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

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

Senior High School

[illustration: teachers and learners studying outdoors and at home, with a globe and a lamp]

CO_Q1_General Mathematic SHS Module 9

General Mathematics Alternative Delivery Mode Quarter 1 – Module 9: Intercepts, Zeroes and Asymptotes of Rational Functions First Edition, 2021

Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.

Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Printed in the Philippines by ________________________

Development Team of the Module

Writers: Jerson D. Jolo Editors: Elizabeth B. Dizon, Anicia J. Villaruel, Roy O. Natividad Reviewers: Fritz A. Caturay, Necitas F. Constante, Celestina M. Alba, Jerome A. Chavez, Edna G. Adel, Lirio G. Parale, Flora P. Segovia Illustrator: Dianne C. Jupiter Layout Artist: Noel Rey T. Estuita, Mark Anthony N. Banta Management Team: Francis Cesar B. Bringas, Job S. Zape Jr., Ramonito Elumbaring, Reicon C. Condes, Elaine T. Balaogan, Fe M. Ong-ongowan, Elias A. Alicaya Jr., Gregorio A. Co Jr., Gregorio T. Mueco, Herbert D. Perez, Lorena S. Walangsumbat, Jee-Ann O. Borines, Asuncion C. Ilao

Department of Education – Region 4A CALABARZON

Office Address: Gate 2 Karangalan Village, Brgy. San Isidro, Cainta, Rizal Telefax: 02-8682-5773/8684-4914/8647-7487 E-mail Address: lrmd.calabarzon@deped.gov.ph

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

Senior High School

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

This module was designed and written to help you determines the intercepts, zeroes and asymptotes of rational functions. Knowing what a rational function is, you are now ready to learn its other properties. It includes finding the intercepts, zeroes and asymptotes. These will be your guide to easily determine the behavior of a rational function and will prepare you for graphing rational function. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lesson is arranged to follow the standard sequence of the course.

In this module you will determine the intercepts, zeroes and asymptotes of rational functions.

The module consists of one lesson namely: Intercepts, Zeroes, and Asymptotes of Rational Functions.

After going through this module, you are expected to:

1. recall the meaning of intercepts, zeroes and asymptotes;
2. identify the intercepts, zeroes and asymptotes of rational functions;
3. solve for the intercepts, zeroes and asymptotes of rational functions.

[illustration: hand pressing a button]

What I Know

In this part, let us see how much you know about the lesson by answering the questions in pre-assessment below. If you obtain 100% or a perfect score, skip the module and immediately move to the next module. While in the event you missed a point, please proceed on the module as it will enrich your knowledge in finding the intercepts, zeroes and asymptotes of rational functions. Let’s get started!

[illustration: hand holding a pencil]

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) = $\frac{x-3}{x+3}$?
   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) = $\frac{x-1}{x}$?
   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) = $\frac{1}{x}$?
   a. R = ${y \mid y = 1}$
   b. R = ${y \mid y = 0}$
   c. R = ${y \mid y \neq 1}$
   d. R = ${y \mid y \neq 0}$

   
   

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 $f(x) = \frac{x+5}{3x^2}$?
   a. y = 3
   b. y = 0
   c. y = – 2
   d. y = – 3

   
   

10. What is the vertical asymptote of $f(x) = \frac{3x+1}{x-5}$?
    a. x = 5
    b. x = 3
    c. x = 1
    d. x = 0

    
    

11. What is the oblique asymptote of $f(x) = \frac{x^2-3x}{x+3}$?
    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 $f(x) = \frac{3}{3+x}$?
    a. does not exist
    b. approaching at x = 3
    c. approaching at y = -3
    d. approaching at y = 0

    
    

14. If the x – intercept of a rational function is at x = 5, what is the zero of the function?
    a. x = 5
    b. x = 0
    c. x = – 5
    d. cannot be determined

    
    

15. What is the y – intercept of $f(x) = \frac{2x^2+x+3}{2x^2+3x+1}$?
    a. 3
    b. 0
    c. – 3
    d. – 6

    
    

Lesson 1: 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.

[illustration: hands holding a puzzle piece]

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) = $\frac{3x^2+1}{x-1}$
2. $\frac{x}{3} = \frac{8}{3}$
3. $\frac{1}{3x-1} + 3 < 0$

B. Find the domain and range of the functions.

1. F(x) = $\frac{x}{x+3}$
2. f(x) = $\frac{3}{x-4}$
3. g(x) = $\frac{x+1}{x^2-1}$

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 $f(x) = \frac{p(x)}{q(x)}$ 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.

