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 199 37.9 1990 19 8.7 1991 1639 5.1 199 1581 55.0 1993 157 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)( + ) Honors 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: a b a 3 3 a a b. Multiply or Divide and Simplify: 8 7 y a. b. 3 3 1y 16 5 10mk 5m 3c d 6c d c. 3 5y d. 3 15y 3 y 6y 0ab 5a b 3 Honors Algebra Chapter 9 Page
5. Simplify: k 3 1 k a. k + 1 k k + 3 d + 6 d + 3 d + d d + 3d + b. c. + + + 3 5 6 9 3d + 9 d + d + d + 3 d + 5d + d. 6. Simplify the comple fraction: 9 y a. 3 y 3 b. a a 9b a a + 3b 1 + 5/ + 1 Simplifying Error: Very Bad News!!!!!! 15 3 3 Note: You cannot simplify through addition or subtraction. Homework: p. 557 1-1 all, 19-3 all, 38, 6-8 all, 60, 71-79 odds Honors 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 + 3. 3 6 zy, 3 9 y z, and z. 3 3 + and + 6 + 9 B. Adding and Subtracting Rational Epressions Eamples 5a 9 1. +. 6b 1a b 3 5 + y 1y 3. + 10 3 + 15 3 15 6 30. + 5 3 + 8 8 C. Simplify Comple Fractions 1 1. 1 3 y. 3 a b 1 b ab Homework p. 565 1-17 all, 9, 31, 37, 51, 59, 71-7 all Honors Algebra Chapter 9 Page
Section 9.3 Graphing Reciprocal Functions Goals: 1. Determine properties of reciprocal functions.. Graph transformations of reciprocal functions. I. Reciprocal Functions A. Key Concept: B. Asymptotes: The lines a rational functions approach. 1. Vertical Asymptote:. Horizontal Asymptote: C. Eamples: Identify the asymptotes, domain, and range of the function. 1.. II. Transformations of Reciprocal Functions B. Key Concept: Honors Algebra Chapter 9 Page 5
C. Eamples Graph the following functions and state their domain and range. 1 1. f ( ) + 3. f ( ) 3 + 1 3 + 6 6 6 6 8 6 8 6 6 8 6 8 3. A commuter train has a nonstop service from one city to another, a distance of about 5 miles. a) Write an equation to represent the travel time between these two cities as a function of rail speed. Then graph the equation. 0 10 0 10 10 0 10 0 b) Eplain any limitations to the range and domain in this situation. Homework: p. 57 1-6 all, 10, 15,, 9-59 odds Honors Algebra Chapter 9 Page 6
Section 9. Graphing Rational Functions Goal: 1. To graph rational functions with vertical and horizontal asymptotes.. To graph rational functions with slant asymptotes and point discontinuity. I. Horizontal and Vertical Asymptotes A. Reminder of Rational Functions B. Vertical Asymptotes Note One eception (point discontinuity) holes in the graph. C. The horizontal asymptotes Produces a slant asymptote. D. Eamples: 1. Determine the equations of any vertical and horizontal asymptotes and the values of for any holes in the graph of: 3 7 a) f ( ) b) f ( ) + 5 + 6 + 8 + 15. Graph and name the asymptotes: a) f ( ) b) f ( ) + 1 + 3 Honors Algebra Chapter 9 Page 7
3. A boat traveled upstream at r1 miles per hour. During the return trip to its original starting point, the boat traveled at r miles per hour. The average speed for the entire r1 r trip R is given by the formula R. r + r a) Draw the graph if r 15 80 60 1 miles per hour. 0 0 60 0 0 0 0 60 80 0 b) What is the R-intercept of the graph? c) What domain and range values are meaningful in the contet of the problem? II. Slant Asymptotes A. Key Concept B. Eample: 1. f ( ). f ( ) + 1 3 10 6 6 6 6 8 6 8 6 6 8 6 8 Honors Algebra Chapter 9 Page 8
III. Point Discontinuity A. Key Point B. Eample: 1. f ( ). f ( ) 16 + 6 6 6 6 8 6 8 6 6 8 6 8 Homework p. 581 3, 13, 1, 16, 3, 38, 39, 1, 51-61 odds Honors Algebra Chapter 9 Page 9
Section 9.5 Variation Functions Goals: 1. To solve problems involving direct and joint variations.. To solve inverse and combined variation problems A. Direct Variation 1. As ( ), y ( ).. Wording 3. General Equation: y k where k is the constant of variation.. 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.. 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, and h 17. b) Suppose y varies jointly as and z. Find y when 3 and z, if y 11 when 5 and z. Honors Algebra Chapter 9 Page 10
C. Inverse Variation 1. As ( ), y ( ). Wording k 3. General Equation: y where k is the constant of variation.. 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 1. 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.. Suppose f varies directly as g, and f varies inversely as h. Find g when f 6 and h 16, if g 10 when h and f 6. Homework p. 590 1-6 all, 11, 16, 1,, 5-7 all,, 57-59 all, 61-65 odds Honors Algebra Chapter 9 Page 11
Section 9.6 Solving Rational Equations and Inequalities Goals: 1. To solve rational equations and inequalities Eamples: 5 1 1. + 3. 5 + 3 1 1 3. p p + 1 p 7 + p p + 1 p 1 p + 1 ( )( ) 1. + + 8 5. Aaron adds an 80% brine (salt and water) solution to 16 ounces of solution that is 10% brine. How much of the solution should be added to create a solution that is 50% brine? 6. Janna adds a 65% base solution to 13 ounces of solution that is 0% base. How much of the solution should be added to create a solution that is 0% base? 7. Tim and Ashley mow lawns together. Tim working alone could complete the job in.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 Honors Algebra Chapter 9 Page 1
8. 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? 9. 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 10. Solve: 1 3k + 9k < 3 11. Solve: 1 + 3 < 5 5 Homework p. 600 1-15 all, - all, 7, 9, 50, 51 Honors Algebra Chapter 9 Page 13