CHAPTER 5 RATIONAL FUNCTIONS

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1 CHAPTER 5 RATIONAL FUNCTIONS Big IDEAS: ) Graphing rational functions ) Performing operations with rational epressions 3) Solving rational equations Section: 5- Model Inverse and Joint Variation Essential Question What are the differences between direct, inverse, and joint variation? Key Vocabulary DIRECT VARIATION A linear function of the form ;. Written Cues: y varies y is to INVERSE VARIATION JOINT VARIATION A function of the form or ;. Written Cues: y varies y is to The relationship that occurs when a quantity varies directly with the product of or more other quantities. Written Cues: z varies p is to CONSTANT OF VARIATION The nonzero constant or in a,, or variation equation. Student Notes Honors Algebra II Chapter 5 Rational Functions Page #

2 Re-Cap: ) y varies directly as, and y 5 when 5, find when y 40. ) The amount of hydrogen produced when sodium is added to water varies directly as the amount of sodium added. If 9 g of sodium produces 4 g of hydrogen, find the amount of sodium needed to produce 0 g of hydrogen. E ) Tell whether and y show direct variation, inverse variation, or neither. 3 a. y3 0 b. c. y 5 y E ) Write an equation for the given relationship. A) r varies inversely with s. D) m varies directly with the square of n and inversely with p. B) z varies jointly with and the square root of y. E) z varies jointly with u and v and inversely with the square of w. C) p varies inversely with the cube of q. E 3) The variables and y vary inversely, and y 5 when relates and y. Then find y when 0.. Write an equation that 3 Student Notes Honors Algebra II Chapter 5 Rational Functions Page #

3 E 4) The driving time between two specific locations varies inversely with the average driving speed. The driving distance between Chicago and Minneapolis is about 400 miles. A) Write a model that gives the driving time t in hours (not including stops) between Chicago and Minneapolis as a function of average driving speed r in miles per hour. B) Make a table showing the driving time for average driving speeds of 3 miles per hour, 40 miles per hour, 50 miles per hour, and 60 miles per hour. E 5) Moving cartons are manufactured in a variety of sizes and shapes to accommodate the variety of objects that need to be packed. The table compares the area A of the bottom of a rectangular carton (in square inches) with the height h for four available cartons that have the same volume. A h A) Write a model that gives h as a function of A. B) Predict the height of a carton with the same volume as those in the table that has a base area of 75 square inches. E 6) The variable z varies jointly with and y. Also, z 60 when 4 and y 5. Find z when 7 and y. Student Notes Honors Algebra II Chapter 5 Rational Functions Page #3

Section: 5- Graph Simple Rational FUNctions Essential Question What does the graph of the rational function y a k h look like? Graph the inverse variation y.

4 Section: 5- Graph Simple Rational FUNctions Essential Question What does the graph of the rational function y a k h look like? Graph the inverse variation y. Key Vocabulary RATIONAL FUNCTION Rational means f p where and are polynomials and. q Graph of function branches. f consisting of two symmetrical HYPERBOLA Domain: Range: VERTICAL ASYMPTOTE(S) Occur when the equals. Rational function of the form occurs at. Rational function of the form at. a y k, vertical asymptote h a b y, vertical asymptote occurs c d HORIZONTAL ASYMPTOTE Rational function of the form occurs at. Rational function of the form occurs at. a y k, horizontal asymptote h a b y c d, horizontal asymptote Student Notes Honors Algebra II Chapter 5 Rational Functions Page #4

5 Graph the function, then state the domain and range. Include all asymptotes! 4 3 E ) y E ) y Domain: Range: Domain: Range: E 3) 3 6 y Domain: Range: E 4) Your long-distance calling plan has a fied monthly fee of $4.95 and costs 5 cents a minute. Write an equation that gives your average cost C (in dollars) per minute m during a given month. Graph the function. Use the graph to estimate when the average cost is $.4 per minute. What happens to the average cost per minute as the number of minutes increase? Student Notes Honors Algebra II Chapter 5 Rational Functions Page #5