[transcriber’s note: fractions are written inline with “/”, and arrows in the printed step list are shown as “→”]

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) = 3y/(y−1) [transcriber’s note: the printed figure shows the (y−1) factor on the left side crossed out]
• Removing common factor, the value of x → x = 3y/(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, ∞)

   
   

How is your review of the rational function? I believed you got it all correct. Are you ready to learn new things about rational functions? Let’s do the next activity.

Notes to the Teacher

The teacher may say that “the domain refers to the set of possible input values and range is the set of possible output values” is related to the saying “you saw what you reap”. Like in our day-to-day activities if we show good deeds to others, in return we will receive the same treatment.

[illustration: hand holding a sprouting plant, icon beside “What’s New”]

What’s New

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.

How was the activity? I believed that you connected it right. In the activity you were asked to find the intercepts and zeroes of rational function. Just like in our lives we are looking for something to achieve our goal in life such as harmonious family relationship, good grades, finish senior high school and college to find a good job all these things are related to each other that will make us happy and contented. So, in this lesson, you will know how to identify intercepts, zeroes and asymptotes of rational function.

[illustration: magnifying glass icon beside “What is It”]

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.

Column A	Column B
1. The intercepts of the graph of a rational function …	M. the x – intercepts
2. To find the x – intercept of a function …	A. let x = 0
3. The zeroes of the function is also …	G. rational function
4. To find the y – intercept of a function …	I. are the points of intersection of its graph and an axis
5. The function of the form f(x) = g(x)/h(x), where g(x) & h(x) are polynomials	C. let y = 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.

[illustration: GeoGebra graph of a rational function with three labels: “y-intercept” (arrow to a black dot on the y-axis), “zero of the function” (arrow to a black dot on the x-axis), and “x-intercept” (arrow to the same dot on the x-axis); Figure 1. x and y intercepts using GeoGebra]

EXAMPLES.

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

   1. f(x) = (3−x)/(x+1)
   2. b. f(x) = 3x/(x+3)
   3. f(x) = (x²−3x+2)/(x²−4)

2. Determine the zeroes of the following rational functions:
   a. g(x) = (x−2)/(x+6)
   b. h(x) = (x−3)/(x²−9)
   c. G(x) = (x²+x−2)/(x²−4)

   
   

SOLUTIONS.

1. To find x – intercept equate the function to 0.
   f(x) = (3 − x)/(x + 1)

   
   
   
   
   • 0 = (3−x)/(x+1) → Equate the function to 0.
   
   
   • (3−x)/(x+1) = 0 → By Symmetric Property of Equality.
   
   
   • 3 – x = 0 → Multiply both sides by (x + 1).
   
   
   • 3 + (−3) – x = 0 + (−3) → By Addition Property of Equality (APE).
   
   
   • − x = − 3 → Simplify.
   
   
   • (−1) (−x) = (−1) (−3) → By Multiplication Property of Equality (MPE).
   
   
   • x = 3
   
   
   
   

   So, the x – intercept is (3, 0).

   
   

   By analyzing the example, we can say that to find the x – intercept simply equate the numerator of the function to 0.

   
   

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

   
   
   
   
   • f(x) = (3−x)/(x+1) → Substitute 0 to x values of the function.
   
   
   • f(x) = (3−0)/(0+1) → Simplifying the fraction.
   
   
   • f(x) = 3/1 = 3 → Value of f(x) or y.
   
   
   
   

   So, the y – intercept is 3 or (0, 3).

   
   

2. f(x) = 3x/(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.
   
   
   • 3x/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.

   
   
   
   
   • f(x) = 3x/(x+3) → Substitute 0 to x values of the function.
   
   
   • f(x) = 3(0)/(0+3) → Simplifying the fraction.
   
   
   • f(x) = 0/3 = 0 → The value of f(x) or y – intercept.
   
   
   
   

   So, the y – intercept is 0 or (0, 0).

   
   

3. f(x) = (x²−3x+2)/(x²−4)

   
   
   
   
   • x² − 3x + 2 = 0 → Equate the numerator to 0.
   
   
   • (x – 2) (x – 1) = 0 → By factoring.
   
   
   • x – 2 = 0 → Solve for x, by Zero product property.
   
   
   • x = 2
   
   
   • x – 1 = 0
   
   
   • 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,

   
   

   x² – 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:

f(x) = (x²−3x+2)/(x²−4), change the x value of the function to 0.

