A Concise Workbook for College Algebra

Transcription

1 City University of New York (CUNY) CUNY Academic Works Oen Educational Resources Queensborough Community College Summer A Concise Workbook for College Algebra Fei Ye Queensborough Community College How does access to this work benefit you? Let us know! Follow this and additional works at: htts://academicworks.cuny.edu/qb_oers Part of the Mathematics Commons Recommended Citation Ye, Fei, "A Concise Workbook for College Algebra" (08). CUNY Academic Works. htts://academicworks.cuny.edu/qb_oers/5 This Other is brought to you for free and oen access by the Queensborough Community College at CUNY Academic Works. It has been acceted for inclusion in Oen Educational Resources by an authorized administrator of CUNY Academic Works. For more information, lease contact AcademicWorks@cuny.edu.

2 A Concise Workbook for College Algebra Fei Ye Deartment of Mathematics and Comuter Science Queensborough Community College - CUNY 08

3 Dr. Fei Ye Deartment of Mathematics and Comuter Science Queensborough Community College of CUNY th Street, Bayside, NY, 6 feye@qcc.cuny.edu A Concise Workbook for College Algebra 08 Fei Ye Creative Commons License. This work is licensed under a Creative Commons Attribution-NonCommercial.0 International (CC BY-NC.0) License. (htts://creativecommons.org/licenses/by-nc/.0/).

4 Preface This workbook is mainly based on the author s worksheets for the college algebra course (MAD-9) at QCC of CUNY. It is intended as a concise introduction to college algebra at the intermediate level. Our motivation is to emhasize the deth of reasoning and understanding rather than multitudes of aroaches to similar tyes of questions. We try to eose only key concets and ideas, which will save time for racticing and reinforcing critical thinking skills which include observing atterns, identifying and analyzing roblems, making logic connections, determining roblem-solving strategies, and solving roblems systematically. For eamle, only the method of undetermined coefficient was introduced for factoring trinomials. To factor the trinomial A C B C C, where A, B and C are integers, we may use trial-and-error method to find integers m, n, and q such that mn D A, q D C and mq C n D B. In ractice, we first factor A and C, and then use the following diagram to check if mq C n D B holds. A D mn m n C D q q n C mq D B This method is based on the observation that A C B C C can be factored into.m C /.n C q/. Indeed, observing and making logic connections are very effective in roblem-solving. Toics are contained in 5 lessons. Each lesson corresonds to roughly one class meeting. A lesson starts with a age on concets, formulas and eamles, and ends with a list of ractice roblems that students are eected to be able to solve and comlete in class. This work was artially funded by the OER Conversion Grant from CUNY. We would like to thank our colleagues and students for their feedback and suort during the develoment of this roject. In articular, we would like to thank Joseh Bertorelli, Beata Ewa Carvajal, Kwai Chiu, Liu Li, Wenjian Liu, Nam Jong Moh, Tian Ren, Kostas Stroumbakis, Evelyn Tam, and Haishen Yao for their encouragement, suort, carefully reading the draft and correcting mistakes. Fei Ye Summer 08

5 Contents Preface ii Linear Inequalities Absolute Value Equations Proerties of Integral Eonents 5 Introduction to Functions 7 5 Linear functions 9 6 Perendicular and Parallel Lines 7 Systems of Linear Equations 8 Factoring Review 5 9 Solve Quadratic Equations by Factoring 7 0 Multily or Divide Rational Eressions 0 Add or Subtract Rational Eressions Comle Rational Eressions Rational Equations 6 Radical Eressions: Concets and Proerties 8 5 Algebra of Radicals 6 Solve Radical Equations 7 Comle Numbers 6 8 Comlete the Square 8 9 Quadratic Formula 0 0 Grahs of Quadratic Functions Rational and Radical Functions Eonential Functions 5 Logarithmic Functions 7 Proerties of Logarithms 9 5 Eonential and Logarithmic Equations 5

6 Lesson. Linear Inequalities Proerties of Inequalities Proerty The additive roerty If a < b, then a C c < b C c. If a < b, then a c < b c. The ositive multilication roerty If a < b and c is ositive, then ac < bc. If a < b and c is ositive, then a c < b c. Eamle If C < 5, then C < 5. Simlifying both sides, we get <. If <, then <. Simlifying both sides, we get <. The negative multilication roerty If a < b and c is negative, then ac > bc. If a < b and c is negative, then a c > b c. Note: These roerties also aly to a b, a > b and a b. If <, then D. / >. / D. If <, then >. Simlifying both sides, we get >. Comound Inequalities A comound inequality is formed by two inequalities with the word and or the word or. For eamle, > and C <. 5 < or > 0. <, which means that and <. Eamle.. Solve the linear inequality C > 0. C > 0 add > divide by > The solution set is. ; /. Eamle.. Solve the comound linear inequality C < and <. C < and < < < < and > That is < <. The solution set is. ; /. Solve the comound linear in- <. Eamle.5. equality < < 6 8 < 0 < 5 The solution set is Π; 5/. A Concise Workbook for College Algebra Eamle.. Solve the linear inequality <. < add < 6 divide by and switch > The solution set is. ; C/. Eamle.. Solve the comound linear inequality C > or 5. C > or 5 > < or That is or <.. ; C/. Solve the comound linear in- C <. Eamle.6. equality The solution set is C < C < 6 6 < > The solution set is. ;.

7 Lesson. Linear Inequalities Practice.. Solve the linear inequality. Write your answer in interval notation. () C 7 () >. (). ;. ().; C/ Practice.. Solve the linear inequality. Write your answer in interval notation. () C 7 > () 6. (). 5; C/. () Œ; C/ Practice.. Solve the comound linear inequality. Write your answer in interval notation. () C > and 7. () 7 < 5 and 5. (). ;. () Œ; / Practice.. Solve the comound linear inequality. Write your answer in interval notation. () C 5 <. () 7 7. () Œ ; /. () Œ ; 5 Practice.5. Solve the comound linear inequality. Write your answer in interval notation. () 5 > or 0. () C 7 < 5 or 8.5 ().; C/.5 (). ; / [ Œ; C/ Practice.6. Solve the comound linear inequality. Write your answer in interval notation. () 5 <. () < C 7..6 () Œ ; 7/.6 (). ; A Concise Workbook for College Algebra