6 Section: Essential Question 5-3 Graph General Rational FUNctions What are the steps for graphing a general rational function? Let pand CHARACTERISTICS OF RATIONAL FUNCTIONS q be polynomials with no common factors other than. p f q. -intercepts occur where the equals zero; p 0.. Vertical asymptotes occur where the equals zero; q Horizontal asymptotes occur based on the degree of the numerator and the degree of the denominator. The graph of f can have at most horizontal asymptote. If the degree of the numerator is the degree of the denominator, then a horizontal asymptote occurs at. f 4, g 3 If the degree of the numerator is the degree of the denominator, then a horizontal asymptote occurs at. h 4 3, r 35 4 If the degree of the numerator is the degree of the denominator, then the graph has horizontal asymptote. Instead, the graph has an oblique ( ) asymptote which can be found using. d 3, k NOTE: Graphs of functions MAY cross and/or asymptotes, but will NEVER cross asymptotes. Student Notes Honors Algebra II Chapter 5 Rational Functions Page #6

7 Graph the function, then state the domain and range. Include all asymptotes! E ) y E ) y 4 Domain: Range: Domain: Range: E 3) y Domain: Range: E 4) A carton manufacturer has a large order for rectangular cartons with square bottoms that have a volume of 6000 cubic inches. Find the dimensions of the carton with that amount of volume that uses the least amount of material. Student Notes Honors Algebra II Chapter 5 Rational Functions Page #7

8 Section: Essential Question 5-4 Multiply and Divide Rational Epressions What are the steps for multiplying and dividing rational epressions? Simplify Key Vocabulary SIMPLIFIED FORM RECIPROCAL When the numerator and denominator have no (other than ). The reciprocal of a number is divided by that number. In other words, switch the and the of a rational epression. Simplifying Rational Epressions- out common Let a, b, and c be epressions with and. ac bc Multiplying Rational Epressions- multiply and multiply Let a, b, c, and d be epressions with and. a c b d Dividing Rational Epressions- multiply the first rational epression by the of the second rational epression Let a, b, c, and d be epressions with, and. a c b d Student Notes Honors Algebra II Chapter 5 Rational Functions Page #8

9 Simplify. E ) 86 4 E ) E 3) y 7 E 4) 4 4 3y 5 y E 5) 4 E 6) E 7) A company makes kitchen canisters with the same base. Style is a cylinder whose height is equal to its radius. Style is a cylinder whose height is twice its radius. Find the surface area and volume of each canister. Calculate the ratio of surface area to volume for each canister. What do the ratios tell you about the efficiencies of the two canisters? Student Notes Honors Algebra II Chapter 5 Rational Functions Page #9

10 Section: Essential Question 5-5 Add and Subtract Rational Epressions What are the steps for adding and subtracting rational epressions with different denominators? Simplify ADDING OR SUBTRACTING RATIONAL EXPRESSIONS. Find the ( ) of the fractions.. Epress each fraction as an fraction with the as its denominator. 3. Combine the. Leave the in factored form. 4. If possible, factor the and the fraction. Simplify. E ) E ) E 3) E 4) Student Notes Honors Algebra II Chapter 5 Rational Functions Page #0

11 5 E 5) 48 E 6) 5 w w SIMPLIFYING COMPLEX FRACTIONS Comple Fraction- a within a METHOD Simplify the numerator and denominator separately. Multiply by the reciprocal. Reduce if necessary. METHOD Multiply the numerator and denominator by the LCD of all fractions appearing in the comple fraction. Reduce if necessary. Simplify. E 7) METHOD METHOD Student Notes Honors Algebra II Chapter 5 Rational Functions Page #

