And Cube Root Functions
Page 1 of 27.5Graphing Square Rootand Cube Root FunctionsWhat you should learnGraph square rootand cube root functions.GOAL 1GOAL 1GRAPHING RADICAL FUNCTIONS3In Lesson 7.4 you saw the graphs of y x and y x . These are examples ofradical functions.GOAL 2 Use square rootand cube root functions tofind real-life quantities, suchas the power of a race car inEx. 48.yyy x13y x1(1, 1)(0, 0)1( 1, 1)x(1, 1)(0, 0) 3xWhy you should learn itREFE To solve real-lifeproblems, such as finding theage of an African elephant inExample 6.AL LIDomain: x 0, Range: y 0Domain and range: all real numbersIn this lesson you will learn to graph functions of the form y a x º h k and3y a x º h k.ACTIVITYDevelopingConcepts1Investigating Graphs of Radical Functions12Graph y a x for a 2, , º3, and º1. Use the graph of y x shown above and the labeled points on the graph as a reference. Describehow a affects the graph.23123Graph y a x for a 2, , º3, and º1. Use the graph of y x shown above and the labeled points on the graph as a reference. Describehow a affects the graph.In the activity you may have discovered that the graph of y a x starts at the origin3and passes through the point (1, a). Similarly, the graph of y a x passes throughthe origin and the points (º1, ºa) and (1, a). The following describes how to graphmore general radical functions.GRAPHS OF RADICAL FUNCTIONS3To graph y a x º h k or y a x º h k, follow these steps.3STEP 1Sketch the graph of y a x or y a x .STEP 2Shift the graph h units horizontally and k units vertically.7.5 Graphing Square Root and Cube Root Functions431
Page 1 of 2EXAMPLE 1Comparing Two GraphsDescribe how to obtain the graph of y x 1 º 3 from the graph of y x .SOLUTIONNote that y x 1 º 3 x º º( 1 ) (º3), so h º1 and k º3. To obtain 1 º 3, shift the graph of y x left 1 unit and down 3 units.the graph of y x EXAMPLE 2Graphing a Square Root FunctionGraph y º3 x º 2 1.ySTUDENT HELPSkills ReviewFor help withtransformations,see p. 921.SOLUTION12Sketch the graph of y º3 x (shown dashed).Notice that it begins at the origin and passesthrough the point (1, º3).Note that for y º3 x º 2 1, h 2 andk 1. So, shift the graph right 2 units and up1 unit. The result is a graph that starts at (2, 1)and passes through the point (3, º2).EXAMPLE 31(2, 1)(0, 0)1x(3, 2)y 3 x 2 1(1, 3)y 3 xGraphing a Cube Root Function3Graph y 3 x 2 º 1.ySOLUTION123Sketch the graph of y 3 x (shown dashed).Notice that it passes through the origin and thepoints (º1, º3) and (1, 3).3Note that for y 3 x 2 º 1, h º2 andk º1. So, shift the graph left 2 units and down1 unit. The result is a graph that passes throughthe points (º3, º4), (º2, º1), and (º1, 2).EXAMPLE 43y 3 x 2 1(1, 3)( 1, 2)1( 2, 1)( 3, 4)(0, 0)13y 3 x( 1, 3)Finding Domain and RangeState the domain and range of the function in (a) Example 2 and (b) Example 3.SOLUTIONa. From the graph of y º3 x º 2 1 in Example 2, you can see that thedomain of the function is x 2 and the range of the function is y 1.3b. From the graph of y 3 x 2 º 1 in Example 3, you can see that the domainand range of the function are both all real numbers.432Chapter 7 Powers, Roots, and Radicalsx
Page 1 of 2GOAL 2FOCUS ONCAREERSUSING RADICAL FUNCTIONS IN REAL LIFEWhen you use radical functions in real life, the domain is understood to be restrictedto the values that make sense in the real-life situation.Modeling with a Square Root FunctionEXAMPLE 5REFELAL IAMUSEMENTRIDE DESIGNERINTAn amusement ride designeruses math and science toensure the safety of therides. Most amusement ridedesigners have a degree inmechanical engineering.NEER TCAREER LINKwww.mcdougallittell.comAMUSEMENT PARKS At an amusement parka ride called the rotor is a cylindrical room thatspins around. The riders stand against the circularwall. When the rotor reaches the necessary speed,the floor drops out and the centrifugal force keepsthe riders pinned to the wall.The model that gives the speed s (in meters persecond) necessary to keep a person pinned to thewall iss 4.95 r where r is the radius (in meters) of the rotor. Use a graphing calculator to graph themodel. Then use the graph to estimate the radius of a rotor that spins at a speed of8 meters per second.SOLUTIONGraph y 4.95 x and y 8. Choose a viewingwindow that shows the point where the graphsintersect. Then use the Intersect feature to find thex-coordinate of that point. You get x 2.61. REFELAL IIntersectionX 2.6119784 Y 8The radius is about 2.61 meters.Modeling with a Cube Root FunctionEXAMPLE 6Biologists have discovered that the shoulder height h (in centimeters) of a maleAfrican elephant can be modeled byBiology3h 62.5 t 75.8where t is the age (in years) of the elephant. Use a graphing calculator to graph themodel. Then use the graph to estimate the age of an elephant whose shoulder heightis 200 centimeters. Source: ElephantsSOLUTION3Graph y 62.5 x 75.8 and y 200. Choose aviewing window that shows the point where thegraphs intersect. Then use the Intersect feature to findthe x-coordinate of that point. You get x 7.85. IntersectionX 7.8473809 Y 200The elephant is about 8 years old.7.5 Graphing Square Root and Cube Root Functions433
