STUDENT NAME CLASS DAYS/TIME MATH 10, COLLEGE ALGEBRA UNIT 3 LECTURE NOTES JILL TRIMBLE, BLACK HILLS STATE UNIVERSITY
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 Denominator 0 30 The Rational Epression is undefined for 1 Denominator 0 10 Comple numbers The Rational Epression is defined for all 56 Denominator 0 5 6 0 and The Rational Epression is undefined for, Math10 Unit3 Unit3 Lecture Note Guide Page 3 of 51
Simplify and find any domain eclusions 5 10 5( ) 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 10 5 ( 5)( 5) ( 5)( 5) 3 3 4 5 3( 1) ( 5)( 1) Denominator 0 ( 5) ( 5) 3 ( 1) ( 5) ( 1) ( 5) ( 5) simplified, Domain restrictions 4 1 ALREADY HAS 6 6 4 1 4 ( 3) 6 ( 3) ( ) Denominator 0 simplified and Math10 Unit3 Unit3 Lecture Note Guide Page 5 of 51
7 9 94 81 16... NEED A 7 9 4 9 9 4 9 4 (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
Solving Rational Equations An equation that is true for ALL values of is called an Identity 3 31 3 3 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
Solve 6 5 6 5 Graph 0 5 5 NO RESTRICTIONS Denominator 0 6 5 10 10() 10 5 ( 6) 0 5( 5) Math10 Unit3 Unit3 Lecture Note Guide Page 8 of 51
1 4 7 5 3 10 1 4 7 Graph : 0 5 3 10 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
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 4 1 4 4 9 1.. 3. 3 9 5 3 7 10 7 7 5 3 5 1. 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
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
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
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
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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
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
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
Find the inverse of the one to one function f ( ) 8 4 1) f( ) 1 8 4 1 ) 3) y 8y 4 y 1 4) f ( ) Math10 Unit3 Unit3 Lecture Note Guide Page 18 of 51
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
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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 3 5 3 1 3 Graph f( ) 3 (Growth) Graph 1 1 f( ) 3 3 3 (Decay) Domain : Range : Math10 Unit3 Unit3 Lecture Note Guide Page 1 of 51
Important Base e the natural number e Graph y e y e Transformations : Parent : y 3 y 3 1... Vertical y 3 5... Vertical y 3... reflection y 3... reflection y y 1 3... horizontal 4 3... horizontal what about y 1 or y 1 BOTH HORIZONTAL LINES Math10 Unit3 Unit3 Lecture Note Guide Page of 51
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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
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
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
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
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
4 log (401) 7 4 5 log ( m) 3 5 m log (16) y 4 y 3 8 51 log ( ) 3 1 15 3 5 log 5 5 1 5 5 log 1 1 log14 1 14 Math10 Unit3 Unit3 Lecture Note Guide Page 9 of 51
log 3 log 3 3 3 1 1 log 3 (3) 1 1 (1) log 3 3 1 1 log log 1 10 100 100 y y y Math10 Unit3 Unit3 Lecture Note Guide Page 30 of 51
1 ln log e y y y e y Math10 Unit3 Unit3 Lecture Note Guide Page 31 of 51
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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) 4 4 4 log (3) log () log (3) 4 4 4 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
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
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
e 0.5 ln e 5 0.5 ln(5) ln( e) ln(5) Note :ln( e) 1 ln(5) log ( 4) Graphlog ( 4) 0 3 3 Math10 Unit3 Unit3 Lecture Note Guide Page 36 of 51
log ( ) log ( 6) log log 64 6 4 0 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
3 9 ln(3 ) ln(9) ln(3) ln(9) ln(9) ln(3) 1 7 343 1 7 7 1 ln(7 ) ln 343 ( 1) ln(7) ln(343) Graph 1 7 343 0 Math10 Unit3 Unit3 Lecture Note Guide Page 38 of 51
10 3.8 log10 log(3.8) Graph10 3.8 0 log(10) log(3.8) 43 e 7 3108 e 43 ln e 43 ln(3115) ln e ln(3115) ln(3115) Math10 Unit3 Unit3 Lecture Note Guide Page 39 of 51
log ( 4) 5 log( 4) log 5( 4) 0 or 0 log(5) log ( 4) 5 4 5 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
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) 31.4 4.e 4.e 0.0765(0) 0.0765( t ) 0 31.4 0.0765( t) 4. ln e ln 4. 31.4 0.0765tln( e) ln 4. t rounded to the nearest year Graph 31.4 4.e 0.0765t 0.0765t 4.e 31.4 0 t Math10 Unit3 Unit3 Lecture Note Guide Page 41 of 51
.0675 5000 115001 ( t) ( t).0675 Graph115001 5000 0 By hand....0675 5000 115001 t t in eact form Math10 Unit3 Unit3 Lecture Note Guide Page 4 of 51
f (5) 8 43ln(5) f ( ) 8 43ln( ) Graph 8 4ln( ) 86 0 86 8 43ln( ) 78 43ln( ) 78 ln( ) 43 78 log e( ) 43 e 78 43 Math10 Unit3 Unit3 Lecture Note Guide Page 43 of 51
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 300 1000e 1000 1600e plug in k e 7k 1000 1600 k (7) e 7k 5 7k ln( e ) ln 8 ln( e) ln k Math10 Unit3 Unit3 Lecture Note Guide Page 44 of 51
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 4 6 3 y 1 ( ) 4(1) 6 3(1) ( ) 1 (1, ) is a solutiontothe system Math10 Unit3 Unit3 Lecture Note Guide Page 45 of 51
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
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
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
Solve the system y 1 3 3 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
Dist ance Rate( time) d rt Solve this system ( y) 10 ( y) p 360 l w 4550 35l10w l w 180 l w (180) 7l w 910 7l w 910 l w Math10 Unit3 Unit3 Lecture Note Guide Page 50 of 51
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