Page of 6 7.5 Graphing Square Root and Cube Root Functions What ou should learn GOAL Graph square root and cube root functions. GOAL Use square root and cube root functions to find real-life quantities, such as the power of a race car in E. 48. Wh ou should learn it To solve real-life problems, such as finding the age of an African elephant in Eample 6. GOAL GRAPHING RADICAL FUNCTIONS In Lesson 7.4 ou saw the graphs of = and = 3. These are eamples of radical functions. 3 (, ) (, ) (0, 0) (, ) (0, 0) 3 Domain: 0, Range: 0 Domain and range: all real numbers In this lesson ou will learn to graph functions of the form = a º h + k and = a 3 º h + k. ACTIVITY Developing Concepts Investigating Graphs of Radical Functions Graph = a for a =,, º3, and º. Use the graph of = shown above and the labeled points on the graph as a reference. Describe how a affects the graph. Graph = a 3 for a =,, º3, and º. Use the graph of = 3 shown above and the labeled points on the graph as a reference. Describe how a affects the graph. In the activit ou ma have discovered that the graph of = a starts at the origin and passes through the point (, a). Similarl, the graph of = a 3 passes through the origin and the points (º, ºa) and (, a). The following describes how to graph more general radical functions. GRAPHS OF RADICAL FUNCTIONS To graph = a º h + k or = a 3 º h + k, follow these steps. STEP STEP Sketch the graph of = a or = a 3. Shift the graph h units horizontall and k units verticall. 7.5 Graphing Square Root and Cube Root Functions 43
Page of 6 E X A M P L E Comparing Two Graphs Describe how to obtain the graph of = + º 3 from the graph of =. Note that = + º 3 = º ( º ) + (º3), so h = º and k = º3. To obtain the graph of = + º 3, shift the graph of = left unit and down 3 units. E X A M P L E Graphing a Square Root Function Skills Review For help with transformations, see p. 9. Graph = º3 º +. Sketch the graph of = º3 (shown dashed). Notice that it begins at the origin and passes through the point (, º3). Note that for = º3 º +, h = and k =. So, shift the graph right units and up unit. The result is a graph that starts at (, ) and passes through the point (3, º). (, ) (0, 0) (3, ) 3 (, 3) 3 E X A M P L E 3 Graphing a Cube Root Function Graph = 3 3 + º. Sketch the graph of = 3 3 (shown dashed). Notice that it passes through the origin and the points (º, º3) and (, 3). Note that for = 3 3 + º, h = º and k = º. So, shift the graph left units and down unit. The result is a graph that passes through the points (º3, º4), (º, º), and (º, ). 3 3 (, ) (, 3) (0, 0) (, ) 3 3 ( 3, 4) (, 3) E X A M P L E 4 Finding Domain and Range State the domain and range of the function in (a) Eample and (b) Eample 3. a. From the graph of = º3 º + in Eample, ou can see that the domain of the function is and the range of the function is. b. From the graph of = 3 3 + º in Eample 3, ou can see that the domain and range of the function are both all real numbers. 43 Chapter 7 Powers, Roots, and Radicals
