STUDENT NAME CLASS DAYS/TIME MATH 102, COLLEGE ALGEBRA UNIT 3 LECTURE NOTES JILL TRIMBLE, BLACK HILLS STATE UNIVERSITY
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1 STUDENT NAME CLASS DAYS/TIME MATH 10, COLLEGE ALGEBRA UNIT 3 LECTURE NOTES JILL TRIMBLE, BLACK HILLS STATE UNIVERSITY
2 Math10 College Algebra Unit 3 Outcome/Homework 1 Students will be able to add, subtract, multiply, divide, and simplify rational epressions and solve rational equations on eaminations, quizzes, and homework problems. (E-tet Section P.6, 1.) Rational Epression Numerator, Denominator 0 Denominator Rational Epressions Common Denominator Rational Epressions No common denominator needed Multiply straight across a c ac b d bd Rational Epressions Multiply by the Reciprocal of the nd Fraction a c a d ad b d b c bc Math10 Unit3 Unit3 Lecture Note Guide Page of 51
3 3 Denominator 0 30 The Rational Epression is undefined for 1 Denominator 0 10 Comple numbers The Rational Epression is defined for all 56 Denominator and The Rational Epression is undefined for, Math10 Unit3 Unit3 Lecture Note Guide Page 3 of 51
4 Simplify and find any domain eclusions ( ) 44 ( )( ) Denominator 0 5 Simplified ( )( ) 0 restriction Multiply, simplify, domain restrictions (Determined Simplifying) 9 3 ( 3)( 3) ( 3) 6 7 ( )( 9)( 3) Denominator 0 ( )( 9)( 3) 0 restrictions :,, ( 3)( 3) ( 3) ( )( 9) ( 3) Math10 Unit3 Unit3 Lecture Note Guide Page 4 of 51
5 ( 5)( 5) ( 5)( 5) ( 1) ( 5)( 1) Denominator 0 ( 5) ( 5) 3 ( 1) ( 5) ( 1) ( 5) ( 5) simplified, Domain restrictions 4 1 ALREADY HAS ( 3) 6 ( 3) ( ) Denominator 0 simplified and Math10 Unit3 Unit3 Lecture Note Guide Page 5 of 51
6 NEED A (9 4)(9 4) 7(94) 9 (94)(94) 7 8 (94)(94) Denominator 0 (9 4)(9 4) 0 simplified and restrictions Math10 Unit3 Unit3 Lecture Note Guide Page 6 of 51
7 Solving Rational Equations An equation that is true for ALL values of is called an Identity IDENTITY True Statement An equation that is NOT TRUE for ANY value of is called Inconsistent 3 03 INCONSISTENT False Statement An equation that is only true for CERTAIN values of is called a Conditional Equation 34 4 CONDITIONAL EQUATION Math10 Unit3 Unit3 Lecture Note Guide Page 7 of 51
8 Solve Graph NO RESTRICTIONS Denominator () 10 5 ( 6) 0 5( 5) Math10 Unit3 Unit3 Lecture Note Guide Page 8 of 51
9 Graph : Restrictions: 5, ( 5) ( ) 1 ( 5) 4 ( ) ( 5) 5 ( ) 7 ( 5) ( ) 5 4( ) 7... Potential Solution restrictions : 5, Math10 Unit3 Unit3 Lecture Note Guide Page 9 of 51
10 Math10 College Algebra Unit 3 Outcome/Homework Students will be able to identify vertical and horizontal asymptotes and graph rational functions. (E-tet Section 3.5) Vertical Asymptote Horizontal Asymptote Rules for Finding Horizontal Asymptotes N D No horizontal asymptote. N D horizontal asymptote is y 0 3. N D horizontal asymptote is the ratio of n the leading coefficients y d Math10 Unit3 Unit3 Lecture Note Guide Page 10 of 51
11 Vertical Asymptote Where does the denominator = 0? X= 6 6 Domain : h( ) 36 ( 6)( 6) Denom inator 0 ( 6)( 6) 0, Vertical Asymptotes, Horizontal Asymptotes N 1 N D D y 0 DomainVertical and Horizontal Asymptotes 16 r( ) 5 Denom inator 0 50 Horizontal Asymptote N N D y D comple number no restrictions no vertical asymptote Math10 Unit3 Unit3 Lecture Note Guide Page 11 of 51