8 Lesson. Absolute Value Equations Absolute Value Equations The absolute value of a real number a, denoted by jaj, is the distance from 0 to a on the number line. In articular, jaj is always greater than or equal to 0, that is jaj 0. The absolute values have the following roerties: j aj D jaj, jabj D jajjbj and ˇˇ aˇ b D jaj jbj. Let X reresent an algebraic eression. If c is a ositive number, then the equation jxj D c is equivalent to X D c or X D c. If c is a negative number, then the solution set of jxj D c is emty. An emty set is denoted by ;. The equation jxj D 0 is equivalent to X D 0. Eamle.. Solve the equation j j D 7. Eamle.. Solve the equation j j D 8. The equation is equivalent to Rewrite the equation into jxj D c form. D 7 or D 7 j j D Solve the equation. D D 0 D or D D or D 5 D 0 D The solutions are D or D 5. In set-builder notation, the solution set is f ; 5g. Eamle.. Solve the equation j 5j D 9. Rewrite the equation into jxj D c form. j 5j D Solve the equation. 5 D or 5 D D D 8 D or D The solutions are D or D. In set-builder notation, the solution set is f; g. Eamle.5. Solve the equation j j D j C j. Because two numbers have the same absolute value only if they are the same or oosite to each other. Then the equation is equivalent to D C or D. C / D D 0 D or D 0 The solutions are D and D 0. In set-builder notation, the solution set is f0; g. D 5 or D 6 The solutions are D 5 or D 6. In set-builder notation, the solution set is f 5; 6g. Eamle.. Solve the equation j j D 7. Rewrite the equation into jxj D c form. j j D 5 Solve the equation. D 5 or D 5 D D 6 D or D The solutions are D or D. In set-builder notation, the solution set is f ; g. Eamle.6. Solve the equation j j D j C 0j. This equation is similar to the one in Eamle.5 ecet that there is an etra number outside. Since is ositive, we are free to move it inside. Moreover, because j Xj D jxj, the equation is equivalent to j j D j C 0j D C 0 or D. C 0/ D 0 or D 8 D The original equation only has one solution D. In set-builder notation, the solution set is f g. A Concise Workbook for College Algebra

9 Lesson. Absolute Value Equations Practice.. Find the solution set for the equation. () j j D 5. () 9 ˇ ˇ D.. () f ; g. () f; 5g Practice.. Find the solution set for the equation. () j 6j C D. () j 5j D 9.. () f ; 5g. () f; g Practice.. Find the solution set for the equation. () j5 0j C 6 D 6. () j j D 5.. () fg. () ; Practice.. Find the solution set for the equation. () j5 j D j j. () j j D 5j. / j.. () f; g. () A Concise Workbook for College Algebra

10 Lesson. Proerties of Integral Eonents Review on Proerties of Eonents An eonent is a number that we ut on the uer right corner of a number, variable or eression, called the base, to tell us how many times we should multily the base with itself. In algebra, we write n D ƒ : n factors of Eonents have the following roerties.. The roduct rule m n D mcn : Eamle:. / D 6 5 :. The quotient rule (Suose 0.) 8 m ˆ< m n if m is greater than n D or equal to n. ˆ: if m is less than n: n m. The zero eonent rule (Suose 0.) 0 D :. The negative eonent rule (Suose 0.) n D n and n D n : 5. The ower to a ower rule: a b D ab : 6. The roduct raised to a ower rule:.y/ n D n y n : Eamle: 5 5 Eamle: Eamle: 5 D ; 6 D. / 0 D ; 0 D. / D. / D 8 ; D D : Eamle: Eamle: D 6 D 6; D 6 :. / D. / D a b D. /.a / b D a 6 b : 7. The quotient raised to a ower rule (Suose y 0.) Eamle: n D n D y y n :. / D a 8 ; b D. a /.b / D a b 6 : Order of Basic Mathematical Oerations In mathematics, the order of oerations reflects conventions about which rocedure should be erformed first. There are four levels (from the highest to the lowest): Parenthesis; Eonent; Multilication and Division; Addition and Subtraction. Within the same level, the convention is to erform from the left to the right. y z 5 Eamle.. Simlify y. y z 5 D y z 5 y 6 D z 5 y 6 D.z 5 /.y 6 /. / D 6z0 y : A Concise Workbook for College Algebra 5

11 Lesson. Proerties of Integral Eonents Practice.. Simlify. Write with ositive eonents. () a a a 5 u 0 v 5 () v 6 (). / (). y/. () a 0. () v. () 8. () y 6 Practice.. Simlify. Write with ositive eonents. ().a b c /.abc /.b c y z 0 / () y (). a b c 0 / (). () a b 6 c 7. () y. () 8a 9 b 6. () 8 Practice.. Simlify. Write with ositive eonents. (). a / 0 y a b 5 () wz () y () 6 y 5 y z. (). () 9a 6 y 9. () 8w z 6 b 5 y. () 9a 6 7y z 6 A Concise Workbook for College Algebra

12 Lesson. Introduction to Functions Functions are Relations A relation is a set of ordered airs. The set of all first comonents of the ordered airs is called the domain. The set of all second comonents of the ordered airs is called the range. A function is a relation such that each element in the domain corresonds to eactly one element in the range. Functions as Equations and Function Notation For a function, we usually use the variable to reresent an element from the domain and call it the indeendent variable. The variable y is used to reresent the value corresonding to and is called the deendent variable. We say y is a function of. When we consider several functions together, to distinguish them we named functions by a letter such as f, g, or F. The notation f./, read as f of or f at, reresents the outut of the function f when the inut is. To find the value of a function at a given number, we substitute the indeendent variable by the given number and then evaluate the eression. We call the rocedure evaluating a function. Eamle.. Find the indicated function value.. f. /, f./ D C f. / D. / C D C D.. g./, g./ D 0 g. / D. / 0 D 0 D 0 D.. h.a t/, h./ D C 5 h.a t/ D.a t/ C 5 D a t C 5. Grahs of Functions and Grahs as Functions The grah of a function is the grah of its ordered airs. A grah of ordered airs.; y/ in the rectangular system defines y as a function of if any vertical line crosses the grah at most once. This test is called the vertical line test. The domain of a grah is the set of all inuts (-coordinates). The range of a grah is the set of all oututs (y-coordinates). To find the domain of a grah, we look for the left and the right endoints. To find the range of a grah, we look for the highest and the lowest ositioned oints. Eamle.. Use the grah on the right to answer the following questions. () Determine whether the grah is a function and elain your answer. () Find the domain of the grah (write the domain in interval notation). () Find the range of the grah (write the range in interval notation). 5 y () The grah is a function. Because every vertical line crosses the grah at most once. () The grah has the left endoint at. ; / and but no right endoint. So the domain is Œ ; C/. () The grah has two lowest ositioned oints. ; / and.; / and but no highest ositioned oint. So the range is Œ; C/. A Concise Workbook for College Algebra 7