• f(x) = ((0)²−3(0)+2)/((0)²−4) → Simplify the fraction.
• f(x) = 2/−4 → Reduce the fraction to lowest term.
• f(x) = −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)/(x²−9) c. G(x) = (x²+x−2)/(x²−4)

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) = (x−3)/(x²−9)

• H(x) = (x−3)/(x²−9) → Simplify by factoring the denominator.
• 1/((x+3)) after removing the common factor (x−3) from (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.

c. G(x) = (x²+x−2)/(x²−4)

• G(x) = ((x+2)(x−1))/((x+2)(x−2)) → Simplify by factoring both the numerator and denominator.
• G(x) = (x−1)/(x−2) → Remove common factors.
• x − 1 = 0 → Equate the numerator to 0.
• x = 1 → Solve for x.

Thus, the zero of G(x) = 1.

ASYMPTOTES

An asymptote is an imaginary line to which a graph gets closer and closer as the x or y increases or decreases its value without limit.

Kinds of Asymptote

• Vertical Asymptote
• Horizontal Asymptote
• Oblique / Slant Asymptote

VERTICAL ASYMPTOTE

The vertical line x = a is a vertical asymptote of a function f if the graph increases or decreases without bound as the x values approach a from the right or left. See illustration below.

[illustration: GeoGebra graph of a rational function with a dashed vertical line labeled “Vertical Asymptote”; Figure 2. Illustration of Vertical Asymptote using geogebra]

EXAMPLES

Consider the function h(x) = (x²+3)/(x−1). Determine the asymptotes.

By looking at the function, h(x) is undefined at x = 1, so the vertical asymptote of h(x) is the line at x = 1.

There is no horizontal asymptote because the degree of the numerator is greater than the degree of the denominator.

If the numerator and denominator of h(x) are divided, we get

h(x) = (x²+3)/(x−1)

        x + 1 r. 4
= x - 1 | x² + 0x +3
        - x² (–) x
          +
          x + 3
        - x (–) 1
              +
              4

[transcriber’s note: the long division layout is reproduced as preformatted text; the struck-through terms and the (–) and (+) signs are shown as printed]

So, the quotient is x + 1 + 4/(x−1).

Thus, the line y = x + 1 is the oblique asymptote of h(x) = (x²+3)/(x−1).

What’s More

Now it’s your turn.

Independent Practice 1

Given the rational function f(x) = (2x+6)/(x−3), answer the following questions:

1. What are the two functions used to form the rational function?
2. What is the x-intercept of the function? Which function did you use to determine the x-intercept? Why?
3. What is the y – intercept of the function? How did you get the y – intercept?
4. What is the zero of the function?

Try This!

Independent Assessment 1

Complete the table below by giving the intercepts and zeroes of rational function.

Rational Function	x – intercept	y – intercept	Zeroes of the function
1. f(x) = (x−9)/(x+3)
2. f(x) = (x²−10x+25)/(x+5)
3. f(x) = (x²+9)/(x²−3)

Independent Practice 2

True or False. Tell whether each of the following is true or false. If the statement is wrong change the underlined word to make it correct. Write your answer on the space provided before each number.

_____ 1. An intercept is a line (or a curve) that the graph of a function gets close to but does not touch.

_____ 2. If n > d, there is no horizontal asymptote.

_____ 3. To determine the vertical asymptote of a rational function, find the zeroes of the numerator.

_____ 4. If n < d, the vertical asymptote is y = 0.

_____ 5. The horizontal asymptote of f(x) = x/(x²−1) is y = 1.

_____ 6. The vertical asymptote of f(x) = ((x−1)(x+3))/(x²−1) are x = 1 and x = 2.

Remember Me!

• An asymptote is an imaginary line to which a graph gets closer and closer as the x or y increases or decreases its value without limit.

  
  

• To find vertical asymptote of a rational function, first reduce the given function to simplest form then find the zeroes of the denominator that are not zeros of the numerator.

  
  

• To determine the horizontal asymptote of a rational function, compare the degree of the numerator n and the degree of the denominator d.

  
  

• If n < d, the horizontal asymptote is y= 0

  
  

• If n = d, the horizontal asymptote y is the ratio of the leading coefficient of the numerator a, to the leading coefficient of the denominator b. That is y = a/b.

  
  

• If n > d, there is no horizontal asymptote.

  
  

• An oblique asymptote is a line that is neither vertical nor horizontal. It occurs when the numerator of f(x) has a degree that is one higher than the degree of the denominator. Divide the numerator by the denominator by either using long division or synthetic division. The oblique asymptote is the quotient with the remainder ignored and set equal to y.

  
  

Independent Assessment 2

Determine the vertical and horizontal asymptotes of the following rational functions.

