Chapter 9 Rational Functions
Lesson 9-4 Rational Epressions
Rational Epression A rational epression is in simplest form when its numerator and denominator are polnomials that have no common divisors.
Eample 1 Page 501, #4 Simplif the rational epression. State an restriction on the variable. z z 49 7 z 7z 7 z 7 z 7; z 7
Eample 1 Page 50, #6 Simplif the rational epression. State an restriction on the variable. 816 4 4 4 4 6 4 ; 6 or 4 6
Eample 3 Page 501, #8 Multipl. State an restrictions on the variables. 5 10 4 where 0 and 0 4 4 3 3
Eample 3 Page 501, #10 Multipl. State an restrictions on the variables. 1 6 3 9 3 8 6 3 3 3 3 8 4 6 8 ; 3 or 3 3 8 3
Eample 3 Page 501, #1 Multipl. State an restrictions on the variables. 5 6 3 4 3 ( 3)( ) ( 1)( ) ( )( ) ( 3)( 1) 1,, 3 or 1
Eample 4 Page 501, #14 Divide. State an restrictions on the variables. 3 5 3 6 5 5 3 3 3 3 5 5 6 5 0 and 0
Eample 4 Page 501, #16 Divide. State an restrictions on the variables. 3 1 6 4 4 4 8 3 1 4 8 4 6 4 3( 4) 4( ) ( ) 6( 4) 1 and 4
Eample 4 Page 50, #31 Multipl or Divide. State an restrictions on the variable. 1 5 5 3 5 53 1 1 1 3 1 4 3 1 4 1,, 4 or 3
Lesson 9-5, Part 1 Adding and Subtracting Rational Epressions
Eample 3 Page 507, #8 Simplif the sum. 1 1 1
Eample 3 Page 507, #1 Simplif the sum. 5 4 4 5 4 4 5 4 4 16 8 4 4 8 4 0 8 4 4 4 8 4
Eample 3 Page 507, #14 Simplif the sum. 3 4 3 4 3 3 3 3 3 3 4 9 6 3 ) 4 33 ( 3 6 41 ( 3)( 3) 1 ( 3)( 3) ( 6) ( 3)( 3) 6 ( 3)( 3)
Lesson 9-5, Part Adding and Subtracting Rational Epressions
Eample 4 Page 508, #16 Simplif the difference. 1 3
Eample 4 Page 508, #18 Simplif the difference. 3 4 3 3 3 1 6
Eample 4 Page 508, #0 Simplif the difference. 3 8 5 5 3 8 5 5 5 3 ) 8 3 8 40 3 8 40 5 5 5 5 5 5 1 ( 5 3 8 5 5 5 40 5 5 5( 8) 5 5
Eample 4 Page 508, #40 Add or Subtract. Simplif where possible. 5 4 9 7 14 5 4 9 7 ( 7) 5 9 7 7 5 ( 7) 9 7 5 9 7 5 9 63 7
Eample 4 Page 508, #40 5 9 63 7 1 63 7 3 4 1 7
Lesson 9-5, Part 3 Adding and Subtracting Rational Epressions
Comple Fractions A comple fraction is a fraction that has a fraction in its numerator or denominator or in both. Simplifing a comple fraction: a b c d a c a d b d b c ad bc
Eample 5 Page 508, # Simplif the comple fractions. 1 1
Eample 5 Page 508, #6 Simplif the comple fractions. 1 1 1 1 1 1 1 1 1
Eample 5 Page 508, #8 Simplif the comple fractions. 5 5 5
Eample 5 Page 508, #44 Simplif the comple fractions. 3 3 3 3 5 7 5 7 5 7 5 7 3 3 3 5 7 5 7 7 5
Eample 5 Page 508, #48 Simplif the comple fractions. 4 3 1 4 4 1 3 4 4 1 3 4 1 4 4 1 8 4 4 3 4 10 4 7 4 4 4 10 7 ( 5) 7
Lesson 9-6 Solving Rational Equations
Eample 1 Page 514, # Solve the equation. Check each solution. 1 1 5 9 5 9 590 4 0 0 Check for etraneous solutions 1 1 5(0) 9(0) No Solution
Eample 1 Page 514, #4 Solve the equation. Check each solution. 4 1 3 6 1 4 6 3 4 5 0 0 5 0 5 310 0
Eample Page 515, #1 Solve the equation. Check the solution. 6 1 5 3 6 1 5 3 6 1 5 3 6 6 61 6 5 3 3 5 4 3 5 8 4 6
Eample Page 516, #50 Solve the equation. Check the solution. 10 7 8 8 16 8 8 10 7 8 8 4 4 4 4 5 7 8 4 4 4 4 4
Eample Page 516, #50 4 4 5 7 8 4 4 4 4 4 4 4 5 4 4 7 8 4 4 4 4 4 4 4 5 4 7 8 4 4 5 0 7 8 4 16 8 4 16 1 6
Eample 3/4 Page 515, # Carlos can travel 40 mi on his motorbike in the same time it takes Paul to travel 15 mi on his biccle. If Paul rides his bike 0 mi/hr slower than Carlos rides his motorbike, find the speed for each bike. d r t d r t Carlos Paul 40 15 0 = = t d r 40 15 0
Eample 3/4 Page 515, # 40 15 0 15 40( 0) 15 40 800 5 800 5 800 5 5 3 Carlos: 3 mi/hr Paul: 3 0 = 1 mi/hr
Eample 3/4 Page 515, #4 Shelle can paint a fence in 8 hours. Karen can do it in 4 hours. How long will it take them to do the job if the work together? Shelle s Rate + Karen s Rate = 1 d r t Time Rate r d t Shelle Karen 8 4 8 4 8 4 1
Eample 3/4 Page 515, #4 8 4 1 8 8 4 1 8 3 8 3 8 3 3 3 It will take them both /3 hrs to complete the job.
