General Mathematics Quarter 1 Module 9, Study notes of Mathematics

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General MathematicsGeneral Mathematics

Quarter 1Quarter 1 – – Module 9:Module 9:

Intercepts, Zeroes andIntercepts, Zeroes and

Asymptotes of RationalAsymptotes of Rational

FunctionsFunctions

General MathematicsGeneral Mathematics AlAl terter natnativive Delie Deliververy Moy Modede

Quarter 1Quarter 1 – – Module 9: InteModule 9: Intercepts, Zercepts, Zeroes and Asymptroes and Asympt otes ofotes of RaRational Functitional Functi onsons

First Edition, 2020First Edition, 2020

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Introductory MessageIntroductory Message

For the facilitator:For the facilitator:

Welcome to the General Mathematics Alternative Delivery Mode (ADM) Module onWelcome to the General Mathematics Alternative Delivery Mode (ADM) Module on Determining the Intercepts, Zeroes and Asymptotes of Rational Functions!Determining the Intercepts, Zeroes and Asymptotes of Rational Functions!

ThisThis modulemodule waswas collaborativcollaborativelyely designed,designed, developeddeveloped andand reviewereviewed bd byy educatorseducators fromfrom

public institutions to assist you, the teacher or facilitator in helping the learnerspublic institutions to assist you, the teacher or facilitator in helping the learners meet the standards set bymeet the standards set by the K tothe K to 12 Curriculum while overcoming their personal,12 Curriculum while overcoming their personal, social, and economic constraints in schooling.social, and economic constraints in schooling.

ThisThis learning resourcelearning resource hopeshopes toto engage theengage the learnerslearners intointo guidedguided andand independenindependentt learning activities at their own pace and time. Furthermore, this also aims to helplearning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration theirlearners acquire the needed 21st century skills while taking into consideration their needs and circumstances.needs and circumstances.

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

As a facilitator you are expected to orient the learners on how to use this 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 manageYou 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 thetheir own learning. Furthermore, you are expected to encourage and assist the learners as they do tlearners as they do the tasks included in the module.he tasks included in the module.

For the learner:For the learner:

Welcome to the General Mathematics Alternative Delivery Mode (ADM) Module onWelcome to the General Mathematics Alternative Delivery Mode (ADM) Module on Determining the Intercepts, Zeroes and Asymptotes of Rational Functions!Determining the Intercepts, Zeroes and Asymptotes of Rational Functions!

The hand is one of the mostThe hand is one of the most symbolized part of the human body. It is often used tosymbolized part of the human body. It is often used to depict skill, action and purpose. Through our hands we may learn, create anddepict 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 learneraccomplish. Hence, the hand in this learning resource signifies that you as a learner is capable and empowered to successfully achieve the relevant competencies andis 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!skills at your own pace and time. Your academic success lies in your own hands!

This module wThis module was designeas designed to provide yd to provide you withou with fun and mefun and meaningful oaningful opportunities forpportunities for guided and independent learning at your own pace andguided and independent learning at your own pace and time. You will betime. You will be enabled toenabled to

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

Notes to the TeacherNotes to the Teacher This conThis containstains helpfulhelpful tips ortips or strategistrategies thates that will help you iwill help you in guiding the learners.n guiding the learners.

vv

At the end of tAt the end of this module you will also find:his module you will also find:

The followingThe following are some remare some reminders in usinders in using this moing this module:dule:

1.1. Use the module with care. Do not put unnecessary mark/s on any part of theUse 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.module. Use a separate sheet of paper in answering the exercises. 2.2. Don’t forget to answerDon’t forget to answer What I Know before moving on to the other activitiesWhat I Know before moving on to the other activities included in the module.included in the module. 3.3. Read the instruction carefully before doing each task.Read the instruction carefully before doing each task. 4.4. Observe honesty and integrity in doing the tasks and checking your answers.Observe honesty and integrity in doing the tasks and checking your answers. 5.5. Finish the task atFinish the task at hand before proceeding to the next.hand before proceeding to the next. 6.6. Return this module to yourReturn this module to your teacher/facteacher/facilitator once you areilitator once you are through with it.through with it. If you encounter any difficulty in answering the tasks in this module, do notIf 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 arehesitate to consult your teacher or facilitator. Always bear in mind that you are not alone.not alone.