Rational Function	Vertical Asymptote	Horizontal Asymptote
1. f(x) = 2/(2x+5)	_____	_____
2. f(x) = (x+3)/(x+7)	_____	_____
3. f(x) = ((x+3)(x−2))/((x+5)(x−4))	_____	_____
4. g(x) = (2 + 3x)/(x²+3x−4)	_____	_____
5. g(x) = (x−3)/(2x²−8)	_____	_____

What I Have Learned

Let us summarize what you have learned from this module by completing the following statements. Write the correct word/s in a separate sheet of paper.

1. _____ of the graph of a rational function are the points of intersection of its graph and an axis.

   
   

2. _____ of a function are the values of x which make the function zero. The numbered zeroes are also _____ of the graph of the function.

   
   

3. _____ 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 _____ equate the function to _____.

   
   

4. _____ 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 _____ simply evaluate the function at x = _____.

   
   

5. An _____ is an imaginary line to which a graph gets closer and closer as the x or y increases or decreases its value without limit.

   
   

6. To find _____ of a rational function, first reduce the given function to simplest form then find the zeroes of the denominator that are not zeros of the numerator.

   
   

7. To determine the _____ of a rational function, compare the degree of the numerator n and the degree of the denominator d.

   • If n < d, the horizontal asymptote is _____

     
     

   • If n = d, the horizontal asymptote y is the ratio of the leading coefficient of the numerator a, to the leading coefficient of the denominator b. That is y = _____.

     
     

   • If n > d, there is _____ horizontal asymptote.

     
     

8. An oblique asymptote is a line that is _____. To determine oblique asymptote, divide the numerator by the denominator by either using long division or synthetic division. The oblique asymptote is the quotient with the remainder ignored and set equal to y.

   
   

What I Can Do

Let’s apply what you have learned from the lesson.

The concentration (C) of a given substance in a mixture is the ratio of the amount of substance to the total quantity. In symbols,

C = S/Q

where C is the concentration, S is the amount of substance, and T is the total quantity. [transcriber’s note: the text says “T is the total quantity” while the formula uses Q] If 8 ounces of punch contains 4 ounces of pure orange juice, the concentration of orange juice in the punch is 4/8 or 50%. The punch is 50% orange juice. Consider the problem where we begin that 8 ounces of punch that is 50% orange juice and want to write a function that gives the orange juice concentration after x ounces of pure orange juice are added.

Questions:

a. How much orange juice do you begin with? Write an expression for the amount of orange juice present after x ounces has been added.

b. Write an expression for the total amount of punch present after x ounces has been added.

c. Using the answers in (a) and (b), write a rational function defining the pineapple juice concentration as a function of x.

d. Give the x and y – intercepts of the rational function.

e. What is the equation of the vertical asymptote and of the horizontal asymptote?

Assessment

Let’s Do This!

1. Which of the following is the set of all values that f(x) take?
   a. Range
   b. Intercept
   c. Domain
   d. Zeroes

   
   

2. What is the y-intercept of f(x) = (x−3)/(x+3)?
   a. 0
   b. – 1
   c. – 3
   d. – 5

   
   

3. What is the x – intercept of f(x) = (x−1)/x?
   a. x = -1
   b. x = 0
   c. x = 1
   d. All real numbers

   
   

4. Complete the sentence: The x- intercept of rational function is also _____ of the graph of the function.
   a. asymptote
   b. range
   c. zero
   d. domain

   
   

5. Which of the following are the points of intersection of the graph and the axes?
   a. Range
   b. Intercept
   c. Domain
   d. Zeroes

   
   

6. What is the domain of f(x) = 3/x?
   a. D = {x|x = 1}
   b. D = {x|x = 0}
   c. D = {x|x ≠ 1}
   d. D = {x|x ≠ 0}

   
   

7. Which of the following is not 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. Domain and Range of rational functions are always equal

   
   

8. If the degree of the leading coefficient of the numerator is less than 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 zero of f(x) = (x+5)/(3x²)?
   a. x = 5
   b. x = 0
   c. x = – 3
   d. x = – 5

   
   

10. What is the horizontal asymptote of f(x) = (3x+1)/(x−5)?
    a. y = 5
    b. y = 3
    c. y = 1
    d. y = 0

    
    

11. What is the y – intercept of f(x) = (x²−3x)/(x+3)?
    a. y = 3
    b. y = 1
    c. y = 0
    d. y = – 2

    
    