13 Lesson. Introduction to Functions Practice.. Find the indicated function values for the functions f./ D C and g./ D. Simlify your answer. () f./ () f. / () g. / () g.f.//. () f./ D. () f. / D. () g. / D. () g.f.// D Practice.. Use the grah on the right to answer the following questions. y () Determine whether the grah is a function and elain your answer. () Find the domain of the grah (write the domain in interval notation). () Find the range of the grah (write the range in interval notation).. () Yes. (). ;. (). ; / Practice.. Find the corresonding -coordinate of a oint on the grah in Practice. whose y-coordinate is.. D 0 8 A Concise Workbook for College Algebra

14 Lesson 5. Linear functions The Sloe-Intercet Form Equation The sloe of a line measures the steeness, in other words, rise over run, or rate of change of the line. Using the rectangular coordinate system, the sloe m of a line is defined as m D y y D rise run, where. ; y / and. ; y / are any two distinct oints on the line. If the line intersects the y-ais at the oint.0; b/, then a oint.; y/ is on the line if and only if y D m C b. This equation is called the sloe-intercet form of the line. Linear Function A linear function is a function whose grah is a line. A linear functions can be written as f./ D m C b, where m is the sloe and.0; b/ is the y-intercet. If a function f./ is a linear function, then b D f.0/ and the sloe Eamle 5.. function. m D f. / f. / D change in outut change in inut. How to Write an Equation for a Linear Function? If f./ is a linear function, with f./ D 5, and f. / D, find an equation for the f./ f. / D 5. / Ste. Find the sloe m: m D. / D D. Ste. Plug D into the sloe-intercet form f./ D C b and then solve for b: 5 D f./ D C b 5 C. / D C. / C b D b Ste. The sloe-intercet form equation of this function is f./ D C. Sketch the Grah of a Linear Function via Plotting Points Eamle 5.. Sketch the grah of the linear function f./ D C. Method : Get oints by evaluating f./. Ste. Choose two or more inut values, e.g. D 0 and D. Ste. Evaluate f./: f.0/ D and f./ D 0. Ste. Plot the oints.0; / and.; 0/ and draw a line through them. Method : Get oints by raise and run. Ste. Plot the y-intercet.0; f.0// D.0; /. Ste. Use rise run D to get one or more oints, e.g, we will get. ; / by taking rise D and run D, i.e. move u unit, then move to the right units. Ste. Plot the oints.0; / and. ; / and draw a line through them. y run=- y rise= A Concise Workbook for College Algebra 9

15 Lesson 5. Linear functions Practice 5.. Finding the sloe of the line assing through.; 5/ and. ; /. 5. m D. Practice 5.. Find the sloe-intercet form equation of the line assing through.6; / and.; 5/. 5. y D C 6 Practice 5.. Determine if the linear functions f./ and h./ with the following values f. / D f.0/ D h.0/ D and h./ D 8 define the same function. Elain your answer. Practice 5.. Grah the function. 5. Yes. Because f./ D C and g./ D C. () f./ D C () f./ D 5 y 5 y A Concise Workbook for College Algebra

16 Lesson 6. Perendicular and Parallel Lines Horizontal and Vertical Lines A horizontal line is defined by an equation y D b. The sloe of a horizontal line is simly zero. A vertical line is defined by an equation D a. The sloe of a vertical line is undefined. A vertical line gives an eamle that a grah is not a function of. Indeed, the vertical line test fails for a vertical line. Get the Elicit Function from an Equation When studying functions, we refer a clearly eressed function rule. For eamle, in f./ D C, the eression C clearly tells us how to roduce oututs. For a function f defined by an equation, for instance, C y D, to find the function rule (that is an eression), we simly solve the given equation for y. C y D y D C y D C : Now, we get f./ D C. Point-Sloe Form Equation of a Line When the sloe m of a line and a oint. 0 ; y 0 / on the line are given, we can write down an equation for the line in the following form, called the oint-sloe form: y y 0 D m. 0 /. This equation comes essentially from the sloe formula m D y y 0 0. Any two vertical lines are arallel. Two nonvertical lines are arallel if and only if they have the same sloe. A line that is arallel to the line y D m C a has an equation y D m C b, where a b. Perendicular and Parallel Lines Horizontal lines are erendicular to vertical lines. Two non-vertical lines are erendicular if and only if the roduct of their sloes is. A line that is erendicular to the line y D mca has an equation y D m C b. Find Equations for Perendicular or Parallel Lines Eamle 6.. Find an equation of the line that is arallel to the line C y D and asses through the oint. ; /. Ste. Find the sloe m of the original line from the sloe-intercet form equation by solving for y. y D C. So m D. Ste. Find the sloe m k of the arallel line. m k D m D. Ste. Use the oint-sloe form. y D. C / y D 5: Eamle 6.. Find an equation of the line that is erendicular to the line y D and asses through the oint. ; /. Ste. Find the sloe m of the original line from the sloe-intercet form equation by solving for y. y D. So m D. Ste. Find the sloe m? of the erendicular line. m? D m D. Ste. Use the oint-sloe form. y D. C / y D C : A Concise Workbook for College Algebra

17 Lesson 6. Perendicular and Parallel Lines Practice 6.. Find the oint-sloe form equation of the line with sloe 5 that asses though. ; /. 6. y D 5. C / Practice 6.. Find the oint-sloe form equation of the line assing thought.; / and.; /. 6. y D. / Practice 6.. Find the sloes of the lines that are arallel and erendicular to the line 5y D. 6. mk D 5 m? D 5 Practice 6.. Find the oint-sloe form and then the sloe-intercet form equations of the line arallel to y D and assing through the oint.; /. 6. y C D. / y D 9 Practice 6.5. Find the sloe-intercet form equation of the line that is erendicular to y C D 0 and assing through the oint.; 5/ 6.5 y D A Concise Workbook for College Algebra

18 Lesson 7. Systems of Linear Equations The Substitution Method and the Addition Method (the Elimination Method) A system of linear equations of two variables consists of two equations. A solution of a system of linear equations of two variables is an ordered air that satisfies both equations. Substitution Method Eamle 7.. Solve the system of linear equations using the substitution method. C y D () C y D () Ste. Solve one variable from one equation. For eamle, one may solve y from equation (). y D : Ste. Plug y D into the second equation and solve for. C. / D C D D Ste. Plug the solution D into the equation in Ste to solve for y. y D D D Ste. Check the roosed solution. Plug.; / into the first equation: C D. Plug.; / into the second equation: C D. Elimination Method Eamle 7.. Solve the system of linear equations using the addition method. 5 C y D 7 () y D () Ste. Eliminate one variable and solve for the other. For eamle, one may choose to eliminate y. In order to eliminate y, we add the oosite. We multily both sides of the second equation by to get the oosite y../ y D P./ 6 y D 6 () Adding equations () and () will eliminate y. 5 C y D 7 C 6 y D 6 C 0 D D Ste. Find the missing variable by lugging D into the first equation and solve for y. 5 C y D 7 5 C y D 7 y D 8 y D Ste. Check the roosed solution. Plug.; / into the first equation: 5 C. / D 5 8 D 7. Plug.; / into the second equation:. / D 9 C D. Note: A linear system may have one solution, no solution or infinitely many solutions. If the lines defined by equations in the linear system have the same sloe but different y-intercets or the elimination method ends u with 0 D c, where c is a nonzero constant, then the system has no solution. If the lines defined by equations in the linear system have the same sloe and the same y-intercet or the elimination method ends u with 0 D 0, then the system has infinitely many solutions. In this case, we say that the system is deendent and a solution can be eressed in the form.; f.// D.; m C b/. A Concise Workbook for College Algebra

