General Mathematics Quarter 1 Module 7: Representations of Rational Functions

Senior High School

General Mathematics Quarter 1 – Module 7: Representations of Rational Functions

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CO_Q1_General Mathematics SHS Module 7

General Mathematics Alternative Delivery Mode Quarter 1 – Module 7: Representations of Rational Functions First Edition, 2020

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

Development Team of the Module

Writer: Jea Aireen Charimae M. De Mesa

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Senior High School

General Mathematics Quarter 1 – Module 7: Representations of Rational Functions

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 with you in mind. It is here to help you represents rational function through table of values, graphs and equations. The scope of this module permits it to be used in many different learning situations. This module will guide you on how to see the essence of rational functions which we think has no value in real life, but in reality, it is in everything that we do.

In this module, you will learn to represent a rational function in three different ways. It is important that you apply the skills you have learned on how to represent a function in the previous module. Good luck!

After going through this module, you are expected to:

Represents rational function using:

a. table of values b. graphs c. equations

What I Know

Before studying this module, let us assess on what you already know about this topic.

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

1. Which family of function does the graph on the right belong to?

a. Trigonometric b. Logarithmic c. Exponential d. Rational

[chart: line graph of a curve with two branches beside question 1; axis values are not labeled]

2. Complete the table using the equation f(x) = (x−1)/(x+1)

x	1	2	3	4
f(x)	0	1/3	1/2	?

a. 3/5 b. 5/3 c. 3 d. 5

3. Which of the following table represents the function y = (2x+1)/x?

a.

x	1	2	3	4
y	-1	-2	-3	-4

b.

x	1	2	3	4
y	1	2	3	4

c.

x	1	2	3	4
y	3	5/2	7/3	9/4

d.

x	1	2	3	4
y	-3	-5/2	-7/3	-9/4

4. From the question in number 3, which represents the graph of the function f(x) = (2x+1)/x?

a. [chart: curve decreasing from left to right, approaching a vertical asymptote near x = 0; points plotted at x = -4 to 0; axis values not legible] c. [chart: straight line with negative slope through points plotted at x = 1 to 4; axis values not legible] b. [chart: curve in the first quadrant decreasing toward the horizontal axis, points plotted at x = 1 to 4; axis values not legible] d. [chart: straight line with positive slope through points plotted at x = 1 to 4; axis values not legible]

5. Reynolds manufactures aluminum cans in the shape of a cylinder with a capacity of 500 cubic centimeters (1/2 liters). The top and bottom of the can are made of special aluminum alloy that costs 0.05 cents per square centimeter. The sides of the can are made of material that cost 0.02 cents per square centimeter. Express the cost of the material for the can as a function of the radius r of the can.

a. 0.10πr² + 0.04πr² b. 0.10πr² − 0.04πr² c. 0.10πr + 0.04πr d. 0.10πr − 0.04πr

6. Which of the following is the graph of f(x) = x/(x−2) using the values x = -2, -1, 0, 1, 2?

a. [chart: curve with two branches in the upper-left and lower-right, plotted points marked on each branch; axis values not legible] c. [chart: curve decreasing from the upper left to the lower right through the origin, plotted points marked; axis values not legible]

b. [chart: curve with two branches in the upper-left and lower-right, plotted points marked; axis values not legible] d. [chart: curve with two branches in the upper-left and upper-right, plotted points marked; axis values not legible]

7. Using the rational function g(x) = (x²−1)/(x+1), what is the value of g(x) when x = 4? (Hint: simplify the function first).

a. 3 b. -3 c. 5 d. -5

For numbers 8-10, consider this situation

A cylindrical soft drink can is to be constructed so that it will have a volume of 21.6 cubic inches.

8. Write the total surface area A of the can as a function of r, where r is the radius of the can in inches.

a. A = (r+2)/(r+3) b. A = (r−2)/(r+3) c. A = (r+2)/(r−3) d. A = (r−2)/(r−3)

9. What should be the value of the function when values of x approaches to 3?

a. 5 b. 5/6 c. 6 d. 6/5

10. What is the graph of the function using x= 1, 2, 3, 4?
    a. [chart: graph of a function with plotted points at x = 1, 2, 3, 4; choice a]
    c. [chart: graph of a function with plotted points at x = 1, 2, 3, 4; choice c]
    b. [chart: graph of a function with plotted points at x = 1, 2, 3, 4; choice b]
    d. [chart: graph of a function with plotted points at x = 1, 2, 3, 4; choice d]

For numbers 11-13, refer to the problem below.