12 Domain horizontal and vertical holes v( ) ( 8) Denom inator 0 ( 8) 0, restrictions Vertical asymptote 1 ( 8) 8 hole at vertical asymptote at Horizontal Asymptote 1 N ( 8) 8 D N D y Math10 Unit3 Unit3 Lecture Note Guide Page 1 of 51
13 Parent Graph y 1 1 y 1 1 Graph g( ) 7 Parent Graph : y 6 TwoTransformations 7 6 Math10 Unit3 Unit3 Lecture Note Guide Page 13 of 51
14 Math10 Unit3 Unit3 Lecture Note Guide Page 14 of 51
15 Math10 College Algebra Unit 3 Outcome/Homework 3 Students will be able to find inverse functions on eaminations, quizzes, and homework problems. (E-Tet section.7) Testing for a Function Vertical Line Test Testing for an Inverse Horizontal Line Test Function and its inverse f ( ) function 1 f notation that means the inverse of the function f ( ) " ( )" if a function has aninverseit is called one to one for f and f 1 ( ) ( ), the domain and range swap the domain of f becomes the range of f 1 ( ) ( ) 1 the range of f ( ) becomes the domain of f ( ) 1 f( ) ( 3, 1),(,7),(3,6),(4,9) 1 1 Domain of f ( ) 3,,3,4 Range of f ( ) 1,7,6,9 f ( ) ( 1, 3),(7,),(6,3),(9,4) Domain of f Range of f ( ) 1,7,6,9 ( ) 3,,3,4 Math10 Unit3 Unit3 Lecture Note Guide Page 15 of 51
16 The graph of a function and its inverse the graphs reflect across the y= line. f ( ) 1 (0, 1),(,3),(3,5),( 1, 3) 1 1 ( 1,0),(3,),(5,3),( 3, 1) 1 f ( ) How to verify functions are inverses of each other f f g 1 ( ) ( ) ( ) ( f g)( ) f g( ) ( g f )( ) g f ( ) How to find an inverse 1) Replace f() with ) Swap the and 3) Solve for 4) Replace with Math10 Unit3 Unit3 Lecture Note Guide Page 16 of 51
17 Determ ine if f ( ) 4 and g( ) areinverses of each other 4 f g( ) f 4 g f g ( ) 4 and areinverses of each other the function f ( ) 7 is one to one. find the inverse 1) f ( ) 7 ) 3) 4) Math10 Unit3 Unit3 Lecture Note Guide Page 17 of 51
18 Find the inverse of the one to one function f ( ) 8 4 1) f( ) ) 3) y 8y 4 y 1 4) f ( ) Math10 Unit3 Unit3 Lecture Note Guide Page 18 of 51
19 Function f a find f 1 ) ( ) 1) 3 ( ) 8 ) 3) 4) b Domain of f range of f 1 ) ( ) ( ) Range of f domain of f 1 ( ) ( ) Math10 Unit3 Unit3 Lecture Note Guide Page 19 of 51
20 Math10 Unit3 Unit3 Lecture Note Guide Page 0 of 51
21 Math10 College Algebra Unit 3 Outcome/Homework 4 Students will be able to know the basic properties of eponential functions and graph eponential functions on eaminations, quizzes, and homework problems. (E-tet Section 4.1) Polynomial Function: Eponential Function: 3 4 or 3 or Graph f( ) 3 (Growth) Graph 1 1 f( ) (Decay) Domain : Range : Math10 Unit3 Unit3 Lecture Note Guide Page 1 of 51
22 Important Base e the natural number e Graph y e y e Transformations : Parent : y 3 y Vertical y Vertical y 3... reflection y 3... reflection y y horizontal horizontal what about y 1 or y 1 BOTH HORIZONTAL LINES Math10 Unit3 Unit3 Lecture Note Guide Page of 51
23 Math10 Unit3 Unit3 Lecture Note Guide Page 3 of 51