19 Lesson 7. Systems of Linear Equations Practice 7.. Solve. Practice 7.. Solve. y D 8 C y D0 5y D y D Practice 7.. Solve. Practice 7.. Solve. 5y D 9 C 5y D9 7 C y D 8y D Practice 7.5. Solve. Practice 7.6. Solve. C 7y D 5 C y D 0 C y D 6 C 5y D8 Practice 7.7. Solve. Practice 7.8. Solve. C y D 6 y D 6 6 C y D 6 C 6y D 7..; 7.. ; / 7.. ; 7.. ; / 7.5.6; 7.6. ; / 7.7 no 7.8.; C / / 5/ / solution A Concise Workbook for College Algebra

20 Lesson 8. Factoring Review Factor by Removing the GCF The greatest common factor (GCF) of two terms is a olynomial with the greatest coefficient and of the highest ossible degree that divides each term. To factor a olynomial is to eress the olynomial as a roduct of olynomials of lower degrees. The first and the easiest ste is to factor out the GCF of all terms. Eamle 8.. Factor y 8 y C y. Ste. Find the GCF of all terms. The GCF of y, 8 y and y is y. Ste. Ste. Write each term as the roduct of the GCF and the remaining factor. y D. y/, 8 y D. y/. y/, and y D y. Factor out the GCF from each term. y 8 y C y D y. y C y /. Factor a Four-term Polynomial by Grouing For a four-term olynomial, in general, we will grou them into two grous and factor out the GCF for each grou and then factor further. Eamle 8.. Factor 6y C z yz. Ste. Grou the first two terms and the last two terms. 6y C z yz Ste. Ste. D. 6y/ C.z yz/ Factor out the GCF from each grou. D. y/ C z. y/ Factor out the binomial GCF. D. y/. C z/: Eamle 8.. Factor a C b a b. Ste. Grou the first term with the third term and grou the second term and the last term. Ste. Ste. Factor Binomials in Secial Forms a C b a b D.a a/ C. b C b/ Factor out the GCF from each grou. Da. / C. b/. / Factor out the binomial GCF. D. /.a b/: Difference of squares: a b D.a b/.a C b/: Eamle 8.. Factor 5 6. Ste. Recognize the binomial as a difference of squares. Ste. 5 6 D.5/ Aly the formula. D.5 /.5 C /: Sum of Cubes: a C b D.a C b/.a ab C b /: Eamle 8.5. Factor C 7. Ste. Recognize the binomial as a sum of cubes. C 7 Ste. D C Aly the formula. D. C /. C / D. C /. C 9/: Difference of cubes: a b D.a b/.a C ab C b /: Eamle 8.6. Factor 5 8. Ste. Recognize the binomial as a difference of cubes. 5 8 Ste. D.5/ Aly the formula. D.5 /..5/ C.5/ C / D.5 /.5 C 0 C /: A Concise Workbook for College Algebra 5

21 Lesson 8. Factoring Review Practice 8.. Factor out the GCF. () 8 y y 6 y () 5. 7/ C y. 7/ 8. () 6y. y C y/ 8. (). 7/.5 C y/ Practice 8.. Factor by grouing. () y 0y C 8 5 () ac 8bc 0ad C 5bd () 5a b 5ay C by 8. ().6 5/.y C / 8. ().a b/.6c 5d/ 8. (). y/.5a b/ Practice 8.. Factor comletely. () 5 () 8 7 () 5y C 8. ().5 /.5 C / 8. (). /. C 6 C 9/ 8. ().5y C /.5y 5y C / Practice 8.. Factor comletely. () 7 C y () 6y () C 8. (). C y/.9 C y / 8. ().y /.y C y C / 8. (). /. C /. C / 6 A Concise Workbook for College Algebra

22 Lesson 9. Solve Quadratic Equations by Factoring Factor Trinomials If a trinomial A C B C C can be factored, then it can be eressed as a roduct of two binomials: A C B C C D.m C n/. C q/: By simlify the roduct using the FOIL method and comaring coefficients, we observe that A D ƒ mn F B D mq ƒ O C C n ƒ I C D nq ƒ L The observation suggests the following strategy called the method of undetermined coefficients. Eamle 9.. Factor C 6 C 8. Ste. Factor A D : D. Ste. Factor C D 8: 8 D 8 D. A D D C D 8 D C D 6 D B Ste. Factor the trinomial C 6 C 8 D. C /. C /. Eamle 9.. Factor C 5. Ste. Factor A D : D. Ste. Factor C D : D. / D. /. Ste. Choose a roer combination of airs Ste. Choose a roer combination of airs of of factors and check if the sum of cross roduct equals B D 6: C D 6. This ste can be checked easily using the following diagram. factors and if the sum of cross roducts equals B D 5: C. / D 5. This ste can be checked easily using the following diagram. A D D C D D. / Ste. C. / D 5 D B Factor the trinomial C 6 D. C /. /. The following eamle shows how to factor a certain higher degree olynomial by substitution. Eamle 9.. Factor the trinomial comletely. Ste. Let D y. Then D y y. Ste. Factor the trinomial in y: y y D.y C /.y /. Ste. Relace y by and factor further. D y y D.y C /.y / D. C /. / D. C /. /. C /: A Concise Workbook for College Algebra 7

