Graph Square Root and Cube Root Functions

Transcription

1 TEKS 6.5 2A.4.B, 2A.9.A, 2A.9.B, 2A.9.F Graph Square Root and Cube Root Functions Before You graphed polnomial functions. Now You will graph square root and cube root functions. Wh? So ou can graph the speed of a racing car, as in E. 8. Ke Vocabular radical function parent function, p. 89 In Lesson 6.4, ou saw the graphs of 5 Ï } and 5 Ï }. These are eamples of radical functions. In this lesson, ou will learn to graph functions of the form 5 aï } 2 h k and 5 a Ï } 2 h k. KEY CONCEPT For Your Notebook Parent Functions for Square Root and Cube Root Functions The parent function for the famil of square root functions is f() 5 Ï }. f () 5 The parent function for the famil of cube root functions is g() 5 Ï }. g() 5 (0, 0) (, ) (2, 2) (, ) (0, 0) Domain: 0, Range: 0 Domain and range: all real numbers E XAMPLE Graph a square root function Graph 5 } Ï }, and state the domain and range. Compare the graph with 2 the graph of 5 Ï }. Make a table of values and sketch the graph REVIEW DOMAIN AND RANGE For help with the domain and range of a function, see p The radicand of a square root must be nonnegative. So, the domain is 0. The range is 0. The graph of 5 } 2 Ï } is a vertical shrink of the graph 5 2 of 5 Ï } b a factor of } Chapter 6 Rational Eponents and Radical Functions

2 E XAMPLE 2 Graph a cube root function Graph 52Ï }, and state the domain and range. Compare the graph with the graph of 5 Ï }. Make a table of values and sketch the graph. REVIEW STRETCHES AND SHRINKS For help with vertical stretches and shrinks, see p The domain and range are all real numbers. The graph of 52Ï } is a vertical stretch of the graph of 5 Ï } b a factor of followed b a reflection in the -ais E XAMPLE TAKS Solve a REASONING: multi-step problem Multi-Step Problem PENDULUMS The period of a pendulum is the time the pendulum takes to complete one back-and-forth swing. The period T (in seconds) can be modeled b T 5.Ï } l where l is the pendulum s length (in feet). Use a graphing calculator to graph the model. How long is a pendulum with a period of seconds? STEP Graph the model. Enter the equation 5.Ï }. The graph is shown below. STEP 2 Use the trace feature to find the value of when 5. The graph shows ø 7.. Trace X=7. Y=.0008 c A pendulum with a period of seconds is about 7. feet long. GUIDED PRACTICE for Eamples, 2, and Graph the function. Then state the domain and range.. 52Ï } 2. f() 5 } 4 Ï }. 52 } 2 Ï } 4. g() 5 4 Ï } 5. WHAT IF? Use the model in Eample to find the length of a pendulum with a period of second. 6.5 Graph Square Root and Cube Root Functions 447

3 TRANSLATIONS OF RADICAL FUNCTIONS The procedure for graphing functions of the form 5 aï } 2 h k and 5 a Ï } 2 h k is described below. KEY CONCEPT For Your Notebook Graphs of Radical Functions To graph 5 aï } 2 h k or 5 aï } 2 h k, follow these steps: STEP Sketch the graph of 5 aï } or 5 aï }. STEP 2 Translate the graph horizontall h units and verticall k units. E XAMPLE 4 Graph a translated square root function Graph 522Ï } 2 2. Then state the domain and range. REVIEW TRANSLATIONS For help with translating graphs, see p. 2. STEP Sketch the graph of 522Ï } (shown in blue). Notice that it begins at the origin and passes through the point (, 22). STEP 2 Translate the graph. For 522Ï } 2 2, h 5 and k 5 2. So, shift the graph of 522Ï } right units and up 2 units. The resulting graph starts at (, 2) and passes through (4, 0) (, 2) (0, 0) (4, 0) (, 22) 522 From the graph, ou can see that the domain of the function is and the range of the function is 2. at classzone.com E XAMPLE 5 Graph a translated cube root function Graph 5 Ï } 4 2. Then state the domain and range. STEP STEP 2 Sketch the graph of 5 Ï } (shown in blue). Notice that it passes through the origin and the points (2, 2) and (, ). Translate the graph. Note that for 5 Ï } 4 2, h 524 and k 52. So, shift the graph of 5 Ï } left 4 units and down unit. The resulting graph passes through the points (25, 24), (24, 2), and (2, 2). From the graph, ou can see that the domain and range of the function are both all real numbers. at classzone.com (24, 2) (25, 24) (2, 2) 5 (, ) (0, 0) (2, 2) 448 Chapter 6 Rational Eponents and Radical Functions