12. When the degree of the leading coefficient of the denominator of a rational function is greater than the degree of the leading coefficient of the numerator, the horizontal asymptote is at y = aₙ/a_d the statement is _____.
    a. Always true
    b. Sometimes true
    c. Never true
    d. Cannot be determined

    
    

13. How will you describe the vertical asymptote of f(x) = ((x−3)(x−2)(x+5))/((x−1)(x−3)(x−2))?
    a. does not exist
    b. approaching at x = 1
    c. approaching at x = -1
    [transcriber’s note: option d has no letter printed] approaching at x = 0

    
    

14. What is the x – intercept of f(x) = (x²−2x−15)/(x²−25)?
    a. x = 5
    b. x = 3
    c. x = – 3
    d. x = – 5

    
    

15. What is the horizontal asymptote of f(x) = (2x²+x+3)/(2x²+3x+1)?
    a. y = 3
    b. y = 2
    c. 1
    d. 0

    
    

Additional Activities

To deepen your knowledge on finding the intercepts, zeroes and asymptotes of rational function you can visit the following websites, https://youtu.be/gDC7XflNbQl and https://youtu.be/GgdGpjiJmkl.

For those who don’t have online connections you can answer the following questions to deepen your understanding about the lesson.

Analyze the given function and determine: a. x – and y- intercepts b. zeros c. Asymptotes

1. f(x) = (x+1)/(x−4)

   
   

2. f(x) = (x²−4x+5)/(x−4)

   
   

Answer Key

[transcriber’s note: this page is printed upside down; the text is transcribed in upright reading order]

What I Know

1. c
2. b
3. b
4. b
5. a
6. d
7. d
8. c
9. b
10. a
11. b
12. a
13. d
14. a
15. a

What I Know

1. I
2. C
3. M
4. A
5. G

[transcriber’s note: the second “What I Know” box is printed with the same title; its answers are the letters I, C, M, A, G]

Independent Practice 1

1. 2x + 6 and x – 3
2. x = – 3 or (–3, 0)
   2x + 6, in getting the x-intercept use the numerator of the function
3. y = –2 or (0, –2)
   substitute 0 to the x value of the function
4. The zero is at x = –3

Independent Assessment 1

Zeroes of f(x)	y-intercept	x-intercept
9	-3	9
5 (multiplicity 2)	5	5 (multiplicity 2)
none	-3	none

Independent Practice 2

1. Asymptote
2. True
3. Denominator
4. Horizontal asymptote
5. y = 0
6. x = –1

Application

1. 4, 4 + x
2. 8 + x
3. C(x) = (4+x)/(8+x)
4. x = – 4 or (–4, 0)
   y = 0.5 or (0, 0.5)
5. y = 1

Independent Assessment 2

	Vertical Asymptote	Horizontal Asymptote
1	x = -5/2 or -2.5	y = 0
2	x = -7	y = 1
3	x = -5 & x = 4	y = 1
4	x = -4 & x = 1	y = 0
5	x = -2 & x = -2	y = 0

Assessment – Post

1. a
2. b
3. c
4. c
5. b
6. d
7. d
8. c
9. d
10. b
11. c
12. c
13. b
14. c
15. c

Additional Activity

1. a. x = -1 and y = -¼
   b. x = -1
   c. VA at x = 4
   HA at y = 1
   SA = none

   
   

2. a. x = none and y = -1.25
   b. none
   c. VA at x = 4
   HA none
   SA none

   
   

References

DIWA Senior High School Series: General Mathematics, DIWA Learning Systems Inc, Makati City, 2016.

General Mathematics Learner’s Materials. Pasig City, Philippines: Department of Education- Bureau of Learning Resources, 2016.

Orines, Fernando B., Next Century Mathematics 11 General Mathematics, Phoenix Publishing House, Quezon City, 2016.

Oronce, Orlando A., General Mathematics, 1st Edition, Rex Book Store, Inc., Sampaloc Manila, 2016.

Santos, Darwin C. and Ma. Garnet P. Biason, Math Activated: Engage Yourself and Our World General Math, Don Bosco Press, Makati City, 2016.

Young, Cynthia, Algebra and Trigonometry, John Wiley & Sons, Inc. New Jersey, 2010.

Internet Source: https://youtu.be/gDC7XflNbQl https://youtu.be/GgdGpjiJmkl.

[transcriber’s note: the printed page number at the foot of this page is 29. The file name refers to rational functions, but this page contains only the References section, and no document title appears on these pages.]

For inquiries or feedback, please write or call:

Department of Education – Bureau of Learning Resources (DepEd-BLR)

Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600

Telefax: (632) 8634-1072; 8634-1054; 8631-4985

Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph