Intercepts, Zeroes and Asymptotes of Rational Functions: A High School Mathematics Module, Lecture notes of Mathematics

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General Mathematics

Quarter 1 – Module 9 :

Intercepts, Zeroes and

Asymptotes of Rational

Functions

General Mathematics

Alternative Delivery Mode

Quarter 1 – Module 9 : Intercepts, Zeroes and Asymptotes of Rational Functions

First Edition, 2020

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iii

Introductory Message

For the facilitator:

Welcome to the General Mathematics Alternative Delivery Mode (ADM) Module on

Determining the Intercepts, Zeroes and Asymptotes of Rational Functions!

This module was collaboratively designed, developed and reviewed by educators from

public institutions to assist you, the teacher or facilitator in helping the learners

meet the standards set by the K to 12 Curriculum while overcoming their personal,

social, and economic constraints in schooling.

This learning resource hopes to engage the learners into guided and independent

learning activities at their own pace and time. Furthermore, this also aims to help

learners acquire the needed 21st century skills while taking into consideration their

needs and circumstances.

In addition to the material in the main text, you will also see this box in the body of

the module:

As a facilitator you are expected to orient the learners on how to use this module.

You also need to keep track of the learners' progress while allowing them to manage

their own learning. Furthermore, you are expected to encourage and assist the

learners as they do the tasks included in the module.

For the learner:

Welcome to the General Mathematics Alternative Delivery Mode (ADM) Module on

Determining the Intercepts, Zeroes and Asymptotes of Rational Functions!

The hand is one of the most symbolized part of the human body. It is often used to

depict skill, action and purpose. Through our hands we may learn, create and

accomplish. Hence, the hand in this learning resource signifies that you as a learner

is capable and empowered to successfully achieve the relevant competencies and

skills at your own pace and time. Your academic success lies in your own hands!

This module was designed to provide you with fun and meaningful opportunities for

guided and independent learning at your own pace and time. You will be enabled to

process the contents of the learning resource while being an active learner.

Notes to the Teacher

This contains helpful tips or strategies that

will help you in guiding the learners.

v

At the end of this module you will also find:

The following are some reminders in using this module:

1. Use the module with care. Do not put unnecessary mark/s on any part of the

module. Use a separate sheet of paper in answering the exercises.

2. Don’t forget to answer What I Know before moving on to the other activities

included in the module.

3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.

If you encounter any difficulty in answering the tasks in this module, do not

hesitate to consult your teacher or facilitator. Always bear in mind that you are

not alone.

We hope that through this material, you will experience meaningful learning and

gain deep understanding of the relevant competencies. You can do it!

References This is a list of all sources used in developing

this module.

I. Choose the letter of the best answer. Write the chosen letter on a separate

sheet of paper.

1. Which of the following is the set of all values that the variable x can take?

a. Range

b. Intercept

c. Domain

d. Zeroes

2. What is the domain of f(x) =

𝑥− 3

𝑥+ 3

a. All real numbers

b. All real numbers except – 3

c. All real numbers except 3

d. Cannot be determined

3. What value/s of x that will make the function meaningless f(x) =

𝑥− 1

𝑥

a. x = - 1

b. x = 0

c. x = 1

d. All real numbers

4. Complete the sentence: The real numbers zeroes are also ____________ of the

graph of the function.

a. Asymptote

b. x – intercepts

c. y – intercepts

d. Range

5. Which of the following is the set of all values that f(x) can take?

a. Range

b. Intercept

c. Domain

d. Zeroes

6. What is the range of f(x) =

1

𝑥

a. R =

b. R =

c. R = {𝑦|𝑦 ≠ 1 }

d. R =

7. Which of the following is a true statement?

a. A rational function is a quotient of functions.

b. Asymptotes are a common characteristic of rational functions.

c. An asymptote is a line that a graph approaches, but does not touch.

d. All of the above.