We hope that through this material, you will experience meaningful learning andWe hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!gain deep understanding of the relevant competencies. You can do it!

ReferencesReferences This is aThis is a list of allist of all sourcesl sources used inused in developingdeveloping this module.this module.

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

1.1. Which of the following is the set of allWhich of the following is the set of all values that the variable x can take?values that the variable x can take? a.a. RangeRange b.b. InterceptIntercept c.c. DomainDomain d.d. ZeroesZeroes

2.2. What is the domain of f(x) =What is the domain of f(x) =−−++??

a.a. All real numbersAll real numbers b.b. All realAll real numbers exceptnumbers except – – 33 c.c. All real numbers except 3All real numbers except 3 d.d. Cannot be dCannot be determineeterminedd

3.3. What value/s of x that will make the function meaningless f(x) =What value/s of x that will make the function meaningless f(x) = −− ??

a.a. x = -1x = - b.b. x = 0x = 0 c.c. x = 1x = 1 d.d. All realAll real numbersnumbers

4.4. Complete the sentence: The real numbers zeroes are also ____________ of theComplete the sentence: The real numbers zeroes are also ____________ of the graph of thegraph of the function.function. a.a. AsymptoteAsymptote b.b. xx – – interceptsintercepts c.c. yy – – interceptsintercepts d.d. RangeRange

5.5. Which of the following is the set of allWhich of the following is the set of all values that f(x) can take?values that f(x) can take? a.a. RangeRange b.b. InterceptIntercept c.c. DomainDomain d.d. ZeroesZeroes

6.6. What is the range of f(x) =What is the range of f(x) =??

a.a. R =R ={{|| = 1= 1}} b.b. R =R ={{|| = 0= 0}} c.c. R =R = {{|| ≠ 1≠ 1}} d.d. R =R = {{|| ≠ 0≠ 0}}

7.7. Which of the following is a true statement?Which of the following is a true statement?

a.a. A rational function is a quotient of functions.A rational function is a quotient of functions. b.b. Asymptotes are aAsymptotes are a common characteriscommon characteristic oftic of rational functionsrational functions.. c.c. An asymptote is a line that aAn asymptote is a line that a graph approacgraph approaches, but does not touch.hes, but does not touch. d.d. All of the above.All of the above.

8.8. If the degree ofIf the degree of the leading coefficient of the numerator is equal tthe leading coefficient of the numerator is equal to theo the degree of the leading coefficient of thedegree of the leading coefficient of the denominator of a rationaldenominator of a rational function, which of the following statements has to befunction, which of the following statements has to be true?true? a.a. The graph hThe graph has no asyas no asymptotemptote b.b. The graph oThe graph of the funcf the function has slanttion has slant asymptoteasymptote c.c. The graph oThe graph of the funcf the function has a htion has a horizontal asorizontal asymptoteymptote d.d. None of the aboveNone of the above

9.9. What is the horizontal asymptote ofWhat is the horizontal asymptote of (()) == +5+5 ??

a.a. y = 3y = 3 b.b. y = 0y = 0 c.c. y = - 2y = - 2 d.d. y = - 3y = - 3

10.10.What is the vertical asymptote ofWhat is the vertical asymptote of (()) ==^ ++−5−5 ??

a.a. x = 5x = 5 b.b. x = 3x = 3 c.c. x = 1x = 1 d.d. x = 0x = 0

11.11.What is the oblique asymptote ofWhat is the oblique asymptote of (()) == ^

−− ++ ?? a.a. y = 3xy = 3x b.b. y = xy = x – – 66 c.c. y = x -3y = x - d.d. y = 3x + 6y = 3x + 6

12.12. Oblique asymptote occurs when there is no horizontal asymptote,Oblique asymptote occurs when there is no horizontal asymptote, the statement is ____________.the statement is ____________.

a.a. Always trueAlways true b.b. Sometimes trueSometimes true c.c. Never trueNever true d.d. Cannot be determinedCannot be determined

13.13. How will you describe the horizontal asymptote ofHow will you describe the horizontal asymptote of (()) == (^) ++ ??

