Unit 4 Rational Functions

Transcription

1 Unit 4 Rational Functions Test date: Name: By the end of this unit, you will be able to Simplify rational expressions Find the LCM for rational expressions Add and subtract rational expressions Solve rational equations Find horizontal and vertical asymptotes for rational functions Graph rational functions Find points of discontinuity (holes) for rational functions Find domain and range of hyperbolas Find domain for any rational function

2 Table of Contents 8.1 Multiplying and Dividing Rational Expressions... 3 Simplified form... 3 Multiplying Rational Expressions... 3 Dividing Rational Expressions... 4 Simplifying Complex Fractions Adding and Subtracting Rational Expressions... 5 Finding the LCM of Polynomials... 5 Adding and Subtracting Rational Expressions Graphing Reciprocal Functions Graphing Rational Functions Oblique Asymptotes and Points of Discontinuity Solving Rational Equations

3 8.1 Multiplying and Dividing Rational Expressions Simplified form of a rational expression: The numerator and denominator have. If the numerator and/or denominator are polynomials,, then. Examples: a. 9: ;<9 b. 9: =<9=> 9 : 9 : =? c. <9(9 : ;A9;B)?D(9=>)(9 : =E) Determine when the expressions above are undefined. Then, some more examples. d. FG H=IF G IF J =F J H e. 9: K=B9 : B9 L =9 L K Multiplying Rational Expressions Examples: f. M9 OK: g. A9=I9 : 9: =A9;B I?K J?>9 J 9 : =<9;> I9 h. B9=IO9 J B9: =A9;? B9 : =I9=? B9 3

4 Dividing Rational Expressions Examples: i.?dpq : BR : S <PL j. M9 : ;?D9=B 4x I x >R : S : A9 : k. 9 2x + 3 A9: =E 9=I 9=I Simplifying Complex Fractions Examples: l. [ : \[ : ]G^: [ J :^]J[ m. _ : _ : ]\`: _ G _aj` 4

5 8.2 Adding and Subtracting Rational Expressions Warm Up: Simplify the following expressions without a calculator. a. B < +? < b. I B? I c.? I B A Examples: d. B O I9 I9 e. B9 + > 9=A 9=A Finding the LCM of Polynomials 1. each expression. 2. The LCM contains each factor it appears as a factor. Examples: 1. Find the LCM of 16p I q B r, 40pq A r I, and 15p B r A. 2. Find the LCM of 3m I 3m 6 and 4m I + 12m 40. Adding and Subtracting Rational Expressions 1. Find the least common denominator (LCD). Rewrite each expression with the. 2. or the. 3. Combine in the numerator. 4. (if possible). 5. (if possible). Examples with Monomial Denominators: f. <F : >H + E?AF : H : g. B9 : IK + <?I9K : 5

6 Steps (for reference): 1. Find the least common denominator (LCD). Rewrite each expression with the LCD. 2. Add or subtract the numerators. 3. Combine any like terms in the numerator. 4. Factor (if possible). 5. Simplify (if possible). Examples with Polynomial Denominators: h. 9;?D B9;?< B9=?< >9=BD i. 9;< B9;M I9=A A9=M j. A + 9 B9 J >9 J ;B9 : Tip! k. 9;?? 9 : ;>9;E 9 : =E l. 9=M + > A9 : ;I?9;<?I9;B 6

7 8.3 Graphing Reciprocal Functions Plot the ordered pairs to graph y 1 x =. x x Why would the calculator give us an error when x=0? What happens as x gets closer and closer to 0? Asymptotes: & Graph them. Domain: Range: 2. Plot the ordered pairs to graph y = I Error (Use the table feature on your calculator to fill in table.) 9=B x y Identify x-values for which f(x) is undefined. Asymptotes: & Draw them as dashed lines. Domain: Range: Why is x=3 a vertical asymptote? 1 y = + 2. (Use the table feature on your calculator to fill in table.) x x y 3. Find the ordered pairs and graph Why is there an Error when x=0 (why is x undefined at 0? Try plugging 0 in for x.) Identify x-values for which f(x) is undefined. Domain: Range: Asymptotes: & How is this equation different than the equation from #1? How does this graph compare to the graph from #1 (describe shifting)? Determine the value of x for which the following function is not defined: f x = A I9;? 7

8 1 y 2 x + 3. x y 4. Find the ordered pairs and graph Identify x-values for which f(x) is undefined. Domain: Range: Asymptotes: & How does this graph compare to the graph from #1 (shifting)? SUMMARIZE: Rational Function Family - Type 1 a The graph of the rational function y = k x- h + has 1) Horizontal translation from negative. a y = : units if h is positive. units if h is x The asymptote is at. 2) Vertical translation from a y = : units if k is positive. units if k is negative. x The asymptote is at. In other words, the function graph is shifted h units or and k units or from the a parent graph y = x 3) a orientation and shape: If, the graph is stretched. If, the graph is stretched. 4) The domain is. 5) The range is. 8

