝟗𝒙𝟐 +𝟒𝒙−𝟐𝟏
𝟑𝒙𝟐
𝟏𝟐𝒙𝟐 +𝟓
𝟒𝒙𝟐 +𝟒𝒙−𝟐𝟏
𝟑𝒙𝟐
𝒙𝟐 +𝟒𝒙−𝟐𝟏