23 Lesson 9. Solve Quadratic Equations by Factoring Solving a Quadratic Equation by Factoring A quadratic equation is a olynomial equation of degree, for eamle, C 5 D 0. The standard form of a quadratic equation is a C b C c D 0, where a, b and c are numbers, and a 0. To solve a quadratic equation, we may first factor the olynomial and then aly the zero roduct roerty: A B D 0 if and only if A D 0 or B D 0. Eamle 9.. Solve the equation C 5 D. Ste. Rewrite the equation into Eression=0 form and factor. C 5 D Ste. Ste. C 5 D 0. /. C / D 0 Aly the zero roduct roerty. D 0 or C D 0: Solve each equation. D 0 or C D 0 D D D or D Ste. The solution set is f ; g. Eamle 9.6. A rectangular garden is surrounded by a ath of uniform width. If the dimension of the garden is 0 meters by 6 meters and the total area is 6 square meters, determine the width of the ath. 6 0 Eamle 9.5. Solve the equation. /. C / D. Ste. Rewrite the equation into Eression=0 form and factor.. /. C / D Ste. Ste. C 6 D C D 0. /. C / D 0 Aly the zero roduct roerty. D 0 or C D 0: Solve each equation. D 0 or C D 0 D or D Ste. The solution set is f ; g. Alications of Quadratic Equations Ste. Suose that the width of the frame is meters. Translate given information into eressions in and build an equation. Total Width: +0 Total Length: +6 Width Length=Total Area: Ste. Ste.. C 0/. C 6/ D 6: Solve the equation.. C 0/. C 6/ D 6 C 5 C 60 D 6 C 5 56 D 0 C D 0. C /. / D 0 D or D So the width of the ath is meter. 8 A Concise Workbook for College Algebra

24 Lesson 9. Solve Quadratic Equations by Factoring Practice 9.. Factor the trinomial. () C C () C 6 7 () 0 () 5 C 6 9. (). C /. C / 9. (). /. C 7/ 9. (). 5/. C / 9. (). /. / Practice 9.. Factor the trinomial. () 5 C 7 C () C 5 () 0 8 () C 5 9. (). C /.5 C / 9. (). C /. / 9. (). /. C / 9. (). /. 5/ Practice 9.. Solve the equation by factoring. () C D 0 () D 5 (). /. C / D 5 9. () f; g 9. () f ; 5 g 9. () f ; g Practice 9.. A aint measuring inches by inches is surrounded by a frame of uniform width. If the combined area of the aint and the frame is 0 square inches, determine the width of the frame. 9. inches. Practice 9.5. A rectangle whose length is meters longer than its width has an area 8 square meters. Find the width and the length of the rectangle. 9.5 width: meters length: meters. Practice 9.6. The roduct of two consecutive negative odd numbers is 5. Find the numbers. 9.6 The numbers are 7 and 5. A Concise Workbook for College Algebra 9

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26 Lesson 0. Multily or Divide Rational Eressions What is a Rational Eression and How to Simlify It? A rational eression is a fraction, where the numerator and the denominator q are both q olynomials and the degree of q is nonzero. A rational eression is simlified if the numerator and the denominator have no common factor other than. If, q and c are olynomials with q 0 and c 0, then c q c D q : Eamle 0.. Simlify C C C C. Ste. Ste. Factor both the to and the bottom. C C. C /. C / D C C. C /. C / : Divide out common factors.. C /. C /. C /. C / D C C : Eamle 0.. Simlify 5. Ste. Ste. How to Multily Rational Eressions? If, q, s, t are olynomials with q 0 and t 0, then q s D s t qt : Factor both the to and the bottom.. C /. / D 5. C /. 5/ : Divide out common factors.. C /. /. C /. 5/ D 5 : Eamle 0.. Multily and then simlify. Eamle 0.. Multily and then simlify. C 6 6 : 8 C 8 C : Ste. Factor numerators and denominators.. C /. / D /. C /. C /. C /. C /. /.5 C /. / C 6 D. 6. /. C / D. C /. /. C /. / Ste. Multily and simlify.. C /. C /.5 C /. /. /. C /. C /. C /. /. /. D D C /. C /. C /.5 C / How to Divide Rational Eressions? If, q, s, t are olynomials where q 0, s and t 0, then q s D t q t s D t qs : Eamle 0.5. Divide and then simlify. C : Ste. Ste. Rewrite the division as a multilication. C D C Factor and simlify. C D 6. C / C /. /. C /. C /. /. C /. /. D D. C /. 7/. /. C /. 7/. /. 7/. / 0 A Concise Workbook for College Algebra

27 Lesson 0. Multily or Divide Rational Eressions Practice 0.. () C Simlify. () C C () 9 8 C C 0. () Practice 0.. Multily and simlify. C 5 () C C () 5 C C 9 0. () 0. () () 6y y y 6y C 9 y y 0. () 5 0. (). / C 0. ().y / y Practice 0.. Divide and simlify. 9 9 () () 6 C C 0 5 C 6 C () y y y y 0. () C 7 0. () C 5 0. () y. C y/ A Concise Workbook for College Algebra

28 Lesson. Add or Subtract Rational Eressions Add or Subtract Rational Eressions with the Same Denominator If P, Q and R are olynomials with R 0, then P R C Q R D P C Q R Eamle.. Add and simlify and C C C C 5 C C. P R Q R D P Q R : Ste. Determine if the rational eressions have the same denominator. If so, the new numerator will be the sum/difference of the numerators. C C C C 5 C C D C 5 C C C : Ste. Simlify the resulting rational eression. C 5 C. C /. C / D C C. C /. C / D C C : Add or Subtract Rational Eressions with Different Denominators To add or subtract rational eressions with different denominators, we need to rewrite the rational eressions to equivalent rational eressions with the same denominator, say the LCD. Ste. Find the least common denominator (LCD). Ste. Rewrite each rational eression into an equivalent rational eression with the LCD as the new denominator. Ste. Add or subtract the resulting rational eressions and simlify. Eamle.. Find the LCD of 6 and 6. Ste. Factor each denominator. 6 D. C /. / D. /. C / Ste. List the factors of the first denominator and add unlisted factors of the second factor to get the final list. First list. C /. / Second list. C /. / Final list. C /. /. / Ste. The LCD is the roduct of factors in the final list.. C /. /. /. Eamle.. Subtract and simlify 8 Ste. Find the LCD. First list. C /. / 8 D. C /. / Second list. C /. / D. /. C / Final list. C /. /. / The LCD is. C /. /. /. Ste. Rewrite each rational eression into an equivalent eression with the LCD as the new denominator. 8 D. /. /. /. C /. /. /. C /. /. / Ste. Subtract and simlify.. /. /. /. C /. /. / D. 5 C 6/. /. C /. /. / D 6 C 0. C /. /. / A Concise Workbook for College Algebra

29 Lesson. Add or Subtract Rational Eressions Practice.. Add/subtract and simlify. C 5 C () C C 5 C 5 () () C. () 5 Practice.. Find the LCD of rational eressions. () and 5 () and 8 C 6. (). C /. /. C /. () 7. / Practice.. Add and simlify. () C C C () C C C C () C 6. () C. C /. C /. () C 6 C. C /. C /. /. () 7 C 6. /. C / Practice.. Subtract and simlify. C 5 y () () 7 C y 5y 6 7 y y 5 () C 0 C C 8. () 7. /. /. () y y C.y 6/.y 5/.y C /. (). /. C /. C 5/ A Concise Workbook for College Algebra