a.a. does not existdoes not exist b.b. approaching at x = 3approaching at x = 3 c.c. approaching at y = -3approaching at y = - d.d. approaching at y = 0approaching at y = 0

LessonLesson

Intercepts, Zeroes, and Asymptotes ofIntercepts, Zeroes, and Asymptotes of Rational FunctionsRational Functions

In the previous lesson, you learned how to find domain and range of a rationalIn the previous lesson, you learned how to find domain and range of a rational function. In this particular lesson, determining intercepts, zeroes and asymptotes offunction. In this particular lesson, determining intercepts, zeroes and asymptotes of rational functions will be done. Knowing fully the concept of the different propertiesrational 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 rationalof rational function will be your guide to easily determine the behavior of a rational function and itfunction and it will prepare you forwill prepare you for the next topic which is aboutthe next topic which is about graphing rationgraphing rationalal function.function.

What’s InWhat’s In

Let’s recall first what you have learned from the previous lesson by answering theLet’s recall first what you have learned from the previous lesson by answering the following questions:following questions:

A.A. Which of the following is an example of rational function?Which of the following is an example of rational function? 1.1. F(x) =F(x) =^ 

++ −− 2.2.^

  ==^

88  3.3.^

 −−  3 <3 < 0^0

B.B. Find the domain and range of the functions.Find the domain and range of the functions.

1.1. F(x) =F(x) = (^) ++ 2.2. f(x)f(x) == (^) −− 3.3. g(x)g(x) == (^) ++−−

Let us see ifLet us see if you got the correct answer in the activity, if your answer inyou got the correct answer in the activity, if your answer in question Aquestion A is number 1, you got it right you have a clear understanding of the concept of rationalis 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 functionfunction but if you are incorrect allow me to help you recall what a rational function is,is, when two polynomial functions are expressed as a quotient and can be written inwhen two polynomial functions are expressed as a quotient and can be written in

the formthe form (()) == ()()()() and q(x) is a not the zero function it is called a rational function.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 andNumbers 2 and 3 are not examples of rational function, it is a rational equation and rationalrational inequality,inequality, respectively.respectively. NumberNumber 11 isis writtenwritten asas thethe quotientquotient ofof twotwo polynomial functions, so it ispolynomial functions, so it is a rational function.a rational function.

For activity B, let us review the meaning of domain and range of the function.For activity B, let us review the meaning of domain and range of the function. DomainDomain is the set of first coordinates of a relation and it is the value of x that willis 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 whilenot make the denominator of the function equal to zero while RangeRange is the set ofis the set of second coordinates. To determine the domain of rational function, simply equate thesecond 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 thedenominator 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 thefunction will not give an undefined or a meaningless function. Example find the

domain of F(x) =domain of F(x) = (^) ++ , equating the denominator to zero, we have x + 3 = 0, so the, equating the denominator to zero, we have x + 3 = 0, so the

value of x = -3, so the domain of the function are allvalue of x = -3, so the domain of the function are all real numbers except -3 remembreal numbers except -3 rememberer we will avoid value/s that will make our denominator equal to zero, so if we willwe 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 ansubstitute -3 to our x in the denominator it will result to 0 and it will give us an undefined function.undefined function. In notation, D= (-In notation, D= (-∞,∞, )) ∪ (∪ (,, ∞)∞)

To find the range of the function, change f(x) to y then, solve for x; rememTo find the range of the function, change f(x) to y then, solve for x; remember rangeber range are real values of y that will make a real value for the function. For example, find theare real values of y that will make a real value for the function. For example, find the

range of F(x) =range of F(x) = (^) ++ ;;

ChangingChanging F(x)F(x) toto y,y, thethe newnew functionfunction isis yy == (^) ++

ByBy doingdoing crosscross multiplicatimultiplicationon wewe havehave y(x+3)y(x+3) == xx

DistribuDistributingting yy wewe nownow havehave xyxy ++ 3y3y == xx

SimplifyiSimplifyingng thethe equationequation willwill givegive xyxy – – x = 3yx = 3y

FactoriFactoringng thethe leftleft sideside ofof thethe equationequation x(yx(y – – 1) = 3y1) = 3y

Dividing the equation by (yDividing the equation by (y – – 1)1) (−)(−)(−)(−) == (^) (−)(−)

Removing common factor, the value of xRemoving common factor, the value of x  ==^ −−

Since we are looking for the value of y that will give a real value for the function soSince 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 twe need to find value/s for y that will not make the denominator equal to 0.hat will not make the denominator equal to 0.

