Name: Date: INTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II Rational functions are simply the ratio of polynomial functions. They take on more interesting properties and have more interesting graphs than polynomials because of the interaction between the numerator and denominator of the fraction. In Common Core Algebra II, we will be primarily concerned with the algebra of these functions. But in this lesson we will eplore some of their characteristics. Eercise #1: Consider the rational function given by Algebraically determine the y-intercept for this function. f 6. (b) Algebraically determine the -intercept of this function. Hint a fraction can only equal zero if its numerator is zero. (c) For what value of is this function undefined? Why is it undefined at this value? (d) Based on (c), state the domain of this function in set-builder notation. Eercise #: Find all values of for which the rational function using your calculator to evaluate this epression for these values. h 5 116 is undefined. Verify by Eercise #: Which of the following represents the domain of the function (1) 4 () and 8 () (4) 6 and f? 616 1
Eercise #4: If g and f 1 then find: 5 f g (b) f g (c) f g Eercise #5: Find formulas for the inverse of each of the following simple rational functions below. Recall that as a first step, switch the roles of and y. y (b) y 8 Eercise #6: The function f is either an even or an odd function. Determine which it is and justify. 4 Based on your answer, what type of symmetry must this function have? Use your calculator to sketch a graph to y verify.
SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II Simplifying a rational epression into its lowest terms is an etremely useful skill. Its algebra is based on how we simply numerical fractions. The basic principle is developed in the first eercise. Eercise #1: Recall that to multiply fractions, one simply multiplies their numerators and denominators. Simplify the numerical fraction by first (b) Simplify the algebraic fraction by first epressing it as a product of two fractions, one of which is equal to one. epressing it as the product of two fractions (factor!), one of which is equal to one. Every time we simplify a fraction, we are essentially finding all common factors of the numerator and denominator and dividing them to be equal to one. Key in this process is that the numerator and denominator must be factored and only common factors cancel each other. This is true whether our fraction contains monomial, binomial, or polynomial epressions. Eercise #: Simplify each of the following monomials dividing other monomials. y 6y 5 6 8 (b) 0 y 4 10 8 (c) 7y 1y 5 8 Eercise #: Which of the following is equivalent to 10y 15 6 6 y? (1) y () y 4 () y 8 9 (4) y
When simplifying rational epressions that are more comple, always factor first, then identify common factors that can be eliminated. Eercise #4: Simplify each of the following rational epressions. 514 4 (b) 4 1 10 5 (c) 148 16 A special type of simplifying occurs whenever epressions of the form y and y are involved. Eercise #5: Simplify each of the following fractions. 9 6 6 9 (b) 15 15 (c) a b b a 10 Eercise #6: Which of the following is equivalent to? 5 (1) 5 () 5 () 5 (4) 5 Eercise #7: Which of the following is equivalent to 69? 18 6 (1) 6 () 9 () 6 (4) 6 4
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II Multiplication of rational epressions follows the same principles as those involved in simplifying them. The process is illustrated in Eercise #1 with both a numerical and algebraic fraction. Notice the parallels. Eercise #1: Simplify each of the following rational epressions by factoring completely. For the numerical fraction, make sure to prime factor all numerators and denominators. 6 10 (b) 8 4 15 6 6 1 The ability to cross-cancel with fractions is a result of the two facts: (1) to multiply fractions we multiply their respective numerators and denominators and () multiplication is commutative. The keys to multiplication, then, are the same as that for simplifying factor and then reduce. Eercise #: Simplify each of the following products. 7 8y 10 (b) 5 6y 6 y 10y 4 y 9 y 5 5 7 (c) 1 4 1 48 6 (d) 9 4 4 6 5