Lina is doing mathematics tutorial for a summer job. For every tutorial, she charges an initial fee of ₱500.00 per month, plus a constant fee of ₱200.00 for each hour of tutorial.

11. Which of the following equation best describes Lina’s fee for each of her tutorials?
    a. $fee = 500 + 200x$
    b. $fee = 500x + 200$
    c. $fee = 500 – 200x$
    d. $fee = 500x – 200$

    
    

12. Which of the following tables best represents Lina’s fee for each of her tutorials?

    
    

a.

No of hours	1	2	3	4
Fee	200	400	600	800

b.

No of hours	1	2	3	4
Fee	700	1200	1700	2200

c.

No of hours	1	2	3	4
Fee	700	1400	2100	2800

d.

No of hours	1	2	3	4
Fee	700	900	1100	1300

13. If Lina spends 15hours on a student and another 13 hours for another student in a month, how much will she earn? Write an equation that will suit best the situation
    a. $fee = [500 – 200(15)] + [500 – 200(13)]$
    b. $fee = [500 + 200(15)] + [500 + 200(13)]$
    c. $fee = [500(15) + 200] + [500(13) + 200]$
    d. $fee = [500(15) – 200] + [500(13) – 200]$

For numbers 14 and 15, refer to the problem below.

There are 1,200 freshmen and 1,500 sophomores at SSG Election Meeting de Avance at noon. After 12 p.m., 20 freshmen arrive at the gymnasium every five minutes while 15 sophomores leave the gymnasium.

14. Which equation best describes the total number of students who attended the SSG Election Meeting de Avance?
    a. $y = [1200x + 20] + [1500x – 15]$
    b. $y = [1200 + 20x] + [1500 – 15x]$
    c. $y = [1200x – 20] + [1500x + 15]$
    d. $y = [1200 – 20x] + [1500 + 15x]$

    
    

15. Simplifying the answer in number 14, we can get
    a. $y = 2700x + 5$
    b. $y = 2700 + 5x$
    c. $y = 2700x – 5$
    d. $y = 2700 – 5x$

    
    

Lesson 1 Representations of Rational Functions

This lesson is about representations of rational function in different ways. We will deal with the application of rational functions that may involve the number of persons who can do a task in a certain amount of time. We can handle these applications involving work in a manner similar to the method we used to solve distance, speed, and time problems.

What’s In

[illustration: hand holding a puzzle piece]

Let us recall on how to represent a polynomial function problem through table, graph, and equation.

Example: A siomai vendor can wrap 4 dumplings every minute. If he wraps a total of 32 dumplings, how much time did he spent wrapping? Use the following to justify your answer:

a. table of Values b. graph c. equation

Solution:

For A, we will complete the table using the given information provided in the problem. Since a vendor can wrap 4 dumplings per minute, we will add 4 dumplings for every minute until we reach 32 dumplings.

Number of dumplings wrapped	4	8	12	16	20	24	28	32
Minutes	1	2	3	4	5	6	7	8

Using this table, we can say that the vendor was able to wrap 32 dumplings in 8 minutes.

For B, let us use the data we have in A and plot these points in a Cartesian plane.

[chart: scatter plot on a Cartesian plane; points labeled A to H lie along a straight line; Values are not labeled on the chart.]

For C, we can solve the problem by formulating a formula where we divide the total number of dumplings made by the number of dumplings per minute. In symbol,

$T = \frac{D}{N}$

Where;

T= Time in Minutes

D= Total number of Dumplings made

N= Number of Dumplings made per minute

Notes to the Teacher

Tell the students to use the proper scaling and labeling. It is important that the graphical representations are neatly made especially if not using a graphing paper.

Teacher can also introduce using tools such as GeoGebra, Desmos, and other applications.

What’s New

[illustration: hand holding a leafy sprout]

Life is a Beach!

Pueblo Por La Playa is a 12.5 hectare Mexican-inspired exclusive leisure club nestled off the calm, clear waters of Pagbilao Quezon. The “Pueblo” offers the total leisure and recreation experience for the entire family. Since it is an exclusive resort, it has a membership fee. Pueblo Por La Playa charges a ₱300,000.00 annual fee, then ₱700.00 for each day you stay there. Find the average cost per day to stay in the resort in 5, 10, 15 and up to 30 days. Graph the function to show whether it forms a straight line or a curve.

a. Define a formula for the average cost for every 5 days to stay in the resort f(x). Hint: Since the problem asks for the average cost, use the formula in getting an average.

b. Based from the situation above, complete the following table to show the average cost every 5 days.