EquatingEquating thethe denominatordenominator toto zerozero yy – – 1 = 01 = 0

So,So, yy == 1.1. The rangeThe range of the funof the function F(x) =ction F(x) = (^) ++ is all real values of y except 1. In notation,is all real values of y except 1. In notation,

R= (-R= (-∞∞,, )) ∪∪ ((,∞),∞)..

The followingThe following are the ansare the answers to Acwers to Activity Btivity B

1.1. Domain = {x/xDomain = {x/x≠ 3}≠ 3} oror (-(-∞,∞, )) ∪ (∪ (,, ∞)∞)

Range = {y/yRange = {y/y ≠ 1}≠ 1} oror (-(-∞∞,),) ∪∪ ((,∞),∞)

2.2. Domain = {x/xDomain = {x/x≠ 4}≠ 4} oror (-(-∞∞,, )) ∪∪ ((,, ∞∞))

Range = {y/yRange = {y/y ≠ 0}≠ 0} oror (-(-∞∞,),) ∪∪ ((,∞),∞)

3.3. Domain = {x/xDomain = {x/x≠ 1   ≠ 1}≠ 1   ≠ 1} oror (-(-∞,∞,)) ∪∪ ((,),) ∪∪ ((,∞),∞)

Range = {y/yRange = {y/y ≠ 0}≠ 0} oror (-(-∞∞,),) ∪∪ ((,∞),∞)

ActivityActivity II – – Connect Mo!Connect Mo!

Connect the given statement/phraConnect the given statement/phrase in column A with the answer in columnse in column A with the answer in column B to complete the statement/phrase in column A. Write the letter of yourB to complete the statement/phrase in column A. Write the letter of your answer in a separate sheet of paper.answer in a separate sheet of paper.

COLUMNCOLUMN AA COLUMNCOLUMN BB

1.1. The intercepThe intercepts of the gts of the graphraph M. the x - inM. the x - interceptstercepts of a rational function …of a rational function …

2.2. To find the xTo find the x – – intercepinterceptt A.A. letlet xx == 0 0 of a function …of a function …

3.3. The zeroes oThe zeroes of the functionf the function G. rationG. rational functional function is also …is also …

4.4. To find the yTo find the y – – interceptintercept I.I. areare thethe pointspoints ofof of a function …of a function … intersection of itsintersection of its graph and an axisgraph and an axis

5.5. The functionThe function of the formof the form C. let y = 0C. let y = 0 , where g(x), where g(x) & h(x) are polynomials& h(x) are polynomials

What’s NewWhat’s New

How wasHow was the activity? I believed that you cthe activity? I believed that you connected it right.onnected it right. So, in thisSo, in this lesson, youlesson, you will know how twill know how to identify intercepts, zeroes and asymptotes of rational function.o identify intercepts, zeroes and asymptotes of rational function.

What is ItWhat is It

INTERCEPTS AND ZEROES OF RATIONAL FUNCTIONSINTERCEPTS AND ZEROES OF RATIONAL FUNCTIONS

The intercepThe intercepts of the graph ots of the graph of a rational funf a rational function are thection are the points of intersepoints of intersection of itsction of its graph and an axis.graph and an axis.

TheThe y-intercepty-intercept of the graph of a rational function r(x) if it exists, occurs at r(0),of the graph of a rational function r(x) if it exists, occurs at r(0), provided that r(x) is defined at xprovided that r(x) is defined at x = 0.= 0. To find y-intercept simply evaluate the functionTo find y-intercept simply evaluate the function at x = 0.at x = 0.

TheThe x-interceptx-intercept of the graph of a rational function r(x), if it exists, occurs at the zerosof 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 xof the numerator that are not zeros of the denominators. To find x – – intercept equateintercept equate the function to 0.the function to 0.

