Graph General Rational Functions. }} q(x) bn x n 1 b n 2 1. p(x) 5 a m x m 1 a m 2 1

Transcription

1 TEKS 8.3 A.0.A, A.0.B, A.0.C, A.0.F Graph General Rational Functions Before You graphed rational functions involving linear polnomials. Now You will graph rational functions with higher-degree polnomials. Wh So ou can solve problems about altitude, as in E. 35. Ke Vocabular end behavior, p. 339 asmptote, p. 78 rational function, p. 558 KEY CONCEPT Graphs of Rational Functions For Your Notebook Let p() and q() be polnomials with no common factors other than 6. The graph of the following rational function has the characteristics listed below. f() 5 p() 5 a m m a m m... a a 0 q() bn n b n n... b b 0. The -intercepts of the graph of f are the real zeros of p().. The graph of f has a vertical asmptote at each real zero of q(). 3. The graph of f has at most one horizontal asmptote, which is determined b the degrees m and n of p() and q(). m < n m 5 n m > n The line 5 0 is a horizontal asmptote. The line 5 a m bn is a horizontal asmptote. The graph has no horizontal asmptote. The graph s end behavior is the same as the graph of 5 a m m n. bn E XAMPLE Graph a rational function (m < n) Graph 5 6. State the domain and range. Solution The numerator has no zeros, so there is no -intercept. The denominator has no real zeros, so there is no vertical asmptote. The degree of the numerator, 0, is less than the degree of the denominator,. So, the line 5 0 (the -ais) is a horizontal asmptote. The graph passes through the points (3, 0.6), (, 3), (0, 6), (, 3), and (3, 0.6). The domain is all real numbers, and the range is 0 < Graph General Rational Functions 565

2 E XAMPLE Graph a rational function (m 5 n) Graph 5 9. REVIEW ZEROS OF FUNCTIONS For help with finding zeros of functions, see p. 5. Solution The zero of the numerator is 0, so 0 is an -intercept. The zeros of the denominator 9 are 63, so 5 3 and 53 are vertical asmptotes. The numerator and denominator have the same degree, so the horizontal asmptote is 5 a m 5 5. bn Plot points between and beond the vertical asmptotes. To the left of Between 53 and To the right of E XAMPLE 3 Graph a rational function (m > n) Graph 5 3. Solution The numerator factors as ( )( ), so the -intercepts are and. The zero of the denominator is, so 5 is a vertical asmptote. The degree of the numerator,, is greater than the degree of the denominator,, so the graph has no horizontal asmptote. The graph has the same end behavior as the graph of 5 5. Plot points on each side of the vertical asmptote To the left of To the right of Chapter 8 Rational Functions

3 GUIDED PRACTICE for Eamples,, and 3 Graph the function f() E XAMPLE TAKS REASONING: Multi-Step Problem MANUFACTURING A food manufacturer wants to find the most efficient packaging for a can of soup with a volume of 3 cubic centimeters. Find the dimensions of the can that has this volume and uses the least amount of material possible. Solution STEP Write an equation that gives the height h of the soup can in terms of its radius r. Use the formula for the volume of a clinder and the fact that the soup can s volume is 3 cubic centimeters. V 5 πr h Formula for volume of clinder 3 5 πr h Substitute 3 for V. 3 5 h Solve for h. πr STEP Write a function that gives the surface area S of the soup can in terms of onl its radius r. S 5 πr πrh Formula for surface area of clinder INTERPRET FUNCTIONS The function for the surface area is a rational function because it can be written as a quotient of polnomials: S 5 πr 3 68 r STEP 3 5 πr πr 3 Substitute pr 3 for h. pr 5 πr 68 r Simplif. Graph the function for the surface area S using a graphing calculator. Then use the minimum feature to find the minimum value of S. You get a minimum value of about 7, which occurs when r ø 3.79 and h ø 3 π(3.79) ø Minimum X= Y=70.77 c So, the soup can using the least amount of material has a radius of about 3.79 centimeters and a height of about 7.58 centimeters. Notice that the height and the diameter are equal for this can. GUIDED PRACTICE for Eample 5. WHAT IF In Eample, suppose the manufacturer wants to find the most efficient packaging for a soup can with a volume of 5 cubic centimeters. Find the dimensions of this can. 8.3 Graph General Rational Functions 567