30 Lesson. Comle Rational Eressions Simlify Comle Rational Eressions A comle rational eression is a rational eression whose denominator or numerator contains a rational eression. A comle rational eression is equivalent to the quotient of its numerator by its denominator. That suggests the following strategy to simlify a comle rational eression. Ste. Ste. Ste. Simlify the numerator and the denominator. Rewrite the eression as the numerator multilying the recirocal of the denominator. Multily and simlify. Eamle.. Simlify C C C Ste. Simlify the numerator and the denominator. C. /. / C C. /. C /. /. C / D C. C /. C /. /. C /. /. C / Ste. Ste.. / C. C /. /. C / D. C C /. /. C / Rewrite as a roduct.. /. C / C. /. C / Multily and simlify.. /. C /. /. C / D C D D D. /. C / C. /. C /. /. C /. /. C / C. /. C / D. /. C /. C /. / C. /. /. /. C /. C /. C /. /. C /. /. C /. C /. /. C / D C A Concise Workbook for College Algebra

31 Lesson. Comle Rational Eressions Practice.. () C Simlify. (). () C. () C Practice.. y () y y Simlify. (). C / C. C /. (). C y/ y. () Practice.. Simlify. 5 () 6 C C () C C C C C C A Concise Workbook for College Algebra 5. (). ()

32 Lesson. Rational Equations Solve Rational Equations by Clearing Denominators A rational equation is an equation that contains a rational eression. One idea to solve a rational equation is to reduce the equation to a olynomial equation by clearing denominators, that is multilying the LCD to both sides of the equation. Ste. Ste. Ste. Find the LCD. Clear denominators by multilying the LCD to both sides. Solve the resulting olynomial equation. Ste. Check if there is an etraneous solution which is a solution that would cause any of the eressions in the original equation to be undefined. Eamle.. Solve 5 9 D Ste. Find the LCD. Since 9 D. C /. /, the LCD is. C /. /. C : Ste. Clear denominators. Multily each rational eression in both sides by. C /. 5. C /. / 9 D. C /. / 5 D. C /. / Ste. Solve the resulting equation. 5 D. C /. / 5 D C 5 0 D / and simlify:. C /. / C Ste. Check for any etraneous solution by lugging the solution into the LCD to see if it is zero. If it is zero, then the solution is etraneous. So D. 0 C /. 0 / 0 0 is a valid solution of the original equation. Eamle.. Solve for from the equation C y D z : Ste. The LCD is yz. Ste. Clear denominators. yz C yz y D yz z Ste. Solve the resulting equation. Ste. The solution is D yz y z. yz C z D y yz C z D y yz D y z yz D.y z/ yz y z D 6 A Concise Workbook for College Algebra

33 Lesson. Rational Equations Practice.. Solve. () C C D () 0 5 D C 5 C 5 Practice.. Solve. () C 8 D C. () D. () D 7 () 5 D C 5. () D. () D 6 or D 7 Practice.. Solve a variable from a formula. () Solve for f from C q D f C c. () Solve for from A D. f. () D f A c A Concise Workbook for College Algebra 7. () f D q C q

34 Lesson. Radical Eressions: Concets and Proerties Radical Eressions If b D a, then we say that b is a square root of a. We denote the ositive square root of a as a, called the rincial square root. For any real number a, the eression a can be simlified as a D jaj: If b D a, then we say that b is a cube root of a. The cube root of a real number a is denoted by a. For any real number a, the eression a can be simlified as a D a: In general, if b n D a, then we say that b is an n-th root of a. If n is even, the ositive n-th root of a, called the rincial n-th root, is denoted by n a. If n is odd, the n-the root n a of a has the same sign with a. In n a, the symbol is called the radical sign, a is called the radicand, and n is called the inde. If n is even, then the n-th root of a negative number is not a real number. For any real number a, the eression n a n can be simlified as n. a n D jaj if n is even. n. a n D a if n is odd. Eamle.. Simlify the radical eression using the definition. ().y / () 8 y 6 ().y / D Œ.y / D jy j. () 8 y 6 D. y / D y. If n a is a real number, then we define a m n Definition and Proerties of Rational Eonents as a m n D n a m D. n a/ m : Rational eonents have the same roerties as integral eonents:. a m a n D a mcn a m. a n D am n. a m n D a m n a m..a m / n D a mn 5..ab/ m D a m b m a m 6. D b Eamle.. Simlify the radical eression or the eression with rational eonents. Write in radical notation. () () () () () () D D C D 7 6 D 6 : 5 6 q D. / D Œ. / D D D : 5 6! D. 5 6 / D. C 5 6 / D. / D D : b m 8 A Concise Workbook for College Algebra

35 Lesson. Radical Eressions: Concets and Proerties Practice.. () 6 Evaluate theqsquare root. If the square root is not a real number, state so. () 5 () 9 C 9 () 5. () 0. () not a real number Practice.. Simlify the radical eression. (). 7/ (). C / () 5 y 6 () 7 (5). (). () 6 8 (6) 5. / 5. () 7. () j C j. () j5y j. (). (5). (6) Practice.. Write the radical eression with rational eonents. ()./ 5 (). 5 y/ 7. ()./ 5. ().y/ 7 5 A Concise Workbook for College Algebra 9

36 Lesson. Radical Eressions: Concets and Proerties Practice.. Write in radical notation and simlify. () () 8. () 8. () 7 Practice.5. Simlify the eression. Write with ositive rational eonents. Assume all variables reresent nonnegative numbers. () (). 5 y /.5 ().5 () y Practice.6. Simlify the eression. Write in radical notation. Assume all variables reresent nonnegative numbers. () 6a y () () ().6 ().6 () y 6.6 () a y.6 () 6.6 (5) 6.6 (6) A Concise Workbook for College Algebra

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38 Lesson 5. Algebra of Radicals Product and Quotient Rules for Radicals If n a and n b are real numbers, then n n a b D n ab. If n a and n b are real numbers and b 0, then n r a a n D n b b : Eamle 5.. Simlify the eression. () 8y 7 y. () y 5 y. () 8y 7 y D.8y /. 7 y/ D 6 8 y 5 D. / y y D y y. () y 5 y D 5 q 96 9 y y D 5 0 y D 5. / 5 y D 5 y. Add/Subtract by Combining Like Radicals Two radicals are called like radicals if they have the same inde and the same radicand. We add or subtract like radicals by combining their coefficients. Eamle Simlify the eression. 8 q. / C 5 : q. / C 5 D C D. C / : Multilying Radicals with Two or More Terms Multilying radical eressions with many terms is similar to that multilying olynomials with many terms. Eamle 5.. Simlify the eression.. C /. /:. C /. / D C 6 D C 6 D D. C /: Rationalizing Denominators Rationalizing denominator means rewriting a radical eression into an equivalent eression in which the denominator no longer contains radicals. Eamle 5.. Rationalize the denominator. C y () () y () In this case, to get rid of the radical in the bottom, we multily the eression by the radicand in the bottom becomes a erfect ower. D D D. () In this case, we use the formula.a b/.a C b/ D a C b. Multily the eression by so that C y C y. C y D y.. C y/ D C y C y : y/. C y/ y A Concise Workbook for College Algebra