Division of rational epressions continues to follow from what you have seen in previous courses. Since division by a fraction can always be thought of in terms of multiplying by it reciprocal, these problems simply involve an additional step. Eercise #: Perform each of the following division problems. Epress all answers in simplest form. 8 15 5 (b) 5 7 6y y 0y 4y 0 8 6 (c) 8 16 816 10 (d) 9 1 5 15 7 5 14 Eercise #4: When 5 is divided by 5 9 the result is (1) 5 7 () 0 () 15 (4) 9 5 6
COMBINING RATIONAL EXPRESSIONS WITH ADDITION AND SUBTRACTION COMMON CORE ALGEBRA II Occasionally it will be important to be able to combine two or more rational epressions by addition. Keep in mind two key principles that dictate fraction addition. TWO GUIDELINES FOR ADDITION AND SUBTRACTION OF FRACTIONS 1. Fractions must have a common denominator.. Denominators can only be changed by multiplying the overall fraction by one. Eercise #1: Combine each of the following fractions by first finding a common denominator. Epress your answers in simplest form. 5 4 (b) 4 6 41 5 (c) 1 5 10 4 Each of the combinations in Eercise #1 should have been reasonably easy because each denominator was monomial in nature. If this is not the case, then it is wise to factor the denominators before trying to find a common denominator. Eercise #: Combine each of the following fractions by factoring the denominators first. Then find a common denominator and add. 4 5 5y15 y 9 (b) 9 0 6 8 7
Subtraction of rational epressions is especially challenging because of errors that naturally arise when students forget to distribute the subtraction (or the multiplication by -1). Still, with careful eecution, these problems are no different than their addition counterparts. Eercise #: Perform each of the following subtraction problems. Epress your answers in simplest form. 7 4 4 (b) 4 1 105 6 (c) 4 8 0 8 (d) 5 4 1 Eercise #4: Which of the following is equivalent to 1 1? 1 (1) 1 1 () 1 () (4) 1 8
COMPLEX FRACTIONS COMMON CORE ALGEBRA II Comple fractions are simply defined as fractions that have other fractions within their numerators and/or denominators. To simplify these fractions means to remove these minor fractions and then eliminate any common factors. The key, as always, is to multiply by the number one in ways that simplify the fraction. Eercise #1: Consider the comple fraction 1 1 9 18 1 What is the least common denominator amongst the three minor fractions?. (b) Multiply the numerator and denominator of the major fraction by your answer in part and then simplify your result. (c) Why is it acceptable to perform the operation in part (b)? What number are you effectively multiplying by? By multiplying the major fraction by the number one, by using the least common denominator, we will always eliminate the minor fractions (by turning them into integer epressions). Eercise #: Simplify each of the following comple fractions. 1 1 10 5 (b) 5 5 (c) 1 8 4 7 4 9
These types of problems can certainly involve more complicated secondary simplification. Don t forget the primary use of factoring in order to simplify. Eercise #: Simplify each of the following comple fractions. 1 (b) 5 1 1 5 (c) 1 1 6 4 1 If the denominators of the minor fractions become more comple, be sure to factor them first, just as you did with the addition and subtraction in the previous lesson. Eercise #4: Simplify each of the following comple fractions. 4 4 1 4 8 1 (b) 6 4 81 10
SOLVING FRACTIONAL EQUATIONS COMMON CORE ALGEBRA II Equations involving fractions or rational epressions arise frequently in mathematics. The key to working with them is to manipulate the equation, typically by multiplying both sides of it by some quantity, that eliminates the fractional nature of the equation. The most common form of this practice is cross-multiplying. Eercise #1: Use the technique of cross multiplication to solve each of the following equations. 4 5 1 (b) 5 1 6 Since this technique should be familiar to students at this point, we will move onto a less familiar method when more than two fractions are involved. The net eercise will illustrate the process. Eercise #: Consider the equation 1 9 4 4. What is the least common denominator for all three fractions in this equations? (b) Multiply both sides of this equation by the LCD to clear the equation of the denominators. Now, solve the resulting linear equation. It is very important to note the similarities and differences between this technique and the one employed to simplify comple fractions. With comple fractions we multiplied by one in creative ways. Here we are multiplying both sides of an equation by a quantity that removes the fractional nature of the equation. Eercise #: Which of the following values of solves: 4 1? 6 10 15 (1) 14 () 8 () 6 (4) 11 11