X	0	5	10	15	20	25	30
Y	0

Hint: Substitute the value of x in your equation

c. Plot the following points on the cartesian plane

To graph, simply plot the points and connect it by a smooth curve line.

What is It

[illustration: magnifying glass over a hand]

The problem presented above is an example of Rational Function. To solve the problem, let us answer each question one by one. Below is the definition of a Rational Function.

Definition

Rational function is written in the form of $f(x) = \frac{p(x)}{q(x)}$. It should follow the following conditions namely:

1. Both p(x) and q(x) are polynomial functions wherein it has no negative and fractional exponents.
2. The denominator or q(x) should not be equal to 0.
3. The domain of all values of x where q(x) ≠ 0.

a. Define a formula for the average cost for every 5 days to stay in the resort f(x). To define the formula, use the formula in getting the average cost.

Let the function be f(x). We can use the formula of getting an average. Average problems use the formula $A = \frac{X}{N}$, where A= Average, X= cost, and s= number of days

Let f(x) represents the average cost per day and x represents the number in days. Note that ₱300,000.00 is a fixed price you need to pay plus the ₱700.00 per day divided by the number of days (x). We will have,

$f(x) = \frac{300000 + 700(x)}{x}$

Observe that it is similar to the structure of our original formula. Note that you will be using a formula depending on the classification of problems given to you.

b. For every 5 day stay in the resort, create a table of values showing the average cost.

Solution: Make a table of values with x-values at 0, 5, 10, 15, 20, 25, 30.

X	0	5	10	15	20	25	30
Y	0	20,965	41,930	62,895	83,860	104,825	125,790

From the table, we can observe that the average cost of stay decreases as the time increases. We can use a graph to determine if the points of this function follow a curve or a line

c. Graph the following points in the Cartesian plane.

[chart: line graph of a decreasing curve on a Cartesian plane with plotted points; Values are not labeled on the chart.]

By connecting the lines, we can clearly see that it follows a curve, thus a Rational Function.

Example 2:

$f(x) = \frac{x}{x + 2}$

a. Since we already have an equation, we can skip this part. Proceed with the table of values

b. Construct table of values from -2 to 2. We can substitute each value on the equation to complete the table. We will get,

X	-2	-1	0	1	2
f(x)	Und	-1	0	0.33	0.5

We can observe that the value of f(x) is undefined in when x= -2. It is because when you substitute -2 in the function it will have an answer of zero whereas in the definition of rational function, we cannot have a denominator equal to zero.

c. Plot the points in the Cartesian plane and determine whether the points form a smooth curve or a straight line.

Represent this rational equation through table and graph. Identify whether the graph forms a straight line or a curve.

$f(x) = \frac{x + 3}{x + 1}$

It can be observed that the function formed a curve.

What’s More

In the following activities, read each situation carefully to solve each problem. Write your answer on a separate sheet of paper.

Practice Activity 1

Represent this rational equation through table and graph. Identify whether the graph forms a straight line or a curve.

$f(x) = \frac{x + 3}{x + 1}$

a. Since we already have an equation, we can skip this part. Proceed with letter b. Note that when given with a word problem, you cannot skip this part.

b. Construct table of values from -2 to 2. Hint: Substitute the value of x to obtain f(x).

x	-2	-1	0	1	2
f(x)	-1	undefined	3	2	5/3

The function is undefined when x= -1 since it makes the denominator zero.

c. Plot the points in the Cartesian plane and determine whether the points form a smooth curve or a straight line.

[chart: line graph of a curve on a Cartesian plane with plotted points at x = -2, -1, 0, 1, 2; Values are not labeled on the chart.]

By plotting the points obtained in B we can get,

[chart: graph of a rational function plotted from the points obtained in B; x-axis labeled -2, -1, 0, 1, 2; y-axis labeled 1 to 4; the curve falls from upper left to lower right, with marked points near (-2, -1), (0, 3), (1, 2), and (2, 1.67). Values are not labeled on the chart.]

Independent Assessment 1

In order to join a voice lesson class, you will pay a ₱1,500.00 pesos fee, then ₱500.00 for each class you go to. What is the average cost per class? Graph the function to show whether it forms a straight line or a curve.

Independent Assessment 2

Represent this rational equation through table and graph. Identify whether the graph forms a straight line or a curve.

P(x) = (x² + 1) / (x + 1)

What I Have Learned

In your own words, how do you represent rational function using

a. Equation

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b. Graph

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c. Table of values

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What I Can Do

Read and analyze the situation below then answer the question given.