4 GUIDED PRACTICE for Eamples 4 and 5 Graph the function. Then state the domain and range Ï } Ï } 8. f() 5 } Ï } Ï } Ï } 2 5. g() 52Ï } EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es., 7, and 7 5 TAKS PRACTICE AND REASONING Es. 9, 25, 27, 7, 4, and 42 5 MULTIPLE REPRESENTATIONS E. 9. VOCABULARY Cop and complete: Square root functions and cube root functions are eamples of? functions. 2. WRITING The graph of 5 Ï } is the graph of 5 aï } 2 h k with a 5, h 5 0, and k 5 0. Predict how the graph of 5 Ï } will change if: a. a 52 b. h 5 2 c. k 5 4 EXAMPLE on p. 446 for Es. 9 SQUARE ROOT FUNCTIONS Graph the function. Then state the domain and range.. 524Ï } 4. f() 5 } 2 Ï } } 5 Ï } Ï } Ï } 8. g() 5 9Ï } 9. MULTIPLE TAKS REASONING CHOICE The graph of which function is shown? A 5 } Ï } 4 B 52} Ï } 4 C 5 } Ï } 2 D 52} Ï } 2 (0, 0) (4, 2) EXAMPLE 2 on p. 447 for Es. 0 5 CUBE ROOT FUNCTIONS Graph the function. Then state the domain and range } 4 Ï }. 5 2 Ï } 2. f() 525 Ï }. h() 52 } 7 Ï } 4. g() 5 6 Ï } } 9 Ï } EXAMPLES 4 and 5 on p. 448 for Es RADICAL FUNCTIONS Graph the function. Then state the domain and range. 6. f() 5 2Ï } ( ) / Ï } } 4 / Ï } h() 52 Ï } Ï } g() 52} Ï } Ï } SHORT TAKS REASONING RESPONSE Eplain wh there are limitations on the domain and range of the function 5 Ï } Graph Square Root and Cube Root Functions 449

5 26. ERROR ANALYSIS A student tried to eplain how the graphs of 522Ï } and 522Ï } 2 are related. Describe and correct the error. The graph of 522 Ï } 2 is the graph of 522 Ï } translated right unit and down units. 27. MULTIPLE CHOICE If the graph of 5 Ï } TAKS REASONING is shifted left 2 units, what is the equation of the translated graph? A 5 Ï } 2 2 B 5 Ï } 2 2 C 5 Ï } 2 D 5 Ï } 2 REASONING Find the domain and range of the function without graphing. Eplain how ou found our answers Ï } Ï } } Ï } } 2 Ï } 7 2. g() 5 Ï } 7. f() 5 } 4 Ï } CHALLENGE Graph 5 Ï 4 }, 5 Ï 5 }, 5 Ï 6 }, and 5 Ï 7 } on a graphing calculator. Make generalizations about the graph of 5 Ï n } when n is even and when n is odd. PROBLEM SOLVING EXAMPLE on p. 447 for Es INDIRECT MEASUREMENT The distance d (in miles) that a pilot can see to the horizon can be modeled b d 5.22Ï } a where a is the plane s altitude (in feet above sea level). Graph the model on a graphing calculator. Then determine at what altitude the pilot can see 8 miles. 6. PENDULUMS Use the model T 5.Ï l } for the period of a pendulum from Eample on page 447. a. Find the period of a pendulum with a length of 2 feet. b. Find the length of a pendulum with a period of 2 seconds. 7. SHORT TAKS REASONING RESPONSE The speed v (in meters per second) of sound waves in air depends on the temperature K (in kelvins) and can be modeled b: v 5.5Î } K } 27.5, K 0 a. Kelvin temperature K is related to Celsius temperature C b the formula K C. Write an equation that gives the speed v of sound waves in air as a function of the temperature C in degrees Celsius. b. What are a reasonable domain and range for the function from part (a)? 5 WORKED-OUT SOLUTIONS 450 Chapter 6 Rational p. WS Eponents and Radical Functions 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

6 8. DRAG RACING For a given total weight, the speed of a car at the end of a drag race is a function of the car s power. For a car with a total weight of 500 pounds, the speed s (in miles per hour) can be modeled b s Ï } p where p is the power (in horsepower). Graph the model. Then determine the power of a 500 pound car that reaches a speed of 200 miles per hour. 9. MULTIPLE REPRESENTATIONS Under certain conditions, a skdiver s terminal velocit v t (in feet per second) is given b v t 5.7Î } W }A where W is the weight of the skdiver (in pounds) and A is the skdiver s crosssectional surface area (in square feet). Note that skdivers can var their cross-sectional surface area b changing positions as the fall. a. Writing an Equation Write an equation that gives v t as a function of A for a skdiver who weighs 65 pounds. b. Making a Table Make a table of values for the equation from part (a). c. Drawing a Graph Use our table to graph the equation. 40. CHALLENGE The surface area S of a right circular cone with a slant height of unit is given b S 5 πr πr 2 where r is the cone s radius. a. Use completing the square to show the following: r 5 } Ï } p Ï } S p } 4 2 } 2 b. Graph the equation from part (a) using a graphing calculator. c. Find the radius of a right circular cone with a slant height of unit and a surface area of } p square units. 4 r unit MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 4.0; TAKS Workbook 4. TAKS PRACTICE Which equation best represents the relationship between and shown in the table? TAKS Obj. A B C D REVIEW Skills Review Handbook p. 996; TAKS Workbook 42. TAKS PRACTICE The two polgons are similar. What is the value of? TAKS Obj. 6 F 24 G 4 H 68 J 204 ( 2 72) EXTRA PRACTICE for Lesson 6.5, p Graph ONLINE Square Root QUIZ and at Cube classzone.com Root Functions 45