12 Student Notes Honors Algebra II Chapter 5 Rational Functions Page # E 8) METHOD METHOD E 9) METHOD METHOD

13 E 0) METHOD METHOD 5 w w 5 w w E ) A college student drives home for a weekend. The distance between her home and her dormitory is d miles, her average speed coming home is r, and her average speed driving back to college is r. Write an epression that represents her average speed for the round trip and simplify. Student Notes Honors Algebra II Chapter 5 Rational Functions Page #3

14 Section: Essential Question 5-6 Solve Rational Equations What are the steps for solving rational equations? Key Vocabulary Cross Multiplying- used to solve a rational equation that is epressed as a (when each side of the equation is a rational epression) Multiplying by the LCD- used to solve a rational equation that is not epressed as a Etraneous Solution- a root (solution, zero, answer) of the transformed equation that is a root of the equation. Caution!!! Multiplying both sides of the equation by the may result in roots! You must verify your answers because they may not be in the of the original problem! Solve. E ) 7 0 E ) 5 8 Student Notes Honors Algebra II Chapter 5 Rational Functions Page #4

15 E 3) Check: E 4) 7 Check: E 5) Check: E 6) Check: Student Notes Honors Algebra II Chapter 5 Rational Functions Page #5

16 E 7) The coolant in a car radiator is a miture of antifreeze and water. The recommended miture for your car is 50% antifreeze. If you have a miture of 7 liters of coolant that is 40% antifreeze, how much pure antifreeze should you add to bring the miture up to the recommended level? E 8) A company produces computer desks. The average cost to produce desks can be modeled by the function C. How many desks should the company produce each month in order to achieve an average cost of $85 per desk? Student Notes Honors Algebra II Chapter 5 Rational Functions Page #6

17 Section: 5-7 Describe and Compare Function Characteristics Essential Question How do you compare functions represented in different ways? Key Vocabulary INCREASING Graph from left to right slope y DECREASING CONSTANT Graph from left to right slope line slope AVERAGE RATE OF CHANGE of a segment determined by the endpoints of an interval m Sketch a graph for the situation described. Label key information. E ) The temperature (in o F) during a snowstorm slowly decreased at the start of the storm, reached its coldest halfway through the storm, quickly rose above the temperature at the start of the storm, and then decreased to the temperature at the start of the storm. E ) The US gross domestic product (GDP) growth rate over a year period can be modeled by a polynomial that began slightly positive before dropping to significantly negative, swinging back to significantly positive, then falling to a growth rate a little higher than at the beginning of the period. Student Notes Honors Algebra II Chapter 5 Rational Functions Page #7

18 E 3) For the function intervals f, find the average rate of change over the 3,, 0 0,. What happens to the average rate 3,,,, and of change as increases? What does this mean for the graph of f? E 4) Compare the properties of the two functions and key characteristics of their graphs. Include information such as domain and range, asymptotes, end behavior, and general appearance of the graphs. Function : y 5 Function : a simple rational function that is undefined when 0 and y 0 and contains the points, 3 and, 3. Student Notes Honors Algebra II Chapter 5 Rational Functions Page #8

19 E 5) A scientist studies the growth of two types of bacteria. The number y of cells of Bacteria after t hours is given by y 00.4 t. The number y of cells of Bacteria after t hours is given by the following data pairs t, y : 0,00,,5,,3, 3,5, 4,75, each type of bacteria over time. 5, 0. Compare the population growth of Key Vocabulary EVEN FUNCTION ODD FUNCTION f f symmetric about the f f symmetric about the Determine whether the function is even, odd, or neither. E 6) f 8 5 E 7) f 4 f 3 7 E 8) 6 4 Student Notes Honors Algebra II Chapter 5 Rational Functions Page #9

Vocabulary: I. Inverse Variation: Two variables x and y show inverse variation if they are related as. follows: where a 0

Vocabulary: I. Inverse Variation: Two variables x and y show inverse variation if they are related as. follows: where a 0 8.1: Model Inverse and Joint Variation I. Inverse Variation: Two variables x and y show inverse variation if they are related as follows: where a 0 * In this equation y is said to vary inversely with x.