These equations can involve quadratic as well as root epressions. The key, though, remains the same multiplying both sides of the equation by the same quantity. Eercise #4: Solve each of the following equations for all values of. 1 1 1 10 5 (b) 1 1 1 1 4 Because fractional equations often involve denominators containing variables, it is important that we check to see if any solutions to the equation make it undefined. These represent further eamples of etraneous roots. Eercise #5: Solve and reject any etraneous roots. 1 18 9 5 8 15 4 1 1 (b) 4 1 6 1
I. Holes and Vertical Asymptotes Discontinuities A rational function is any function that can be written as f() = where 0. Discontinuities are in the graph of a function. The graph of a polynomial function is (no jumps or breaks). However, the graph of a rational function may be. To find these values, factor the top and bottom and see if any factors cancel. There are two cases: (1) A factor cancels, which creates a. () A factor left in the denominator creates a. E. 1: Identifying Holes in a Graph E. : Identifying Vertical Asymptotes Identify the vertical asymptote(s) of 1
GRAPHING RATIONAL FUNCTIONS ALL ASYMPTOTES! II. Horizontal Asymptotes The graph of a function has at most horizontal asymptote. To find a horizontal asymptote, compare the degree of the polynomial on top with the degree of the polynomial on the bottom. There are three Cases: (1) If the degree of the denominator (bottom) is greater than the degree of the numerator, then is the horizontal asymptote. () IF the degrees are equal, then the graph has a horizontal asymptote at. Note: a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. () If the degree of denominator is less than the degree of numerator, then there is horizontal asymptote. Want a quick and easy way to remember Horizontal Asymptotes?? Ask yourself where is the degree LARGER? E. 4: a) y = Find any Horizontal Asymptotes: 4 1 5 b) y = c) y = 4 4 14
TRY THESE!! Name the values where the function is discontinuous. Identify any holes, vertical and/ or horizontal asymptotes. Then sketch the graph. 1. y = 18. y = 6 9 15
Work Problems: RULES FOR SOLVING WORK PROBLEMS: Together + Alone Together = 1 job completed Alone 1. To solve work problems, you need to work with the same unit of measure within each problem. For eample, you cannot mi hours and minutes in the same equation.. You need to find the fractional part of the job that would be done in one unit of time, such as 1 minute or 1 hour. If a person can do a complete job in days, he can do 1/ of it in 1 day. Eample: Mark can dig a ditch in 4 hours. Greg can dig the same ditch in hours. How long would it take them to dig it together? 1. Write an equation: Together + Alone Together = 1 Alone OR T + A T = 1 A. Substitute your values in the equation:. Solve! Try These!!! 1. Jaron can paint a fence in hours and his friend Tyrek can paint it in 5 hours. How long would it take them to do the job if they work together?. It takes Ashly days to cultivate the garden. It takes Ivanna twice as long to do the same job. How long does it take them to do the job if they work together? 16
. Heckel can plow a field in 6 hours. If his brother Jeckel helps him, it will take 4 hours. How long would it take Jeckel to do the job alone? 4. Jordan can paint a fence in 8 hours. Kejuan can paint the same fence in 4 hours. How long will it take them to do the job if they work together? 5. If Dal can build a doghouse in 5 hours, and together he and Yves can build it in hours, how many hours would it take Yves alone to build the same doghouse? 6. One machine can complete a job in 10 minutes. If this machine and an older machine do the same job together, the job can be completed in 6 minutes. How long would it take the older machine to do the job? 7. Working alone, Janell can pick 40 bushels of apples in 11 hours. Shanice can pick the same amount in 14 hours. How long would it take them if they worked together? 17