8. If the degree of the leading coefficient of the numerator is equal to the

degree of the leading coefficient of the denominator of a rational

function, which of the following statements has to be true?

a. The graph has no asymptote

b. The graph of the function has slant asymptote

c. The graph of the function has a horizontal asymptote

d. None of the above

9. What is the horizontal asymptote of 𝑓(𝑥) =

𝑥+ 5

3 𝑥

2

a. y = 3

b. y = 0

c. y = - 2

d. y = - 3

10. What is the vertical asymptote of 𝑓

3 𝑥+ 1

𝑥− 5

a. x = 5

b. x = 3

c. x = 1

d. x = 0

11. What is the oblique asymptote of 𝑓

𝑥

2

− 3 𝑥

𝑥+ 3

a. y = 3x

b. y = x – 6

c. y = x - 3

d. y = 3x + 6

12. Oblique asymptote occurs when there is no horizontal asymptote,

the statement is ____________.

a. Always true

b. Sometimes true

c. Never true

d. Cannot be determined

13. How will you describe the horizontal asymptote of 𝑓

3

3 +𝑥

a. does not exist

b. approaching at x = 3

c. approaching at y = - 3

d. approaching at y = 0

Lesson

Intercepts, Zeroes, and Asymptotes of

Rational Functions

In the previous lesson, you learned how to find domain and range of a rational

function. In this particular lesson, determining intercepts, zeroes and asymptotes of

rational functions will be done. Knowing fully the concept of the different properties

of rational function will be your guide to easily determine the behavior of a rational

function and it will prepare you for the next topic which is about graphing rational

function.

What’s In

Let’s recall first what you have learned from the previous lesson by answering the

following questions:

A. Which of the following is an example of rational function?

1. F(x) =

3 𝑥

2

• 1

𝑥− 1

𝑥

3

8

3

1

3 𝑥− 1

B. Find the domain and range of the functions.

1. F(x) =

𝑥

𝑥+ 3

2. f(x) =

3

𝑥− 4

3. g(x) =

𝑥+ 1

𝑥

2

− 1

Let us see if you got the correct answer in the activity, if your answer in question A

is number 1, you got it right you have a clear understanding of the concept of rational

function but if you are incorrect allow me to help you recall what a rational function

is, when two polynomial functions are expressed as a quotient and can be written in

the form 𝑓

𝑝(𝑥)

𝑞(𝑥)

and q(x) is a not the zero function it is called a rational function.

Numbers 2 and 3 are not examples of rational function, it is a rational equation and

rational inequality, respectively. Number 1 is written as the quotient of two

polynomial functions, so it is a rational function.

For activity B, let us review the meaning of domain and range of the function.

Domain is the set of first coordinates of a relation and it is the value of x that will

not make the denominator of the function equal to zero while Range is the set of

second coordinates. To determine the domain of rational function, simply equate the

denominator to zero and then solve for x, this value should be avoided so that the

function will not give an undefined or a meaningless function. Example find the

domain of F(x) =

x

x+ 3

, equating the denominator to zero, we have x + 3 = 0, so the

value of x = - 3, so the domain of the function are all real numbers except - 3 remember

we will avoid value/s that will make our denominator equal to zero, so if we will

substitute - 3 to our x in the denominator it will result to 0 and it will give us an

undefined function. In notation, D= (- ∞, −𝟑) ∪ (−𝟑, ∞)

To find the range of the function, change f(x) to y then, solve for x; remember range

are real values of y that will make a real value for the function. For example, find the

range of F(x) =

𝑥

𝑥+ 3

Changing F(x) to y, the new function is y =

𝑥

𝑥+ 3

By doing cross multiplication we have y(x+3) = x

Distributing y we now have xy + 3y = x

Simplifying the equation will give xy – x = 3y

Factoring the left side of the equation x(y – 1) = 3y

Dividing the equation by (y – 1)

𝑥(𝑦− 1 )

(𝑦− 1 )

3 𝑦

(𝑦− 1 )

Removing common factor, the value of x 𝑥 =

3 𝑦

𝑦− 1

Since we are looking for the value of y that will give a real value for the function so

we need to find value/s for y that will not make the denominator equal to 0.

Equating the denominator to zero y – 1 = 0

So, y = 1.

The range of the function F(x) =

𝑥

𝑥+ 3

is all real values of y except 1. In notation,

R= (- ∞, 𝟏) ∪ (𝟏, ∞).

The following are the answers to Activity B

1. Domain = {x/x≠ − 3 } or (- ∞, −𝟑) ∪ (−𝟑, ∞)

Range = {y/y ≠ 1 } or (- ∞, 𝟏) ∪ (𝟏, ∞)

2. Domain = {x/x≠ 4 } or (- ∞, 𝟒) ∪ (𝟒, ∞)

Range = {y/y ≠ 0 } or (- ∞, 𝟎) ∪ (𝟎, ∞)

3. Domain = {x/x≠ − 1 𝑜𝑟 𝑥 ≠ 1 } or (- ∞, −𝟏) ∪ (−𝟏, 𝟏) ∪ (𝟏, ∞)

Range = {y/y ≠ 0 } or (- ∞, 𝟎) ∪ (𝟎, ∞)

Activity

I – Connect Mo!