Rational function is one of the functions that is underappreciated because they say we cannot use it in real life. No. Rational function can be seen in most of our daily activities, we just didn’t know it.

Basketball League

In an inter-barangay basketball league, the team from Hermana Fausta has won 12 out of 25 games, a winning percentage of 48%. We have seen that they need to win 8 games consecutively to raise their percentage to 60%. What will be their winning percentage if they win 10, 20, 30, 50, 100 games? Will they reach a 100% winning percentage? (hint: try to substitute 300 games). Write your interpretation on the space provided.

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Here are the steps to solve the problem and the rubric that will guide you in giving the correct representations to the problem.

Steps in Problem Solving	Possible Highest Points	Your Score
1. Give the Appropriate model or equation	3 points
2. Create a table of Values	3 points
3. Graph	4 points
4. Essay	5 points
Total	15 points

Assessment

Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.

1. It is in the form of f(x) = p(x)/q(x) where p(x) and q(x) are polynomial functions and q(x) is not equal to zero.
   a. Rational Equality
   b. Rational Inequality
   c. Rational Function
   d. None of these

For questions 2 and 3, refer to this situation.

Martha has won 19 out of 28 tennis matches this season.

2. Which equation models suggest how many more games she needs to win to average 75% wins over loses?
   a. 19/(28+x) = 0.75
   b. (19+x)/(28+x) = 0.75
   c. 19/(28+x) = 75
   d. (19+x)/(18+x) = 75

   
   

3. In order to get a college tennis scholarship, Martha needs to bring her winning average to 80%. What is the number of matches she needs to win given that she already won 19 out of 28?
   a. 3
   b. 4
   c. 17
   d. 22

   
   

For questions 4-6, refer to this situation.

Joel is working on his chemistry project and he has 300mL of 12% acid solution.

4. If he needed to decrease the acidity of the solution, which of the following is correct function that would show the new acidity of the solution given x mL of water added?
   a. f(x) = 36/(300+x)
   b. f(x) = 0.36/(300+x)
   c. f(x) = 12/(300+x)
   d. f(x) = 0.12/(300+x)

   
   

5. If Joel decided to decrease the acidity of the solution by adding 15 more than at every interval, which table of values is correct?

   
   

a.

X	1	2	3	4
f(x) in %	11.43%	10.91%	10.43%	10%

b.

X	1	2	3	4
f(x) in %	0.1143%	0.1091%	0.1043%	0.10%

c.

X	1	2	3	4
f(x) in %	3.81%	3.64%	3.48%	3.33%

d.

X	1	2	3	4
f(x) in %	3.81%	3.64%	3.48%	3.33%

6. Which graph shows the decrease of acidity in Joel’s solution?

a. [chart: line graph; nearly flat red line; axis labels are [illegible]. Values are not labeled on the chart.]

c. [chart: line graph; nearly flat red line; axis labels are [illegible]. Values are not labeled on the chart.]

b. [chart: line graph; red line sloping gently downward across the plot; axis labels are [illegible]. Values are not labeled on the chart.]

d. [chart: line graph; red line sloping gently downward across the plot; axis labels are [illegible]. Values are not labeled on the chart.]

7. Which of the following is the correct table of values of the rational function f(x) = x/(x+1)?

a.

x	-1	0	1
Y	Und	0	0.5

b.

X	-1	0	1
Y	0.5	und	0.5

c.

x	-1	0	1
y	-0.5	0	und

d.

x	-1	0	1
y	-0.5	Und	-0.5

8. When is the graph of the function undefined in a certain value of x?
   a. When the value of the numerator is zero.
   b. When the value of the denominator is zero.
   c. When the value of the function is zero.
   d. None of the above.

   
   

9. Which of the following equations satisfies the table of values below?

   
   

X	-2	-1	0	1	2
Y	-1	Und	3	2	1.67

a. y = (x+1)/(x+3) b. y = (x+3)/(x+1) c. y = (x-3)/(x+1) d. y = (x-1)/(x+3)

10. Which table of values satisfies the graph presented on the right side?

a.

X	-2	-1	0	1	2
Y	0	0.5	1	1.5	2

b.

X	-2	-1	0	1	2
Y	-2	-1	0	1	2

c.

X	-2	-1	0	1	2
Y	0	-0.5	-1	-1.5	-2

d.