39 Lesson 5. Algebra of Radicals Practice 5.. Multily. () 5 () C 7 7 (Assume that 7.) 5. () 0 5. () 9 Practice 5.. Simlify the radical eression. Assume all variables are ositive. () 50 () 8 y 5 () y z 8 () 0y y (5) 6 5 (6) 5 8 y z 5 8y z 8 5. () 5 5. () y 5. () z 5 y z 5. () y 5y 5. (5) 0 5. (6) yz 5 y z Practice 5.. Add or subtract. Assume all variables are ositive. Answers must be simlified. () 5 6 C 6 () 0 5 () C 5 8 () 7 C 5 6 (5) 5 y C 7 5 y (6) 9y y 6y C 5y 5. () () () 5. () C 6 5. (5).5 C y/ 5 y 5. (6) y y A Concise Workbook for College Algebra

40 Lesson 5. Algebra of Radicals Practice s 5.. Divide. Assume all variables are ositive. Answers must be simlified. 9 () r () 0 5 a () y 8 () 6 b b 5. () 5. () 5. () 5 5. () a b y Practice 5.5. Multily and simlify. Assume all variables are ositive. (). / (). 5 C 7/. 5 7/ (). C / (). 6 5/. 6 C 5/ (5). C /. C C / (6). C 6/. C / 5.5 () () C () 5 C () 5.5 (5) 5.5 (6) C 8 C 6 Practice 5.6. are ositive. q () 5 Simlify the radical eression and rationalize the denominator. Assume all variables () q 7y () y () y (5) 6 (6) 5 5C (7) C C (8) y 5.6 () () y 7y 5.6 () 9y y 5.6 (5) 9 C 5.6 (6) C (7) 6 C 5.6 () y (8) y C y y A Concise Workbook for College Algebra

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42 Lesson 6. Solve Radical Equations Solve Radical Equations by Taking the n-th Power The idea to solve a radical equation n X D a is to first take n-th ower of both sides to get rid of the radical sign, that is X D a n and then solve the resulting equation. Note: In general, before alying the above idea, a radical eression has to be isolated first. Eamle 6.. Solve the equation C D. Ste. Arrange terms so that one radical is isolated on one side of the equation. D C Ste. Square both sides to eliminate the square root.. / D C Ste. Solve the resulting equation. C D C D 0. / D 0 D 0 or D 0 D 0 or D Ste. Check all roosed solutions. Plug D 0 into the original equation, we see that the left hand side is 0 0 C D 0 D 0 D which is not equal to the right hand side. So D 0 cannot be a solution. Plug D into the original equation, we see that the left hand side is C D D. So D is a solution. D Eamle 6.. Solve the equation 6 D. Ste. Isolated one radical. D 6 C Ste. Square both sides to remove radical sign and then isolate the remaining radical. D. 6/ C 6 C D 5 C 6 D 6 D 6: Ste. Square both sides to remove the radical sign and then solve. 6 D 6 D D 0: Since 0 > 0 and 0 6 > 0, D 0 is a valid solution. Indeed, D 9 D D : Eamle 6.. Solve the equation D 6. Ste. Isolated the radical. D Ste. Cube both sides to eliminate the cube root and then solve the resulting equation. D. / D 7 D The solution is D. A Concise Workbook for College Algebra

43 Lesson 6. Solve Radical Equations Practice 6.. Solve each radical equation. () C D () 5 D 0 () 5 C D C () D C 7 (5) 6 C 7 D (6) C C D (7) C 5 D (8) D 6 6. (5) D or D 6. (6) D 6. (7) D 6. (8) D 6. () D 5 6. () D 6. () D 0 or D 6. () D or D A Concise Workbook for College Algebra 5

44 Lesson 7. Comle Numbers Comle Numbers The imaginary unit i is defined as i D. Hence i D. If b is a ositive number, then b D i b. Let a and b are two real numbers. We define a comle number by the eression a C bi. The number a is called the real art and the number b is called the imaginary art. If b D 0, then the comle number a C bi D a is just the real number. If b 0, then we call the comle number a C bi an imaginary number. If a D 0 and b 0, then the comle number a C bi D bi is called a urely imaginary number. Adding, subtracting, multilying, dividing or simlifying comle numbers are similar to those for radical eressions. In articular, adding and subtracting become similar to combining like terms. Eamle 7.. Simlify and write your answer in the form a C bi, where a and b are real numbers and i is the imaginary unit. () C5i ().i /. C i/ () () i i (5) i 08 () D i i D i D : () () () (5).i /. C i/ D i. / C i i C. /. / C. / i C 5i i D 8i C. / C 6 C. i/ D i: D. C 5i/i i i D i 5 D D 5 C i: i C 5i i D. C i/. i/. C i/ D C i.i/ D C i 5 D 5 C 5 i: i 08 D i 50C D.i / 50 i D : i 6 A Concise Workbook for College Algebra

45 Lesson 7. Comle Numbers Practice 7.. Add, subtract, multily comle numbers and write your answer in the form a C bi. () () 8 ().5 i/ C. C i/ (). C 6i/. i/ (5). C i/. C 5i/ (6).7 i/. C 6i/ (7). /. C / (8). C i/ 7. () 6 7. () i 7. () 8 C i 7. () 0 C 0i 7. (5) 7 C 9i 7. (6) 9 C 8i 7. (7) C 9 7. (8) 5 C i Practice 7.. i () C i Divide the comle number and write your answer in the form a C bi. 5 i C i C 7i () () () C i i i 7. () C i 7. () 6i 7. () 0 C i 0 7. () 7 C i Practice 7.. Simlify the eression. (). i/ 8 () i 5 () i 07 () i 08 () () i () i () A Concise Workbook for College Algebra 7