4 8.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 7, 5, and 33 5 TAKS PRACTICE AND REASONING Es. 6,,, 35, 37, and 38 5 MULTIPLE REPRESENTATIONS E. 33. VOCABULARY Cop and complete: The graph of a rational function f has no when the degree of the function s numerator is greater than the degree of its denominator.. WRITING Let f() 5 p() where p() and q() are polnomials with no q() common factors other than 6. Describe how to find the -intercepts and the vertical asmptotes of the graph of f. EXAMPLES,, and 3 on pp for Es. 3 3 MATCHING GRAPHS Match the function with its graph A. B. 8 C. 6. TAKS REASONING The graph of which function is shown A 5 3 B 5 3 C 5 D at classzone.com ANALYZING GRAPHS Identif the -intercept(s) and vertical asmptote(s) of the graph of the function f() g() ERROR ANALYSIS Describe and correct the error in finding the vertical asmptote(s) of f() The vertical asmptote occurs at the zero of the numerator. So, the vertical asmptote is Chapter 8 Rational Functions

5 . TAKS REASONING What is the horizontal asmptote of the graph of the function 5 5 A 5 0 B 5 C 5 D 5 GRAPHING FUNCTIONS Graph the function f() g() h() TAKS REASONING Write two different rational functions whose graphs have the same end behavior as the graph of 5 3. GRAPHING CALCULATOR Use a graphing calculator to find the range of the rational function CHALLENGE The graph of a function of the form f() 5 values of a and b a is shown. Find the b 30. (0, ) (, ) (, ) (, ) (3, ) (, 6) PROBLEM SOLVING EXAMPLE on p. 567 for Es. 3 3 GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving eercises. 3. AGRICULTURE A farmer makes clindrical bales of ha that have a volume of 00 cubic feet. Each bale is to be wrapped in plastic to keep the ha dr. a. Using the formula for the volume of a clinder, write an equation that gives the length l of a bale in terms of the radius r. b. Write a function that gives the surface area of a bale in terms of onl the radius r. l r c. Find the dimensions of a bale that has the given volume and uses the least amount of plastic possible when the bale is wrapped. 8.3 Graph General Rational Functions 569

6 3. AQUARIUM DESIGN A manufacturer is designing an aquarium whose base is a regular heagon. The aquarium should have a volume of cubic feet and use the least amount of material possible. Let s be the length (in feet) of a side of the base, and let h be the height (in feet). a. Write an equation that gives h in terms of s. (Hint: The volume of the aquarium is given b V 5 3Ï 3 s h.) s h b. Find the dimensions s and h that minimize the amount of material used. (Hint: The surface area of the aquarium is given b S 5 3Ï 3 s 6sh.) 0 N 33. MULTIPLE REPRESENTATIONS The mean temperature T (in degrees Celsius) of the Atlantic Ocean between latitudes 08N and 08S can be modeled b T 5 7,800d 0,000 3d 70d 000 where d is the depth (in meters). a. Making a Table Make a table of values showing the mean temperature for depths from 000 meters to 300 meters in 50 meter intervals. b. Using a Graph Graph the model. Use our graph to estimate the depth at which the mean temperature is 8C. 0.0 C 3.3 C 9. C 0 m 00 m 00 m 600 m 800 m 000 m 00 m 00 m 0 S 3. MULTI-STEP PROBLEM From 993 to 00, the number n (in billions) of shares of stock sold on the New York Stock Echange can be modeled b n 5 05t t 00 where t is the number of ears since 993. a. Graph the model. b. Describe the general trends shown b the graph. c. Estimate the ear when the number of shares of stock sold was first greater than 00 billion. 35. TAKS REASONING The acceleration due to gravit g (in meters per second squared) changes as altitude changes and is given b the function g h ( )h ( ) where h is the altitude (in meters) above sea level. a. Graph Graph the function. b. Appl A mountaineer is climbing to a height of 8000 meters. What is the value of g at this altitude c. Appl A spacecraft reaches an altitude of kilometers above Earth. What is the value of g at this altitude d. Eplain Describe what happens to the value of g as altitude increases. This spacecraft reached an altitude of km in WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