9 Rational Functions - Type 2 The equations may be written in a different form. 3x - 3 Find the ordered pairs and graph y = 4x x y By looking at the graph and table of values, find the following. Domain: Range: Asymptotes: & Graph them. How can we find the domain and vertical asymptote from the equation (not graph)? How can we find the range and horizontal asymptote from the equation (not graph)? Use the answers to the above questions to fill in the following concepts. SUMMARIZE: Rational Function Family Type 2 ax + b The graph of the rational function y = cx + d has the following characteristics. 1) The domain is. 2) The range is. 3) The line is the asymptote. 4) The line is the asymptote. 9

10 8.4 Graphing Rational Functions Definition: A rational function is a function of the form where p x and q x are polynomials and q x 0. The shape of the graph is called a One form: 2 o o Graph two points on either side of the VA Draw the branches Another form: VA: HA: Graph two points on either side of the VA Draw the branches Example: Graph y = A=B9 A9=B. VA = HA = D: R: 10

11 Example: Graph y = 9=I B9;B. VA = HA = D: R: Applications: 1. The senior class is sponsoring the homecoming dance. The cost is $45 per person plus a $2500 deposit. Write and graph an equation to represent the average cost per person. Let y represent the average cost and x represent the number of people. How many people must attend for the average cost to drop below $55? 2. You are making calendars to sell to make money for the basketball team. It costs an initial fee of $50 to use the equipment and then each calendar costs $3.50 to manufacture. Write an equation to model this situation. Let y represent the average cost and x represent the number of calendars. How many calendars do you need to produce for the average cost of them to drop below $5? 11

12 No Horizontal Asymptote: Examples: 1. 9 J 9;? VA = HA = D: R: 2. 9 J I9=? VA = HA = D: R: Why are there no horizontal asymptotes? 12

13 8.4 Oblique Asymptotes and Points of Discontinuity Oblique asymptote (aka ): A function will have an oblique asymptote if. The equation of the oblique asymptote is the polynomial part of the quotient. Point of Discontinuity: If f x = F 9 H 9 where a x, b(x) are polynomials and b x 0 and x c is a factor of both a(x) and b(x), then there is a point of discontinuity (aka ) at. Example: f x = 9J ;B9;I 9;? Steps to Graph when the Degree of the Polynomial is Greater than 1: 1. Find the 2. Identify 3. Find 4. Find (3 cases) a. b. c. 5. Make a 6. Graph 7. Check solutions, find on calculator Examples: Graph, then state the domain and range. 1. f x = 9J =B9=A 9=I x-intercepts: VA= HA = D: R: 13

14 2. f x = 9: 9;? x-intercepts: VA= HA = D: R: 3. f x = 9J =B9=?D 9=A x-intercepts: VA= HA = D: R: 4. f x = 9: =A 9=I x-intercepts: VA= HA = D: R: 5. f x = 9: =?> 9;A x-intercepts: VA= HA = D: R: 14

15 8.6 Solving Rational Equations 1. I9 9: =9=?D = B 9;< 9 : ;M9;?< 9;B A quick way to solve a rational equation is to This will Beware of! 2. I9=? 9;I 1 = =O B9 : ;O9;I 3. <9 = 4 < 9;? 9;? 4. B 9 + A 9;I = 2 5. B =? 9 : ;A9 9;A 15

16 Applications: 1. Drew 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? 2. Michael adds a 60% acid solution to 15 milliliters of a solution that is 10% acid. How much of the 60% acid solution should be added to create a solution that is 40% acid? 3. Hannah adds a 65% base solution to 13 ounces of solution that is 20% base. How much of the solution should be added to create a solution that is 40% base? 4. Maggie swims for 5 hours in a stream that has a current of 1 mile per hour. She leaves the dock and swims upstream for 2 miles, then swims back to the dock. What is her swimming speed in still water? 16

17 5. The current in a river is 6 miles per hour. In her motorboat, Emmeline can travel 12 miles upstream or 16 miles downstream in the same amount of time. What is the speed of her motorboat in still water? Is this a reasonable answer? Explain. 6. A direct flight from Chicago, IL to Washington, D.C. is approximately 600 miles. On a particular day, the wind speed is 50 mph at the cruising altitude of a passenger jet. A jet leaves for this flight flying with the wind. Then, the jet makes the return flight against the wind. The total in-air time for the jet for both flights was 2.7 hours. What is the speed of the jet in no wind? 7. Ryan swims for 1 hour in a stream that has a current of 2 mph. He leaves the dock and swims upstream for 3 miles, then swims back to the dock. What is his swimming speed in still water? 8. Carson and Delaney mow lawns together. Carson working alone could complete a particular job in 4.5 hours, and Delaney could complete it alone in 3.7 hours. How long does it take to complete the job when they work together? 9. Garrett and Grant paint rooms together. Garrett working alone could complete a particular job in 6.4 hours, and Grant could complete it alone in 4.8 hours. How long does it take them to complete the job when they work together? 17