Page 1 of 2GUIDED PRACTICEVocabulary CheckConcept Check 1. Complete this statement: Square root functions and cube root functions are? functions.examples of 2. ERROR ANALYSIS Explain why the11graph shown at the far right is not the3 2 º 3.graph of y x -1(3, 2)(2, 1)3. ERROR ANALYSIS Explain why theSkill Checkyygraph shown at the near right is not the 1 2.graph of y x ºx1(3, -2)(2, -3)(1, -4)xEx. 2Ex. 3Describe how to obtain the graph of g from the graph of ƒ.3 ,5 ƒ(x) x 4. g(x) x 35. g(x) x º 10, ƒ(x) x Graph the function. Then state the domain and range.6. y º x 7. y x 12 310. y x 311. y x º 638. y x º 29. y 2 x 3 º13312. y x 513. y º3 x º 7 º4ELEPHANTS Look back at Example 6. Use a graphing calculator to graph14.the model. Then use the graph to estimate the age of an elephant whoseshoulder height is 250 centimeters.PRACTICE AND APPLICATIONSSTUDENT HELPExtra Practiceto help you masterskills is on p. 950.COMPARING GRAPHS Describe how to obtain the graph of g from the graphof ƒ.15. g(x) x 41 , ƒ(x) x 316. g(x) 5 x º 01 º 3, ƒ(x) 5 x 3317. g(x) º x º 10, ƒ(x) º x 318. g(x) x 6 º 5, ƒ(x) x MATCHING GRAPHS Match the function with its graph.19. y x º 1A.20. y x 1B.y21. y x º11C.y2y21 (1, 0)(0, 1)( 1, 0)1x(0, 1)2(0, 0)1x( 1, 1)STUDENT HELPHOMEWORK HELPExample 1:Example 2:Example 3:Example 4:Example 5:Example 6:434Exs. 15–18Exs. 19–30Exs. 31–39Exs. 22–45Exs. 46, 47Exs. 48, 49SQUARE ROOT FUNCTIONS Graph the function. Then state the domainand range.22. y º5 x 123. y x 3124. y x1/2 425. y x1/2 º 226. y x 627. y (x º 7)1/228. y (x º 1)1/2 729. y 2 x 5 º1230. y º x º 3 º25Chapter 7 Powers, Roots, and Radicalsx
Page 1 of 2CUBE ROOT FUNCTIONS Graph the function. Then state the domainand range.1 331. y x 23334. y x 41 1/337. y x º 25INTSTUDENT HELPNEER THOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor help with problemsolving in Exs. 40–45.333. y x º 732. y º2x1/3 2 1/336. y x 3335. y x º 5338. y º3 x 4339. y 2 x º 4 3CRITICAL THINKING Find the domain and range of the function withoutgraphing. Explain how you found your solution. 31 40. y x º41. y 2 x º 242. y º x º 3 º72 3343. y x 844. y º x º 545. y 4 x 4 73GRAPHING MODELS In Exercises 46–49, use a graphing calculator tograph the models. Then use the Intersect feature to solve the problems.346.OCEAN DISTANCES When you look at the ocean, the distance d (in miles)you can see to the horizon can be modeled by d 1.22 a where a is youraltitude (in feet above sea level). Graph the model. Then determine at whataltitude you can see 10 miles. Source: Mathematics in Everyday ThingsCONNECTION In a right circular cone witha slant height of 1 unit, the radius r of the cone is given47. GEOMETRY1 π by r 1 S 4 º 2 where S is the surface area ofπ112the cone. Graph the model. Then find the surface areaof a right circular cone with a slant height of 1 unit and12a radius of unit.48.RACING Drag racing is an acceleration contest over a distance of a quartermile. For a given total weight, the speed of a car at the end of the race is afunction of the car’s power. For a total weight of 3500 pounds, the speed s3(in miles per hour) can be modeled by s 14.8 p where p is the power(in horsepower). Graph the model. Then determine the power of a car thatreaches a speed of 100 miles per hour. Source: The Physics of Sports49.STORMS AT SEA The fetch ƒ (in nautical miles) of the wind at sea is thedistance over which the wind is blowing. The minimum fetch required to create a3f 01 11.1 where s is thefully developed storm can be modeled by s 3.1 speed (in knots) of the wind. Graph the model. Then determine the minimum fetchrequired to create a fully developed storm if the wind speed is 25 knots.FOCUS ONCAREERS Source: OceanographyCOAST GUARDINTMembers of theCoast Guard have a varietyof responsibilities. Someparticipate in search andrescue missions that involverescuing people caught instorms at sea, discussedin Ex. 49.NEER TCAREER LINKwww.mcdougallittell.comindREFELAL Iwlimitsof stormwave directionfetch7.5 Graphing Square Root and Cube Root Functions435