7 36. CHALLENGE You need to build a clindrical water tank using 00 cubic feet of concrete. The sides and the base of the tank must be foot thick. a. Write an equation that gives the tank s inner height h in ft terms of its inner radius r. b. Write an equation that gives the volume V of water that the tank can hold as a function of r. c. Graph the equation from part (b). What values of r and h maimize the tank s capacit r h MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson.7; TAKS Workbook 37. TAKS PRACTICE Doris plants a 75 square foot rectangular garden. She uses 36 feet of fencing to enclose the garden. What are the approimate dimensions of the garden TAKS Obj. 5 A 5.6 ft b 3. ft B 5.7 ft b.3 ft C 6.0 ft b.0 ft D 6.6 ft b. ft REVIEW Skills Review Handbook p. 006; TAKS Workbook 38. TAKS PRACTICE The circle graph represents 80 students. The red section of the circle graph represents the number of students who ride a bus to school everda. How man students ride a bus to school everda TAKS Obj. 8 Bus riders 08 Drivers Walkers Bike riders F 76 G 350 H 90 J 53 QUIZ for Lessons The variables and var inversel. Use the given values to write an equation relating and. Then find when 5. (p. 55). 5 8, , , , 53 Graph the function (p. 558) 6. 5 (p. 558) 7. f() 5 (p. 558) (p. 565) (p. 565) 0. g() (p. 565). SOFTBALL A pitcher throws 6 strikes in her first 38 pitches. The table shows how the pitcher s strike percentage changes if she throws consecutive strikes after the first 38 pitches. Write a rational function for the strike percentage in terms of. Graph the function. How man consecutive strikes must the pitcher throw to reach a strike percentage of 0.60 (p. 558) Total strikes Total pitches Strike percentage EXTRA PRACTICE for Lesson 8.3, p. 07 ONLINE QUIZ at classzone.com 57

8 MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons MULTIPLE CHOICE. EFFICIENT PACKAGING A food manufacturer wants to find the most efficient packaging for a clindrical canister of oatmeal with a volume of 663 cubic centimeters. An equation that gives the canister s surface area S in terms of its radius r is S 5 πr 336 r. Use a graphing calculator to graph the equation. What is the approimate radius r of the canister that uses the least material possible TEKS A.0.B A 5. inches B 6. inches C 6.6 inches D 8. inches. BODY MASS INDEX The bod mass inde b of a person varies directl with the person s weight w (in kilograms) and inversel with the square of the person s height h (in meters). A person who is.6 meters tall and weighs 5. kilograms has a bod mass inde of 0. What is the approimate height of a person who weighs 5 kilograms and has a bod mass inde of 0 TEKS A.0.G F. meters G. meters H.5 meters J.3 meters 3. CANDY SALES The number of boes of cand a manufacturer sells each month varies inversel with the price (in dollars). In one month, the manufacturer sells 800 boes of cand at a price of $5 per bo. About how man boes of cand will the manufacturer sell at a price of $7 per bo TEKS A.0.G A 57 boes B 57 boes C 63 boes D 686 boes. INVERSE VARIATION Which equation represents inverse variation TEKS A.0.G F 5 3 G 5 H 5 3 J 5 5. PLAYGROUND AREA You are designing a rectangular plaground that has an area of 00 square ards. A building borders the length of the plaground. You use fencing for the other three sides. Which length l and width w minimize the amount of fencing needed TEKS A.0.D A l 5 ards; w 5 ards B l 5 0 ards; w 5 0 ards C l 5 5 ards; w 5 8 ards D l 5 8 ards; w 5 7 ards 6. PHOTO PRINTING Your famil bus a photo printer. The printer costs $00. The ink and paper cost about $.60 for each photo ou print. Which equation gives the average cost C of a printed photo as a function of the number of photos printed TEKS A.0.B F C 5 G C H C J C classzone.com GRIDDED ANSWER SOUND INTENSITY The intensit I of a sound (in watts per square meter) varies inversel with the square of the distance d (in meters) from the source of the sound. At a distance of meter from the stage, the intensit of the sound of a rock concert is about 0 watts per square meter. What is the intensit in watts per square meter of the sound ou hear if ou are 5 meters from the stage Write our answer as a decimal rounded to the nearest hundredth. TEKS A.0.G 8. MOTORCYCLE VALUE The value M (in dollars) of a motorccle t ears after it was purchased new can be estimated using the function M(t) where t. Estimate t the motorccle s value 8 ears after it was purchased. Round our answer to the nearest dollar. TEKS A.0.D 57 Chapter 8 Rational Functions