Chapter 9. Rational Functions
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1 Chapter 9 Rational Functions
2 Lesson 9-4 Rational Epressions
3 Rational Epression A rational epression is in simplest form when its numerator and denominator are polnomials that have no common divisors.
4 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
5 Eample 1 Page 50, #6 Simplif the rational epression. State an restriction on the variable ; 6 or 4 6
6 Eample 3 Page 501, #8 Multipl. State an restrictions on the variables where 0 and
7 Eample 3 Page 501, #10 Multipl. State an restrictions on the variables ; 3 or
8 Eample 3 Page 501, #1 Multipl. State an restrictions on the variables ( 3)( ) ( 1)( ) ( )( ) ( 3)( 1) 1,, 3 or 1
9 Eample 4 Page 501, #14 Divide. State an restrictions on the variables and 0
10 Eample 4 Page 501, #16 Divide. State an restrictions on the variables ( 4) 4( ) ( ) 6( 4) 1 and 4
11 Eample 4 Page 50, #31 Multipl or Divide. State an restrictions on the variable ,, 4 or 3
12 Lesson 9-5, Part 1 Adding and Subtracting Rational Epressions
13 Eample 3 Page 507, #8 Simplif the sum
14 Eample 3 Page 507, #1 Simplif the sum
15 Eample 3 Page 507, #14 Simplif the sum ) 4 33 ( ( 3)( 3) 1 ( 3)( 3) ( 6) ( 3)( 3) 6 ( 3)( 3)
16 Lesson 9-5, Part Adding and Subtracting Rational Epressions
17 Eample 4 Page 508, #16 Simplif the difference. 1 3
18 Eample 4 Page 508, #18 Simplif the difference
19 Eample 4 Page 508, #0 Simplif the difference ) ( ( 8) 5 5
20 Eample 4 Page 508, #40 Add or Subtract. Simplif where possible ( 7) ( 7)
21 Eample 4 Page 508, #
22 Lesson 9-5, Part 3 Adding and Subtracting Rational Epressions
23 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
24 Eample 5 Page 508, # Simplif the comple fractions. 1 1
25 Eample 5 Page 508, #6 Simplif the comple fractions
26 Eample 5 Page 508, #8 Simplif the comple fractions
27 Eample 5 Page 508, #44 Simplif the comple fractions
28 Eample 5 Page 508, #48 Simplif the comple fractions ( 5) 7
29 Lesson 9-6 Solving Rational Equations
30 Eample 1 Page 514, # Solve the equation. Check each solution Check for etraneous solutions 1 1 5(0) 9(0) No Solution
31 Eample 1 Page 514, #4 Solve the equation. Check each solution
32 Eample Page 515, #1 Solve the equation. Check the solution
33 Eample Page 516, #50 Solve the equation. Check the solution
34 Eample Page 516, #
35 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 = = t d r
36 Eample 3/4 Page 515, # ( 0) Carlos: 3 mi/hr Paul: 3 0 = 1 mi/hr
37 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
38 Eample 3/4 Page 515, # It will take them both /3 hrs to complete the job.
39 Lesson 9-3, Part 1 Rational Functions and Their Graphs
40 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
41 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
42 Eample 1 Step Find the -intercept (if there is one). Let equal 0. (0)
43 Eample 1 Step 3 Find an vertical asmptotes 1 Set the denominator equal to zero and solve 10 1
44 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
45 Eample 1
46 Eample 1 Step 5 Plot points between and beond each -intercept and vertical asmptote
47 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
48 Eample Step Find the -intercept (if there is one). Let equal (0)
49 Eample Step 3 Find an vertical asmptotes 3 1 Set the denominator equal to zero and solve 10 ( 1)( 1) 0 1
50 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
51 Eample
52 Eample Step 5 Plot points between and beond each -intercept and vertical asmptote
53 Graphing, Part Rational Functions and Their Graphs
54 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
55 Eample 3 Step Find the -intercept (if there is one). Let equal (0)
56 Eample 3 Step 3 Find an vertical asmptotes Set the denominator equal to zero and solve ( )( ) 0
57 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 Degree n n n Horizontal Asmptote d 0 d d None LCN LCD
58 Eample 3
59 Eample 3 Step 5 Plot points between and beond each - intercept and vertical asmptote
60 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
61 Eample 4 Step Find the -intercept (if there is one). 9 3 Let equal 0. 9 (0)
62 Eample 4 Step 3 Find an vertical asmptotes Set the denominator equal to zero and solve
63 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
64 Eample 4
65 Eample 4 Step 5 Plot points between and beond each - intercept and vertical asmptote ERROR 4 7
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