9.1 Exercises. Section 9.1 The Square Root Function 879. In Exercises 1-10, complete each of the following tasks.

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1 Section 9. The Square Root Function Eercises In Eercises -, complete each of the following tasks. i. Set up a coordinate sstem on a sheet of graph paper. Label and scale each ais. ii. Complete the table of points for the given function. Plot each of the points on our coordinate sstem, then use them to help draw the graph of the given function. iii. Use different colored pencils to project all points onto the - and -aes to determine the domain and range. Use interval notation to describe the domain of the given function.. f() = f(). f() = f() 3. f() = + 7 f() 5. f() = f() 6. f() = f() 7. f() = f() 8. f() = f() 9. f() = f(). f() = f() 4. f() = f() Coprighted material. See:

2 880 Chapter 9 Radical Functions In Eercises -0, perform each of the following tasks. i. Set up a coordinate sstem on a sheet of graph paper. Label and scale each ais. Remember to draw all lines with a ruler. ii. Use geometric transformations to draw the graph of the given function on our coordinate sstem without the use of a graphing calculator. Note: You ma check our solution with our calculator, but ou should be able to produce the graph without the use of our calculator. iii. Use different colored pencils to project the points on the graph of the function onto the - and -aes. Use interval notation to describe the domain and range of the function.. f() = + 3. f() = f() = 4. f() = 5. f() = f() = 7. f() = f() = f() = f() = + 3. To draw the graph of the function f() = 3, perform each of the following steps in sequence without the aid of a calculator. i. Set up a coordinate sstem and sketch the graph of =. Label the graph with its equation. ii. Set up a second coordinate sstem and sketch the graph of =. Label the graph with its equation. iii. Set up a third coordinate sstem and sketch the graph of = ( 3). Label the graph with its equation. This is the graph of f() = 3. Use interval notation to state the domain and range of this function.. To draw the graph of the function f() = 3, perform each of the following steps in sequence. i. Set up a coordinate sstem and sketch the graph of =. Label the graph with its equation. ii. Set up a second coordinate sstem and sketch the graph of =. Label the graph with its equation. iii. Set up a third coordinate sstem and sketch the graph of = ( + 3). Label the graph with its equation. This is the graph of f() = 3. Use interval notation to state the domain and range of this function. 3. To draw the graph of the function f() =, perform each of the following steps in sequence without the aid of a calculator. i. Set up a coordinate sstem and sketch the graph of =. Label the graph with its equation. ii. Set up a second coordinate sstem and sketch the graph of =. Label the graph with its equation. iii. Set up a third coordinate sstem and sketch the graph of = ( + ). Label the graph with its equation. This is the graph of f() =. Use interval notation to state the domain and range of this function.

3 Section 9. The Square Root Function To draw the graph of the function f() =, perform each of the following steps in sequence. i. Set up a coordinate sstem and sketch the graph of =. Label the graph with its equation. ii. Set up a second coordinate sstem and sketch the graph of =. Label the graph with its equation. iii. Set up a third coordinate sstem and sketch the graph of = ( ). Label the graph with its equation. This is the graph of f() =. Use interval notation to state the domain and range of this function. In Eercises 5-8, perform each of the following tasks. i. Draw the graph of the given function with our graphing calculator. Cop the image in our viewing window onto our homework paper. Label and scale each ais with min, ma, min, and ma. Label our graph with its equation. Use the graph to determine the domain of the function and describe the domain with interval notation. ii. Use a purel algebraic approach to determine the domain of the given function. Use interval notation to describe our result. Does it agree with the graphical result from part (i)? In Eercises 9-40, find the domain of the given function algebraicall. 9. f() = f() = f() = f() = f() = f() = f() = f() = f() = f() = f() = f() = f() = f() = 7 7. f() = 4 8. f() = +

4 88 Chapter 9 Radical Functions 9. Answers. Domain = [0, ), Range = (, 0] f() Domain = [0, ), Range = [, ) f() f()= + f()= 3. Domain = [, ), Range = [0, ). 7 f() 0 3 f()= + 7. Domain = [ 3, ), Range = [, ). 3 6 f() f()= +3+