24 A P1 r n nt A (Accumulated Amount) = balance in the account when P (principal) dollars are invested at an annual interest r for t years where n is the number of compounding periods. Daily.n= Semiannually..n= Quarterly n= Continuous. Switch to a different formula! Continuously Compounding Interest A Pe rt annually..n= monthly..n= b. Compounded Quarterly (n=4) d. Compounded Continuously Math10 Unit3 Unit3 Lecture Note Guide Page 4 of 51
25 8% compounded daily ( n ) r A P1 n A nt 7.88% compounded continuously A Per A t The 7.88% account, compounded continuously yields the amount. Math10 Unit3 Unit3 Lecture Note Guide Page 5 of 51
26 Math10 College Algebra Unit 3 Outcome/Homework 5 Students will be know the basic properties of logarithmic functions and graph logarithmic functions on eaminations, quizzes, and homework problems. (E-tet Section 4.) Logarithmic Functions y y log ( ) b b log ( ) y Log Form b y b Eponential y log ( ) log( ) Common Log 10 y log ( ) the natural log e y ln( ) Log Graph You cannot take the log of a negative number! Domain: Range: Math10 Unit3 Unit3 Lecture Note Guide Page 6 of 51
27 Properties of Logs log ( ) 1 b? b b b? log b(1) 0 b 1 log ( ) b? b b b log ( b ) b 1 log ( b) b b log b ( ) Log Conversion log ( a) b log( a) ln( a) log( b) ln( b) log (81) 9 log (81) 9 Math10 Unit3 Unit3 Lecture Note Guide Page 7 of 51
28 Transformations g( ) 9 log ( ) Vertical shift 5 g( ) log ( 9) 7 " 7" Vertical shift " 9" Horizontal shift g( ) ln( ) " 1" reflection across Graph 3ln( ) g( ) log ( 5) 5 Math10 Unit3 Unit3 Lecture Note Guide Page 8 of 51
29 4 log (401) log ( m) 3 5 m log (16) y 4 y log ( ) log log 1 1 log Math10 Unit3 Unit3 Lecture Note Guide Page 9 of 51
30 log 3 log log 3 (3) 1 1 (1) log log log y y y Math10 Unit3 Unit3 Lecture Note Guide Page 30 of 51
31 1 ln log e y y y e y Math10 Unit3 Unit3 Lecture Note Guide Page 31 of 51
32 Math10 Unit3 Unit3 Lecture Note Guide Page 3 of 51
33 Math10 College Algebra Unit 3 Outcome/Homework 6 Students will be able to solve eponential and logarithmic equations using the rules of logarithms and eponents on eaminations, quizzes, and homework problems. (E-tet Section 4.3, 4.4) Properties of Logs log ( MN) log ( M ) log ( N) b b b log (3) log () log (3) log (3) log () log (3) us ing log conversion log(3) log() log(3) log(4) log(4) log(4) M logb log b( M) log b( N) N p log M p log ( M ) b b Math10 Unit3 Unit3 Lecture Note Guide Page 33 of 51
34 Change of Base log ( M ) b log ( M) log ( M) log( M) ln( M) log ( b) log ( b) log( b) ln( b) a 10 a 10 log (4 ) log (4) log ( ) log ( ) 3 logt log t( ) 7 log3 log log y log log( ) log( ) 64 log4 log log 8 3 log log ( ) log log ( ) Math10 Unit3 Unit3 Lecture Note Guide Page 34 of 51
35 log(41) log( ) log 1 ln( ) ln( y) 7 ln( ) ln( y) ln log (43) log log b M b N M N ln( b ) ln( b ) ln( b) ln( b) Solve for ln(9) ln(8) Math10 Unit3 Unit3 Lecture Note Guide Page 35 of 51
36 e 0.5 ln e ln(5) ln( e) ln(5) Note :ln( e) 1 ln(5) log ( 4) Graphlog ( 4) Math10 Unit3 Unit3 Lecture Note Guide Page 36 of 51
37 log ( ) log ( 6) log log a b c b b 4ac a, NOT A SOLUTION ( Cannot take the log of a negative number) log ( ) log ( 6) log ( ) log ( 6) 0 log( ) log( 6) 0 log() log() Math10 Unit3 Unit3 Lecture Note Guide Page 37 of 51
38 3 9 ln(3 ) ln(9) ln(3) ln(9) ln(9) ln(3) ln(7 ) ln 343 ( 1) ln(7) ln(343) Graph Math10 Unit3 Unit3 Lecture Note Guide Page 38 of 51
