Section 9.1A Introduction to Rational Functions Work Together How many pounds of peanuts do you think and average person consumed last year? Us the table at the right. What was the average peanut consumption per person in the US in 1993? Year You can use the data in the table to model the average US consumption per person between 1970 and 1993. Let 0 represent 1970. Input the data into three lists on your calculator. Find a cubic function f () to model the annual consumption of peanuts. US Peanut US Consumption Population (millions of pounds) (millions) 1970 1118 03.3 1980 1087 6.5 1985 1499 37.9 1990 149 48.7 1991 1639 5.1 199 1581 55.0 1993 1547 57.8 Find a quadratic function g() to represent the US population since 1970. Write a function h() to model the average peanut consumption in the US since 1970. Graph the function on your calculator you wrote to estimate the number of peanuts that were consumed by the average person last year. Wrap Up A rational function occurs when one polynomial is divided by another polynomial. Eample: 3 1 f ( ) ( 3)( + 4) Algebra Chapter 9 Page 1
Section 9.1 Multiplying and Dividing Rational Epressions Goals: 1. To simplify rational epressions.. To determine when a rational function is undefined. 3. To simplify comple fractions. Eamples 1. Simplify and determine when the rational epression is undefined: 3y ( y + 7) ( + 5) a. b. y + 7 y 9 + 5 16 ( )( ) ( )( ). For what values of p is p + p 3 p p 15 undefined? 3. Simplify: 4 4 a b a 3 3 a a b 4. Multiply and Simplify: 8 7 y a. b. 3 3 1y 16 5a c 4bc 1b 15a b 4 3 c. 3 5y d. 3 15y 6 y 10 y 5y 3yz 3 Algebra Chapter 9 Page
5. Divide and Simplify: 10 ps 5ps a. b. 3c d 6c d 3 y 6y 0ab 5a b 3 6. Simplify: k 3 k 4k + 3 a. k + 1 1 k d + 6 d + 3 d + d d + 3d + b. c. + + + 3 5 6 9 3d + 9 d + d + 4d + 3 d + 5d + 4 d. 7. Simplify the comple fraction: 9 4y a. 3 y 3 b. a a 9b 4 a a + 3b 1 + 5/ + 1 Simplifying Error: 15 3 3 Note: You cannot simplify through addition or subtraction. Homework: p. 557 1-11 all, 13, 17, 5-37, 41 Algebra Chapter 9 Page 3
Section 9. Adding and Subtracting Rational Epressions Goals: 1. To find the least common denominator.. To add and subtract rational epressions. A. Find the Least Common Multiple (or LCD) Eamples 3 5 3 6 1. 15a bc, 16b c and 0a c. 3 and 4 + 4 3. 3 6 zy, 3 9 y z, and 4 z 4. 3 3 + and + 6 + 9 B. Adding and Subtracting Rational and Comple Epressions Eamples 5a 9 3 5 1. +. + 6b 14a b y 1y 3. + 10 3 + 15 3 15 6 30 4. + 5 3 + 8 4 4 8 5. 1 1 + a b 1 1 b 6. 1 1 a b 1 1 + b a Homework p. 565 1-17 all, 60, 71-73 Algebra Chapter 9 Page 4
Section 9.4 Graphing Rational Functions Goal: 1. To graph rational functions.. To locate asymptotes of a function. I. Definition of Rational Functions A. A function occurs when one is by another. 3 1 B. Eample: f ( ) ( 3)( + 4) II. Asymptotes A. The lines a rational functions. B. The asymptote occurs when the equals. Note One eception in the. C. The asymptote occurs when and the y value a certain. D. Eamples: 1. Determine the equations of any vertical asymptotes and the values of for any holes in the graph of: 4 9 a) f ( ) b) f ( ) + 5 + 6 + 8 + 15. Graph and name the asymptotes or any holes in the graphs: a) f ( ) b) f ( ) + 1 + 3 4 4 4 4 4 4 4 4 c) f ( ) 4 d) f ( ) 16 + 4 4 4 4 4 4 4 4 4 Homework: Worksheet Algebra Chapter 9 Page 5
Algebra Worksheet Section 9.4 Name: Algebra Chapter 9 Page 6
Algebra Chapter 9 Page 7
Section 9.5 Direct, Inverse and Joint Variation Goals: 1. To solve problems involving direct, inverse, and joint variations. A. Direct Variation 1. As ( ), y ( ).. Wording 3. General Equation: y k where k is the constant of variation. 4. This is similar to the equation y m where m is the slope. 5. Eample: a) If y varies directly as and y 15 when 5, find y when 3. Method to solve variation problems: 1.. 3. 4. b) If y varies directly as and y 1 when 3, find y when 7. B. Joint Variation 1. Wording. General Equation: y kz where k is the constant of variation. 3. Eample: a) The volume of a cone varies jointly as the square of the radius of the base and the height. Find the equation of joint variation if V 85, r 4, and h 17. b) Suppose y varies jointly as and z. Find y when 3 and z, if y 11 when 5 and z. Algebra Chapter 9 Page 8
C. Inverse Variation 1. As ( ), y ( ). Wording k 3. General Equation: y where k is the constant of variation. 4. Eamples a) If y varies inversely as and y when 6, find y when 7. b) If temperature is constant, gas volume varies inversely with its pressure. If an airfilled balloon has a volume of.6 dm when pressure is 10 kpascals, what is the 3 3 pressure if the volume is 3. dm? D. Combined Variations The force is directly related to the product of the masses of two objects and inversely related to the square of the distance between them. Homework p. 590 1-3 all, 5, 6, 11, 16, 1,, 44, 47, 57-59 all Algebra Chapter 9 Page 9
Section 9.6 Solving Rational Equations and Inequalities Goals: 1. To solve rational equations and inequalities Eamples: 5 1 1. + 4 3 4. 5 + 3 1 1 3. p p + 1 p 7 + p p + 1 p 1 p + 1 ( )( ) 4. + 4 + 8 5. Tim and Ashley mow lawns together. Tim working alone could complete the job in 4.5 hours, and Ashley could complete it alone in 3.7 hours. How long does it take to complete the job when they work together? Note: Work Rate How much of the can be in a given of. W + W + W W 1 3 Total Algebra Chapter 9 Page 10
6. Libby and Nate clean together. Nate working alone could complete the job in 3 hours, and Libby could complete it alone in 5 hours. How long does it take to complete the job when they work together? 7. A car travels 300 km in the same time that a freight train travels 00 km. The speed of the car is 0 km/h faster than the train. Find the speed of the train. Note: Distance 8. Solve: 1 3s + 9s < 3 9. Solve: 1 + 3 < 5 5 Homework Day 1: p. 600 1-8 all, 17-1 odds, 47, 51 Day : p. 600 10-15 all, 4-7 all, 31, 41, 4 Algebra Chapter 9 Page 11