X	-2	-1	0	1	2
Y	2	1	0	-1	-2

[chart: straight line graph on a Cartesian plane, with the line passing through several marked points; values are not labeled on the chart.]

11. Using values from -10 to 10 with an interval of 5. Which of the following best describes the table of values of the function g(x) = 2x³ + 4x − 19?

a.

X	-10	-5	0	5	10
g(x)	2021	251	-19	-289	-2059

b.

X	-10	-5	0	5	10
g(x)	-2059	-289	-19	251	2021

c.

X	-10	-5	0	5	10
g(x)	2059	289	19	-251	-2021

d.

X	-10	-5	0	5	10
g(x)	-2021	-251	19	289	2059

12. In a Bread and Pastry class, a certain recipe calls for 3 kgs of sugar for every 6 kgs of flour. If 60 kgs of this sweet must be prepared, how much sugar is required? Which equation satisfies the problem?
    a. x = (60+3)/(6(3))
    b. x = 6(3)/(60+3)
    c. x = 60(3)/(6+3)
    d. x = (6+3)/(60(3))

    
    

13. How many kilograms of sugar is needed for 90 kilograms of sweets?
    a. 20
    b. 25
    c. 30
    d. 35

    
    

For questions number 14 and 15, refer to the problem below.

In a business math class, Teacher Alex assigned his students a business project. For the business to be established, a certain establishment needs to pay for a semestral fee (5 months) of ₱50.00 pesos and a weekly tax of ₱10.00 which the proceeds will go to their Christmas Party expenses.

14. What is the average amount collected per group in his class? Formulate an equation for this.
    a. f(x) = (50-10x)/x
    b. f(x) = x/(50+10x)
    c. f(x) = (50+10x)/x
    d. f(x) = x/(50-10x)

    
    

15. How much will be collected in each group for a period of 13weeks?
    a. ₱170.00
    b. ₱180.00
    c. ₱190.00
    d. ₱200.00

    
    

Additional Activities

Do the following to enhance your learning.

1. Construct a table of values

   
   

2. the following functions using the interval of -5 to 5.

   
   

   a. g(x) = (x³+3x−5)/x²
   b. j(t) = 1/(t²−2t+1)

   
   

3. Using the data from the table of values on number 1, plot the points on the cartesian plane and connect the points of

   
   

   a. g(x) = g(x) = (x³+3x−5)/x²
   b. j(t) = j(t) = 1/(t²−2t+1)

   
   

An application of rational functions may involve the number of persons who can do a task in a certain amount of time. We can handle these applications involving work in a manner similar to the method we used to solve distance, speed, and time problems. Work = Rate x Time. Suppose you can finish a report in 2 hours. Your classmate can finish the same report in 4 hours. How long will it take to finish the report if both of you work together?

We have a saying that “Two heads are better than one”, would you rather work alone or with a team? Why? Use 3-5 sentences to explain your answer.

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Answer Key

<!– transcriber’s note: this page is printed rotated 90 degrees, with the three columns upside down; the content below is transcribed in reading order –>

What I Know

1. D
2. A
3. C
4. B
5. D
6. B
7. A
8. B
9. B
10. A
11. A
12. D
13. C
14. B
15. B

What’s More

Independent Activity 1

A. f(x) = (1500 + 500x) / x

B.

x	1	2	3	4	5
f(x)	2000	1250	1000	875	800

C.

[chart: graph; Independent Activity 1, item C (no printed title)] Values are not labeled on the chart.

Independent Activity 2

A.

X	-2	-1	0	1	2
f(x)	-6	Und	1	1	1.67

B.

[chart: graph; Independent Activity 2, item B (no printed title)] Values are not labeled on the chart.

Assessment

1. C
2. B
3. C
4. A
5. A
6. C
7. A
8. B
9. B
10. A
11. B
12. C
13. C
14. D
15. B

References

Books:

Oronce, Orlando and Mendoza, Marilyn O. 2016. General Mathematics. Rex Bookstore, Inc.,

Oronce, Orlando. 2016. General Mathematics. Rex Bookstore, Inc.

Verzosa, Debbie Marie, et.al. 2016. General Mathematics: Learner’s Material, First Edition. Philippines: Lexicon Press Inc.

Online Sources:

https://www.geogebra.org/graphing?lang=en

https://www.desmos.com/calculator

Guinness World Records. 2011. Top 5 Records from the Philippines. Retrieved at https://www.guinnessworldrecords.com/news/australasia-news/2011/9/top-five-records-from-the-philippines-324697?fb_comment_id=897379786984508_1871579676231176

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