Connect the given statement/phrase in column A with the answer in column

B to complete the statement/phrase in column A. Write the letter of your

answer in a separate sheet of paper.

COLUMN A COLUMN B

1. The intercepts of the graph M. the x - intercepts

of a rational function …

2. To find the x – intercept A. let x = 0

of a function …

3. The zeroes of the function G. rational function

is also …

4. To find the y – intercept I. are the points of

of a function … intersection of its

graph and an axis

5. The function of the form C. let y = 0

, where g(x)

& h(x) are polynomials

What’s New

How was the activity? I believed that you connected it right. So, in this lesson, you

will know how to identify intercepts, zeroes and asymptotes of rational function.

What is It

INTERCEPTS AND ZEROES OF RATIONAL FUNCTIONS

The intercepts of the graph of a rational function are the points of intersection of its

graph and an axis.

The y-intercept of the graph of a rational function r(x) if it exists, occurs at r(0),

provided that r(x) is defined at x = 0. To find y-intercept simply evaluate the function

at x = 0.

The x-intercept of the graph of a rational function r(x), if it exists, occurs at the zeros

of the numerator that are not zeros of the denominators. To find x – intercept equate

the function to 0.

The zeroes of a function are the values of x which make the function zero. The

numbered zeroes are also x-intercepts of the graph of the function.

EXAMPLES.

1. Find the x- and y – intercepts, of the following rational functions:

a. f(x) =

3 −𝑥

𝑥+ 1

b. f(x) =

3 𝑥

𝑥+ 3

c. f(x) =

𝑥

2

− 3 𝑥+ 2

𝑥

2

− 4

y-intercept

x-intercept

Figure 1. x and y intercepts using GeoGebra

zero of the

function

2. f(x) =

3 𝑥

𝑥+ 3

To find the x – intercept, simply equate the numerator to 0,

0 = 3x Equate the numerator to 0.

3x = 0 By Symmetric Property of Equality.

3 𝑥

3

=

0

3

Simplifying the fraction by multiplying

both sides by 1/3.

x = 0

So, the x – intercept is 0 or (0, 0).

To find the y – intercept, change the x value of the function to 0.

3 𝑥

𝑥+ 3

Substitute 0 to x values of the function.

3 ( 0 )

0 + 3

Simplifying the fraction.

0

3

= 0 The value of 𝑓(𝑥) or y – intercept.

So, the y – intercept is 0 or (0, 0).

𝑥

2

− 3 𝑥+ 2

𝑥

2

− 4

2

− 3 𝑥 + 2 = 0 Equate the numerator to 0.

(x – 2) (x – 1) = 0 By factoring.

x – 2 = 0 x – 1 = 0 Solve for x, by Zero product property.

x = 2 x = 1

So, the x – intercepts are x = 2 and x = 1. But by looking at the denominator

of the original function if we substitute 2 to the value of x,

x

2

• 4 = (2)

2

• 4 = 0,

The denominator will become 0, the function becomes meaningless.

So, we will only accept x – intercept at x = 1 or (1, 0).

To find the y – intercept:

𝑥

2

− 3 𝑥+ 2

𝑥

2

− 4

, change the x value of the function to 0.

( 0 )

2

− 3 ( 0 )+ 2

( 0 )

2

− 4

Simplify the fraction.

f(x) =

2

− 4

Reduce the fraction to lowest term.

1

2

The value of f(x) or y.

So, the y – intercept is −

1

2

or (0, −

1

2

2. Determine the zeroes of the following rational functions:

a. g(x) =

𝑥− 2

𝑥+ 6

b. H(x) =

𝑥− 3

𝑥

2

− 9

c. G(x) =

𝑥

2

+𝑥− 2

𝑥

2

− 4

To find the zeroes of a rational function, equate the function to 0 or solve for the x –

intercept of the function by equating the numerator to 0.

a. g(x) =

𝑥− 2

𝑥+ 6

x – 2 = 0 Equate the numerator to 0.

x = 2 Solve for x.

Thus, the zero of g(x) is 2.

b. 𝐻

𝑥− 3

𝑥

2

− 9

𝑥− 3

𝑥

2

− 9

Simplify by factoring the denominator.

𝑥− 3

(𝑥− 3 )(𝑥+ 3 )

Remove common factors.

1

𝑥+ 3

= 0 Equate the numerator to 0.

1 = 0 False statement.

So, there is no zero of the function. Which means that no point on the

graph touches the x – axis.