TheThe zeroes of a functionzeroes of a function areare the valuesthe values of xof x which makewhich make the function zero.the function zero. TheThe

numbered zeroes are also xnumbered zeroes are also x-interce-intercepts of thepts of the graph of thegraph of the function.function.

EXAMPLES.EXAMPLES.

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

a.a. f(x) =f(x) = −−++ b.b. f(x)f(x) == (^) ++ c.c. f(x)f(x) == 

−+−+ −−

y-intercepty-intercept

x-interceptx-intercept

Figure 1.Figure 1. x andx and yy intercepts using GeoGebraintercepts using GeoGebra

zero of thezero of the functionfunction

2.2. f(x) =f(x) = (^) ++

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

0 0 == 3x3x EquateEquate thethe numeratornumerator toto 0 0 ..

3x3x == 0 0 ByBy SymmetricSymmetric PropertyProperty ofof EqualitEquality.y.

  ==^

  SimplifyingSimplifying thethe fractionfraction byby multiplyinmultiplyingg both sides by 1/3.both sides by 1/3. x = 0x = 0

So, the xSo, the x – – intercept is 0 or (0, 0).intercept is 0 or (0, 0).

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

(()) == (^) ++ SubstitutSubstitutee 00 toto xx valuesvalues ofof thethe functionfunction..

()() ++ (^) SimplifyingSimplifying thethe fraction.fraction.

(()) ==  == 0 0 TheThe valuevalue ofof ()() or yor y – – intercept.intercept.

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

−+−+ −−

^  33  2 =2 = 0 0 EquateEquate thethe numeratornumerator toto 0.0.

(x(x – – 2) (x2) (x – – 1)1) == 0 0 ByBy factoring.factoring.

xx – – 2 2 == 0 0 xx – – 1 1 == 0 0 SolveSolve forfor x,x, byby ZeroZero productproduct propeproperty.rty. x = 2x = 2 x = 1x = 1

So, the xSo, the x – – intercepts are x = 2 and x = 1. But by looking at the denominatorintercepts are x = 2 and x = 1. But by looking at the denominator of the original function if we substitute 2 to the value ofof the original function if we substitute 2 to the value of x,x,

xx^22 – – 4 = (2)4 = (2)^22 – – 4 = 0,4 = 0,

The denominThe denominator will becator will become 0, theome 0, the functionfunction becomes meanbecomes meaningless.ingless. So, we will only accept xSo, we will only accept x – – intercept atintercept at x = 1 or (1, 0).x = 1 or (1, 0).

To find the yTo find the y – – iintercepntercept:t:

−+−+ −− , change the x value of the function to 0., change the x value of the function to 0.

−()+−()+ (())−− SimplifySimplify thethe fraction.fraction.

f(x) =f(x) = (^) −− ReduceReduce thethe fractionfraction toto lowestlowest term.term.

(()) ==   The value of fThe value of f(x) or y.(x) or y.

So, the ySo, the y – – intercept isintercept is   oror (0,(0,  ).).

2.2. Determine the zeroes ofDetermine the zeroes of the following rational functions:the following rational functions:

a.a. g(x) =g(x) = −−+6+6 b.b. H(x)H(x) == (^) −−−9−9 c.c. G(x)G(x) == 

+−+− −−

To find the zeroesTo find the zeroes of a rational fuof a rational function, equnction, equate the funate the function to 0 or solvection to 0 or solve for the xfor the x – – intercept of the function byintercept of the function by equating the numerator to 0.equating the numerator to 0.

a.a. g(x) =g(x) = −−+6+6,,

xx – – 2 2 == 0 0 EquateEquate thethe numeratornumerator toto 0 0 ..

x = 2x = 2 Solve for x.Solve for x.

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

b.b. (()) ==

−− −9−

(()) == (^) −−−9−9 SimplifySimplify byby factoringfactoring thethe denominatordenominator..

−− (−)(+)(−)(+) RemoveRemove commoncommon factors.factors.

 ++ == 0^0 EquateEquate thethe numeratornumerator toto 0.0.

1 1 == 0 0 FalseFalse statement.statement.

So, there isSo, there is no zerono zero of theof the function.function. Which meansWhich means that nothat no point onpoint on thethe graph touches the xgraph touches the x – – axis.axis.