39 log10 log(3.8) Graph log(10) log(3.8) 43 e e 43 ln e 43 ln(3115) ln e ln(3115) ln(3115) Math10 Unit3 Unit3 Lecture Note Guide Page 39 of 51
40 log ( 4) 5 log( 4) log 5( 4) 0 or 0 log(5) log ( 4) e 9ln( ) 45 9ln 45 5 ln( ) 9ln e 45 ln ( ) (9)ln( e) 45 e Math10 Unit3 Unit3 Lecture Note Guide Page 40 of 51
41 Math10 College Algebra Unit 3 Outcome/Homework 7 Students will be able to solve application problems involving eponential and logarithmic functions using the rules of logarithms and eponents on eaminations, quizzes, and homework problems. (Section 4.4) a) A 4. e b) e 4.e (0) ( t ) ( t) 4. ln e ln tln( e) ln 4. t rounded to the nearest year Graph e t t 4.e t Math10 Unit3 Unit3 Lecture Note Guide Page 41 of 51
42 ( t) ( t).0675 Graph By hand t t in eact form Math10 Unit3 Unit3 Lecture Note Guide Page 4 of 51
43 f (5) 8 43ln(5) f ( ) 8 43ln( ) Graph 8 4ln( ) ln( ) 78 43ln( ) 78 ln( ) log e( ) 43 e Math10 Unit3 Unit3 Lecture Note Guide Page 43 of 51
44 Eponential Growth/Decay A A e 0 kt A = Amount Remaining A 0 = Initial Amount K = growth rate/ decay rate k>0 = growth and k<0 = decay t= time in years t=0 our amount is A 0 t=any other time out amount is A kt A A0e find k kt e e plug in k e 7k k (7) e 7k 5 7k ln( e ) ln 8 ln( e) ln k Math10 Unit3 Unit3 Lecture Note Guide Page 44 of 51
45 Math10 College Algebra Unit 3 Outcome/Homework 8 Students will be able to solve a system of two linear equations on eaminations, quizzes, and homework problems. (Section 5.1) Graph No solution S ingle solution Infinite solutions Inconsisten system consistent system dependent system How to solve Linear Systems 1) Graphing ) Substitution 3) Elimination Determine if (1,-) is a solution to the system y 46 3 y 1 y y 1 ( ) 4(1) 6 3(1) ( ) 1 (1, ) is a solutiontothe system Math10 Unit3 Unit3 Lecture Note Guide Page 45 of 51
46 Determine if (,-7) is a solution to the system 5 y 4 3y 7 5 y 4 3y 7 Solve the system y 9 y 5 Substitution substitute y 5 in y 9 (, 7)is a solutiontothe system 9 find the y value by plugging in 1 in either equation y 5 y 5 y, is a solution to the system Math10 Unit3 Unit3 Lecture Note Guide Page 46 of 51
47 El im ination solve the system 5y 1 43y 1 y ( ) (5 ) (1) 4 3y 1 4 3y 1 y, is the solution to the system Check! Graphing solve both equations for Solve the system y 6 y y 4 1 y 4 1 Math10 Unit3 Unit3 Lecture Note Guide Page 47 of 51
48 Solve the system 5 y 3 5 y 3 5 y 7 5 y 7 El im ination / Addition Method (5 ) ( y) ( 3) 5 y 7 5 y 7 false statement parallel lines Solve the system by substitution 6y5 1y 10 plug 6y 5 in 1y 10 1y 10 true statement, number of solutions Math10 Unit3 Unit3 Lecture Note Guide Page 48 of 51
49 Solve the system y clear the fraction by y 3 y 9 y 9 ( y) ( 3) y 9 y 9 Find y by substituting in either equation, Math10 Unit3 Unit3 Lecture Note Guide Page 49 of 51
50 Dist ance Rate( time) d rt Solve this system ( y) 10 ( y) p 360 l w l10w l w 180 l w (180) 7l w 910 7l w 910 l w Math10 Unit3 Unit3 Lecture Note Guide Page 50 of 51
51 Math10 Unit3 Unit3 Lecture Note Guide Page 51 of 51
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