Lesson 9-3, Part 1 Rational Functions and Their Graphs
Rational Function A rational function f() is a function that can be written as f( ) P ( ) Q ( ) Where P() and Q() are polnomial functions and Q() 0
Eample 1 Sketch the graph of the rational function 1 Step 1 Find the -intercepts (if there is an). Set the numerator equal to zero and solve 0 0
Eample 1 Step Find the -intercept (if there is one). Let equal 0. (0) 1 0 1 0
Eample 1 Step 3 Find an vertical asmptotes 1 Set the denominator equal to zero and solve 10 1
Eample 1 Step 4 Find the horizontal asmptotes 1 Identif the degree of the leading coefficient (LC) of the numerator (n) and denominator (d). numerator: n 1 denominator: d 1 1 Degree n n n Horizontal Asmptote d 0 d d None LCN LCD
Eample 1
Eample 1 Step 5 Plot points between and beond each -intercept and vertical asmptote. 1 1.333 4 5.5
Eample Sketch the graph of the rational function 3 1 Step 1 Find the -intercepts (if there is an). Set the numerator equal to zero and solve 3 0 0
Eample Step Find the -intercept (if there is one). Let equal 0. 3 3(0) 1 0 1 0
Eample Step 3 Find an vertical asmptotes 3 1 Set the denominator equal to zero and solve 10 ( 1)( 1) 0 1
Eample Step 4 Find the horizontal asmptotes 3 1 Identif the degree of the leading coefficient (LC) of the numerator (n) and denominator (d). numerator: n 1 denominator: d 0 Degree n n Horizontal Asmptote d 0 d n d None LCN LCD
Eample
Eample Step 5 Plot points between and beond each -intercept and vertical asmptote. 3 1.5.5
Graphing, Part Rational Functions and Their Graphs
Eample 3 Sketch the graph of the rational function 3 4 Step 1 Find the -intercepts (if there is an). Set the numerator equal to zero and solve 3 0 0 0
Eample 3 Step Find the -intercept (if there is one). Let equal 0. 3 3(0) 4 0 4 0
Eample 3 Step 3 Find an vertical asmptotes Set the denominator equal to zero and solve 3 4 40 ( )( ) 0
Eample 3 3 Step 4 Find the horizontal asmptotes Identif the degree of the leading coefficient (LC) of the numerator (n) and denominator (d). 4 numerator: n denominator: d 3 1 3 Degree n n n Horizontal Asmptote d 0 d d None LCN LCD
Eample 3
Eample 3 Step 5 Plot points between and beond each - intercept and vertical asmptote. 3 4 4 4 1 1 1 1 4 4
Eample 4 Sketch the graph of the rational function 9 3 Step 1 Find the -intercepts (if there is an). Set the numerator equal to zero and solve 90 ( 3)( 3) 3
Eample 4 Step Find the -intercept (if there is one). 9 3 Let equal 0. 9 (0) 9 9 3 0 3 3 3
Eample 4 Step 3 Find an vertical asmptotes Set the denominator equal to zero and solve 9 3 30 3
Eample 4 Step 4 Find the horizontal asmptotes 9 3 Identif the degree of the leading coefficient (LC) of the numerator (n) and denominator (d). numerator: n denominator: d 1 Degree n n Horizontal Asmptote d 0 d LCN LCD n d None
Eample 4
Eample 4 Step 5 Plot points between and beond each - intercept and vertical asmptote. 9 3 3 ERROR 4 7