Page 3 of 6 FOCUS ON CAREERS GOAL USING RADICAL FUNCTIONS IN When ou use radical functions in real life, the domain is understood to be restricted to the values that make sense in the real-life situation. E X A M P L E 5 Modeling with a Square Root Function AMUSEMENT RIDE DESIGNER An amusement ride designer uses math and science to ensure the safet of the rides. Most amusement ride designers have a degree in mechanical engineering. CAREER LINK INTERNET AMUSEMENT PARKS At an amusement park a ride called the rotor is a clindrical room that spins around. The riders stand against the circular wall. When the rotor reaches the necessar speed, the floor drops out and the centrifugal force keeps the riders pinned to the wall. The model that gives the speed s (in meters per second) necessar to keep a person pinned to the wall is s = 4.95 r where r is the radius (in meters) of the rotor. Use a graphing calculator to graph the model. Then use the graph to estimate the radius of a rotor that spins at a speed of 8 meters per second. Graph = 4.95 and = 8. Choose a viewing window that shows the point where the graphs intersect. Then use the Intersect feature to find the -coordinate of that point. You get.6. The radius is about.6 meters. Intersection X=.69784 Y=8 E X A M P L E 6 Modeling with a Cube Root Function Biolog Biologists have discovered that the shoulder height h (in centimeters) of a male African elephant can be modeled b h = 6.5 3 t + 75.8 where t is the age (in ears) of the elephant. Use a graphing calculator to graph the model. Then use the graph to estimate the age of an elephant whose shoulder height is 00 centimeters. Source: Elephants Graph = 6.5 3 + 75.8 and = 00. Choose a viewing window that shows the point where the graphs intersect. Then use the Intersect feature to find the -coordinate of that point. You get 7.85. The elephant is about 8 ears old. Intersection X=7.8473809 Y=00 7.5 Graphing Square Root and Cube Root Functions 433
Page 4 of 6 GUIDED PRACTICE Vocabular Check Concept Check Skill Check. Complete this statement: Square root functions and cube root functions are eamples of? functions.. ERROR ANALYSIS Eplain wh the graph shown at the near right is not the graph of = º +. 3. ERROR ANALYSIS Eplain wh the graph shown at the far right is not the graph of = 3 + º 3. Describe how to obtain the graph of g from the graph of ƒ. 4. g() = + 5, ƒ() = 5. g() = 3 º 0, ƒ() = 3 (3, ) (, ) - (3, -) (, -3) (, -4) E. E. 3 Graph the function. Then state the domain and range. 6. = º 7. = + 8. = º 9. = + 3 º 0. = 3 3. = 3 º 6. = 3 + 5 3. = º3 3 º 7 º 4 4. ELEPHANTS Look back at Eample 6. Use a graphing calculator to graph the model. Then use the graph to estimate the age of an elephant whose shoulder height is 50 centimeters. PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 950. COMPARING GRAPHS Describe how to obtain the graph of g from the graph of ƒ. 5. g() = + 4, ƒ() = 6. g() = 5 º 0 º 3, ƒ() = 5 7. g() = º 3 º 0, ƒ() = º 3 8. g() = 3 + 6 º 5, ƒ() = 3 MATCHING GRAPHS Match the function with its graph. 9. = º 0. = +. = + º A. B. C. (0, ) (, 0) (0, 0) (, 0) (0, ) (, ) HOMEWORK HELP Eample : Es. 5 8 Eample : Es. 9 30 Eample 3: Es. 3 39 Eample 4: Es. 45 Eample 5: Es. 46, 47 Eample 6: Es. 48, 49 SQUARE ROOT FUNCTIONS Graph the function. Then state the domain and range.. = º5 3. = 3 4. = / + 4 5. = / º 6. = + 6 7. = ( º 7) / 8. = ( º ) / + 7 9. = + 5 º 30. = º º 3 º 5 434 Chapter 7 Powers, Roots, and Radicals
Page 5 of 6 CUBE ROOT FUNCTIONS Graph the function. Then state the domain and range. 3. = 3 3. = º /3 33. = 3 º 7 34. = 3 + 3 4 35. = 3 º 5 36. = + 3 /3 37. = 5 /3 º 38. = º3 3 + 4 39. = 3 º 4 + 3 HOMEWORK HELP Visit our Web site for help with problem solving in Es. 40 45. INTERNET FOCUS ON CAREERS CRITICAL THINKING Find the domain and range of the function without graphing. Eplain how ou found our solution. 40. = º 3 4. = º 4. = º º 3 º 7 43. = 3 + 8 44. = º 3 3 º 5 45. = 4 3 + 4 + 7 GRAPHING MODELS In Eercises 46 49, use a graphing calculator to graph the models. Then use the Intersect feature to solve the problems. 