46 Lesson 8. Comlete the Square The Square Root Proerty Suose u is an algebraic eression and d is a real number. If u D d, then u D d or u D d. Comlete the Square For a binomial C b, one can obtain a erfect square trinomial by adding b C b C D C b : This rocedure is called comleting the square. Solve by Comleting the Square Eamle 8.. Solve the equation C D 0. Ste. Isolate the constant. C D Ste. With b D, add to both sides to comlete a square for the binomial C b. C C D C C D C Ste.. C / D Solve the resulting equation using the square root roerty. C D or C D D C or D Note that the solution can also be written as D. Eamle 8.. Solve the equation C 8 9 D 0. Ste. Isolate the constant. C 8 D 9 Ste. Divide by to rewrite the equation in C B D C form D 9 b : Ste. With b D, add D to both sides to comlete the square for the binomial. C D 9 C. / D Ste. Solve the resulting equation and simlify. D i or D i The solutions are D i. D C i or D i 8 A Concise Workbook for College Algebra

47 Lesson 8. Comlete the Square Practice 8.. Solve the quadratic equation by the square root roerty. () D 0 () 6 D 0 (). / D 0 (). C / C 5 D 0 8. () D 5 8. () D 8. () D 0 8. () D 5i Practice 8.. Solve the quadratic equation by comleting the square. () 6 C 5 D 0 () C D 0 () 5 D 0 () C D 0 (5) C 8 C D 0 (6) C 6 D 0 8. () D i 8. () D 7 8. () 9 D 8. () 5 D 8. (5) D 6 or D 8. (6) D. A Concise Workbook for College Algebra 9

48 Lesson 9. Quadratic Formula The Quadratic Formula The solutions of a quadratic equation in the standard form a C b C c D 0 with a 0 are given by the quadratic formula D b b ac : a The quantity b ac is called the discriminant of the quadratic equation. If b ac > 0, the equation has two real solutions. If b ac D 0, the equation has one real solution. If b ac < 0, the equation has two imaginary solutions (no real solutions). Eamle 9.. Determine the tye and the number of solutions of the equation. /. C/ D. Ste. Rewrite the equation in the form a C b C c D 0. Ste. Find the values of a, b and c. Ste. Find the discriminant b ac. The equation has two imaginary solutions.. /. C / D C C D 0 a D ; b D and c D : b ac D D : Eamle 9.. Solve the equation C 7 D 0. Ste. Find the values of a, b and c. Ste. Find the discriminant b ac. a D ; b D and c D 7: b ac D. / 7 D 6 56 D 0: Ste. Aly the quadratic formula and simlify. D b b ac D. / 0 D 0i 0 D a i: Eamle 9.. Find the base and the height of a triangle whose base is three inches more than two times its height and has an area of 5 square inches. Round to the nearest tenth of an inch. Ste. We may suose the height is inches. The base can be eressed as C inches. Ste. Ste. By the area formula for a triangle, we have an equation.. C / D 5: Rewrite the equation in a C b C c D 0 form.. C / D 0 C 0 D 0: Ste. By the quadratic formula, we have D. 0/ D 89. Since can not be negative, D C 89 :6 and C 6:. The height and base of the triangle are aroimately :6 inches and 6. inches resectively. 0 A Concise Workbook for College Algebra

49 Lesson 9. Quadratic Formula Practice 9.. Determine the number and the tye of solutions of the given equation. () C 8 C D 0 () C D 0 () C D 0 9. () Two real solutions 9. () Two imaginary solutions 9. () One real solution Practice 9.. Solve using the quadratic formula. () C 7 D 0 () D C 5 () D 9. () D i 9. () D 9. () D 7 Practice 9.. A triangle whose area is 7:5 square meters has a base that is one meter less than trile the height. Find the length of its base and height. Round to the nearest hundredth of a meter. 9. The height is C 8 6 : meters and the base is C 8 6: meters. Practice 9.. A rectangular garden whose length is feet longer than its width has an area 66 square feet. Find the dimensions of the garden, rounded to the nearest hundredth of a foot. 9. The width is 67 7:9 feet and the length is 67 C 9:9 feet. A Concise Workbook for College Algebra

50 Lesson 0. Grahs of Quadratic Functions The Grah of a Quadratic Function The grah of a quadratic function f./ D a C b C c, a 0, is called a arabola. A quadratic function f./ D a C b C c can be written in the form f./ D a. h/ C k, where h D b b and k D f.h/ D f. a a Ste. The line D h D b is called the ais of symmetry of the arabola. Ste. a The oint.h; k/ D b a ; f b a is called the verte of the arabola. Minimum and Maimum of a Quadratic Function Consider the quadratic function f./ D a C b C c, a 0. If a > 0, then the arabola oens uward and f has a minimum f If a < 0, then the arabola oens downward and f has a maimum f b at the verte. a b at the verte. a Intercets of a Quadratic Function Consider the quadratic function f./ D a C b C c, a 0. The y-intercet is.0; f.0// D.0; c/. The -intercets, if eist, are the solutions of the equation a C b C c D 0. Eamle 0.. Does the function f./ D 6 have a maimum or minimum? Find it. Ste. Since a >, the function oens uward and has a minimum. Ste. Find the line of symmetry D b a : D. / D. Ste. Find the minimum by lugging D into the function f. The minimum is f. f./ D 6 D 8. b a / D Eamle 0.. Consider the function f./ D C C 6. Find values of such that f./ D. Ste. Set u the equation for. Ste. Solve the equation C C 6 D. We get D or D. C C 6 D The values of such that f./ D are and. Eamle 0.. A quadratic function f whose the verte is.; / has a y-intercet.0; /. Find the equation that defines the function. Ste. Write down the general form of f using only the verte. Quadratic functions with the verte at.; / are defined by y D a. real number. / C, where a is a nonzero Ste. Determine the unknown a using the remaining information. Since.0; / is on the grah of the function, the number a must satisfy the equation D a.0 / C. Ste. Solving for a from the equation, we get a D 5. The quadratic function f is given by f./ D 5. / C. A Concise Workbook for College Algebra

51 Practice 0.. Lesson 0. Grahs of Quadratic Functions For the given quadratic function, A. determine the coordinates of the -intercets, the coordinates of the y-intercet, the equation of the ais of symmetry and the coordinates of the verte. B. sketch the grah of the quadratic function using the information in art A. () f./ D. / C () f./ D C 5 y 5 y 5 Practice 0.. below. Consider the arabola in the grah Practice 0.. below. Consider the arabola in the grah 5 y 5 y () For what values of is y negative? Eress your answer in interval notation. () Find the domain of the function. () Find the range of the function. () Determine the coordinates of the -intercets. (5) Determine the coordinates of the y-intercet. (6) Determine the coordinates of the verte. (7) For what values of is f./ D. (8) Find the equation of the function () For what values of is y negative? Eress your answer in interval notation. () Find the domain of the function. () Find the range of the function. () Determine the coordinates of the -intercets. (5) Determine the coordinates of the y-intercet. (6) Determine the coordinates of the verte. (7) For what values of is f./ D. (8) Find the equation of the function. A Concise Workbook for College Algebra