Page 1 of 2TestPreparation50. MULTI-STEP PROBLEM Follow the steps below to graph radical functions ofthe form y ƒ(ºx).a. Graph ƒ1(x) x and ƒ2(x) º x . How are the graphs related?33b. Graph g1(x) x and g2(x) º x . How are the graphs related?3c. Graph ƒ3(x) º (x º )2 º 4 and g3(x) 2 º (x )1 5 using what youlearned from parts (a) and (b) and what you know about the effects of a, h,3 h k and y a x º h k.and k on the graphs of y a x ºd.Writing Describe the steps for graphing a function of the form3 (x º )h k or g(x) a º (x º )h k.ƒ(x) a º ChallengeANALYZING GRAPHS Write an equation for the function whose graphis shown.51.52.y1x53.yy111x1EXTRA CHALLENGE1xwww.mcdougallittell.comMIXED REVIEWSOLVING EQUATIONS Solve the equation. (Review 5.3 for 7.6)54. 2x2 3255. (x 7)2 1056. 9x2 3 5157. x2 º 5 132158. (x 6)2 22459. 2(x º 0.25)2 16.5SPECIAL PRODUCTS Find the product. (Review 6.3 for 7.6)60. (x 4)261. (x º 9y)262. (2x3 7)263. (º3x 4y4)264. (6 º 5x)265. (º1 º 2x2)2COMPOSITION OF FUNCTIONS Find ƒ(g(x)) and g(ƒ(x)). (Review 7.3)66. ƒ(x) x 7, g(x) 2x67. ƒ(x) 2x 1, g(x) x º 368. ƒ(x) x2 º 1, g(x) x 269. ƒ(x) x2 7, g(x) 3x º 370.MANDELBROT SET To determine whether a complex number c belongs tothe Mandelbrot set, consider the function ƒ(z) z2 c and the infinite list ofcomplex numbers z0 0, z1 ƒ(z0), z2 ƒ(z1), z3 ƒ(z2), . . . . If the absolute values z0 , z1 , z2 , z3 , . . . are all less than some fixednumber N, then c belongs to the Mandelbrot set. If the absolute values z0 , z1 , z2 , z3 , . . . become infinitely large,then c does not belong to the Mandelbrot set.Tell whether the complex number c belongs to the Mandelbrot set. (Review 5.4)a. c 3i436Chapter 7 Powers, Roots, and Radicalsb. c 2 2ic. c 6
7.5 Graphing Square Root and Cube Root Functions 431 Graph square root and cube root functions. Use square root and cube root functions to find real-life quantities, such as the power of a race car in Ex. 48.
Chapter - 7: Cubes and Cube Roots Cube Concept Sheet: Cube refers to the geometrical shape of cube. Cube roots refers the side of Cube. CUBE ROOT Root – The origin, which results in cube. As square root is the inverse operation of squaring, finding cube root is the inverse operation of finding cube. Cube
of the cube root is the last digit of the cube of the last digit of the cube!) For the first digit, we look at the magnitude of the millions. Example. Let’s find the cube root of the perfect cube 377,933,067. The first digit of the cube root must be 7
43 4 The cube root of 64 is 4. Symbol for cubing is the exponent 3. cube root is 3 —. Work with a partner. Use a cube root symbol to write the edge length of the cube. Then fi nd the cube root. Check your answer by multiplying. Volume a. Sample: s 3 — 343 3 — 73 7 inches The edge le
Cube: Cube is a persistent data representation of a cube object. Data administrator builds a cube from Cube Object. End User can browse a cube on OLAP Viewer or View reports that draws data from a Cube. OLAP Viewer: A browser based viewer with grid and charting options that helps end user to slice and dice a cube.
The cube root of a number is the number that is multiplied together three times to produce the original number. The symbol Ï 3 ·· find the cube root. 6 Look at Solve It on the previous page. The equation shows a variable cubed equal to a perfect cube. Use the cube root to complete the solution. Solution:
6. Find the cube root of – 3375. 7. Find the cube root of the number 614125 using prime factorization method. 8. Find the cube of square of 0.2 3 Marks Questions 1. Find the smallest natural number by which 392 must be multiplied so that the product is a perfect cube . 2. Find the cube root of 27 x ( -2
Determine the dimensions of the cube, x, by calculating the cube root of the volume, or x 3 _ V. Method 1: Use Prime Factorization Determine the cube root of 512. Record the prime factorization for 512. Then, identify the factors that can be cubed to form 512. The cube root of 512 is 8. The
TANK DESIGN & DETAILING Introduction The API 650 standard is designed to provide the petroleum industry with tanks of adequate safety and reasonable economy for use in the storage of petroleum, petroleum products, and other liquid products commonly handled and stored by the various branches of the industry. This standard does not present or establish a fixed series of allowable tank sizes .