46. OCEAN DISTANCES When ou look at the ocean, the distance d (in miles) ou can see to the horizon can be modeled b d =. a where a is our altitude (in feet above sea level). Graph the model. Then determine at what altitude ou can see 0 miles. Source: Mathematics in Everda Things 47. GEOMETRY CONNECTION In a right circular cone with a slant height of unit, the radius r of the cone is given b r = S + π 4 º where S is the surface area of π the cone. Graph the model. Then find the surface area of a right circular cone with a slant height of unit and a radius of unit. 48. RACING Drag racing is an acceleration contest over a distance of a quarter mile. For a given total weight, the speed of a car at the end of the race is a function of the car s power. For a total weight of 3500 pounds, the speed s (in miles per hour) can be modeled b s = 4.8 3 p where p is the power (in horsepower). Graph the model. Then determine the power of a car that reaches a speed of 00 miles per hour. Source: The Phsics of Sports 49. STORMS AT SEA The fetch ƒ (in nautical miles) of the wind at sea is the distance over which the wind is blowing. The minimum fetch required to create a full developed storm can be modeled b s = 3. 3 f + 0 +. where s is the speed (in knots) of the wind. Graph the model. Then determine the minimum fetch required to create a full developed storm if the wind speed is 5 knots. Source: Oceanograph COAST GUARD Members of the Coast Guard have a variet of responsibilities. Some participate in search and rescue missions that involve rescuing people caught in storms at sea, discussed in E. 49. CAREER LINK INTERNET wave direction fetch wind limits of storm 7.5 Graphing Square Root and Cube Root Functions 435
Page 6 of 6 Test Preparation 50. MULTI-STEP PROBLEM Follow the steps below to graph radical functions of the form = ƒ(º). a. Graph ƒ () = and ƒ () = º. How are the graphs related? b. Graph g () = 3 and g () = 3 º. How are the graphs related? c. Graph ƒ 3 () = º ( º ) º 4 and g 3 () = 3 º ( + ) + 5 using what ou learned from parts (a) and (b) and what ou know about the effects of a, h, and k on the graphs of = a º h + k and = a 3 º h + k. Challenge d. Writing Describe the steps for graphing a function of the form ƒ() = a º ( º h ) + k or g() = a 3 º ( º h ) + k. ANALYZING GRAPHS Write an equation for the function whose graph is shown. 5. 5. 53. EXTRA CHALLENGE MIXED REVIEW SOLVING EQUATIONS Solve the equation. (Review 5.3 for 7.6) 54. = 3 55. ( + 7) = 0 56. 9 + 3 = 5 57. º 5 = 3 58. 4 ( + 6) = 59. ( º 0.5) = 6.5 SPECIAL PRODUCTS Find the product. (Review 6.3 for 7.6) 60. ( + 4) 6. ( º 9) 6. ( 3 + 7) 63. (º3 + 4 4 ) 64. (6 º 5) 65. (º º ) COMPOSITION OF FUNCTIONS Find ƒ(g()) and g(ƒ()). (Review 7.3) 66. ƒ() = + 7, g() = 67. ƒ() = +, g() = º 3 68. ƒ() = º, g() = + 69. ƒ() = + 7, g() = 3 º 3 70. MANDELBROT SET To determine whether a comple number c belongs to the Mandelbrot set, consider the function ƒ(z) = z + c and the infinite list of comple numbers z 0 = 0, z = ƒ(z 0 ), z = ƒ(z ), z 3 = ƒ(z ),.... If the absolute values z 0, z, z, z 3,... are all less than some fied number N, then c belongs to the Mandelbrot set. If the absolute values z 0, z, z, z 3,... become infinitel large, then c does not belong to the Mandelbrot set. Tell whether the comple number c belongs to the Mandelbrot set. (Review 5.4) a. c = 3i b. c = + i c. c = 6 436 Chapter 7 Powers, Roots, and Radicals