5 Section 9. The Square Root Function Domain = (, 3], Range = [0, ). 5. Domain = [ 5, ), Range = [, ). 6 3 f() 3 0 f()= 3 f()= Domain = [ 4, ), Range = (, 0].. Domain = [0, ), Range = [3, ). f()= +3 f()= Domain = [0, ), Range = (, 3]. 3. Domain = [, ), Range = [0, ). f()= f()= +3

6 884 Chapter 9 Radical Functions. Domain = (, 3], Range = [0, ). 7. Domain = (, 3] f()= 4 f()= Domain = (, ], Range = [0, ). f()= [ 9, ) ( ], 3 8 ( ], 4 3 ( ], 7 [, ) ( ], Domain = [ 7/, ) f()=

7 Section 9. Multiplication Properties of Radicals Eercises. Use a calculator to first approimate 5. On the same screen, approimate. Report the results on our homework paper.. Use a calculator to first approimate 7. On the same screen, approimate 70. Report the results on our homework paper Use a calculator to first approimate 3. On the same screen, approimate 33. Report the results on our homework paper. 4. Use a calculator to first approimate 5 3. On the same screen, approimate 65. Report the results on our homework paper. In Eercises 5-0, place each of the radical epressions in simple radical form. As in Eample 3 in the narrative, check our result with our calculator In Eercises -6, use prime factorization (as in Eamples and in the narrative) to assist ou in placing the given radical epression in simple radical form. Check our result with our calculator In Eercises 7-46, place each of the given radical epressions in simple radical form. Make no assumptions about the sign of the variables. Variables can either represent positive or negative numbers (6 ) 4 Coprighted material. See:

8 90 Chapter 9 Radical Functions h 8 5f 5j 8 6m 5a (7 + 5) 9w f 4 (3 + 6) (9 8) for =. 49. Given that < 0, place the radical epression 7 in simple radical form. Check our solution on our calculator for =. 50. Given that < 0, place the radical epression 44 in simple radical form. Check our solution on our calculator for =. In Eercises 5-54, follow the lead of Eample 7 in the narrative to simplif the given radical epression and check our result with our graphing calculator. 5. Given that < 4, place the radical epression in simple radical form. Use a graphing calculator to show that the graphs of the original epression and our simple radical form agree for all values of such that < e 4p 5 5q 6 6h 47. Given that < 0, place the radical epression 3 6 in simple radical form. Check our solution on our calculator for =. 48. Given that < 0, place the radical epression 54 8 in simple radical form. Check our solution on our calculator 5. Given that, place the radical epression in simple radical form. Use a graphing calculator to show that the graphs of the original epression and our simple radical form agree for all values of such that. 53. Given that 5, place the radical epression + 5 in simple radical form. Use a graphing calculator to show that the graphs of the original epression and our simple radical form agree for all values of such that Given that <, place the radical epression + + in simple radical form. Use a graphing calculator to show that the graphs of the original epression and our simple radical form agree for all values of such that <.

9 Section 9. Multiplication Properties of Radicals 903 In Eercises 55-7, place each radical epression in simple radical form. Assume that all variables represent positive numbers. In Eercises 73-80, place each given radical epression in simple radical form. Assume that all variables represent positive numbers d f 5 8f k 74. 3s 3 43s k 7 3k n 9 8n j 77. e 9 8e j n 9 5n m 79. 3z 5 7z e t 7 7t c z 65. 5h 66. 5b 67. 9s e p d q 7. 4w 7

10 904 Chapter 9 Radical Functions 9. Answers m 33. (7 + 5) (3 + 6) p 4 p 45. 5q q (6 ) 9. 5 f

11 Section 9. Multiplication Properties of Radicals The graphs of = + 4 and = follow. Note that the agree for < p q f k e z The graphs of = 5 and = + 5 follow. Note that the agree for d 6 d j 5 j 6. 5m 63. c c 65. 5h s 3 s

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13 Section 9.3 Division Properties of Radicals Eercises. Use a calculator to first approimate 5/. On the same screen, approimate 5/. Report the results on our homework paper.. Use a calculator to first approimate 7/ 5. On the same screen, approimate 7/5. Report the results on our homework paper. 3. Use a calculator to first approimate /. On the same screen, approimate 6. Report the results on our homework paper. 4. Use a calculator to first approimate 5/ 5. On the same screen, approimate 3. Report the results on our homework paper. In Eercises 5-6, place each radical epression in simple radical form. As in Eample in the narrative, check our result with our calculator In Eercises 7-8, place each radical epression in simple radical form. As in Eample 4 in the narrative, check our result with our calculator Coprighted material. See:

14 9 Chapter 9 Radical Functions In Eercises 9-36, place the given radical epression in simple form. Use prime factorization as in Eample 8 in the narrative to help ou with the calculations. As in Eample 6, check our result with our calculator In Eercises 37-44, place each of the given radical epressions in simple radical form. Make no assumptions about the sign of an variable. Variables can represent either positive or negative numbers In Eercises 45-48, follow the lead of Eample 8 in the narrative to craft a solution. 45. Given that < 0, place the radical epression 6/ 6 in simple radical form. Check our solution on our calculator for =. 46. Given that > 0, place the radical epression 4/ 3 in simple radical form. Check our solution on our calculator for =.

15 Section 9.3 Division Properties of Radicals Given that > 0, place the radical epression 8/ 8 5 in simple radical form. Check our solution on our calculator for =. 48. Given that < 0, place the radical epression 5/ 0 6 in simple radical form. Check our solution on our calculator for =. In Eercises 49-56, place each of the radical epressions in simple form. Assume that all variables represent positive numbers

16 94 Chapter 9 Radical Functions 9.3 Answers /50 30/4 330/ / / 4. /( 4 ) 43. 5/( 4 ) /4 55/ / / / 5. 5 / 9. /6 53. /(). 3/ /6 5. 7/ /(3 ) /6 5/. 3 / / /4 9. 6/4

17 Section 9.4 Radical Epressions Eercises In Eercises -4, place each of the radical epressions in simple radical form. Check our answer with our calculator. 8. 3(4 3 ) 9. ( + ). (5 7). 3( 3) 3. 3( 5) (4 6) ( + 4) 3( 5 33) (3 7) 3(5 6) ( 3 ) In Eercises 3-30, combine like terms. Place our final answer in simple radical form. Check our solution with our calculator. 7. ( 5)( 3 3) 8. ( 5 )( 7) 9. ( 4 3)( 6) ( 5)( 3 ) ( 3). ( 3 5) 3. ( 5 ) 4. (7 ) In Eercises 5-, use the distributive propert to multipl. Place our final answer in simple radical form. Check our result with our calculator In Eercises 3-40, combine like terms where possible. Place our final answer in simple radical form. Use our calculator to check our result. 5. (3 + 5) (4 7) 7. ( ) Coprighted material. See:

18 94 Chapter 9 Radical Functions In Eercises 4-48, simplif each of the given rational epressions. Place our final answer in simple radical form. Check our result with our calculator In Eercises 49-60, multipl to epand each of the given radical epressions. Place our final answer in simple radical form. Use our calculator to check our result. 49. ( + 3)(3 3) 50. (5 + )( ) 5. (4 + 3 )( 5 ) 5. ( )( 3) 53. ( + 3 )( 3 ) 54. (3 + 5)(3 5) 55. ( )( 3 3 ) 56. (8 + 5)(8 5) 57. ( + 5) 58. (3 ) 59. ( 3 5) 60. ( ) In Eercises 6-68, place each of the given rational epressions in simple radical form b rationalizing the denominator. Check our result with our calculator

19 Section 9.4 Radical Epressions In Eercises 69-76, use the quadratic formula to find the solutions of the given equation. Place our solutions in simple radical form and reduce our solutions to lowest terms = = 78. Given f() = +, evaluate the epression f() f(3), 3 and then rationalize the numerator. 79. Given f() =, evaluate the epression f( + h) f(), h and then rationalize the numerator. 80. Given f() = 3, evaluate the epression f( + h) f(), h and then rationalize the numerator = = = = = = 40 In Eercises 77-80, we will suspend the usual rule that ou should rationalize the denominator. Instead, just this one time, rationalize the numerator of the resulting epression. 77. Given f() =, evaluate the epression f() f(), and then rationalize the numerator.

20 944 Chapter 9 Radical Functions 9.4 Answers / (4 ± 3)/3 7. ( ± 6)/5 73. (3 ± 3)/7 75. ± h /

21 Section 9.5 Radical Equations Eercises For the rational functions in Eercises - 6, perform each of the following tasks. i. Load the function f and the line = k into our graphing calculator. Adjust the viewing window so that all point(s) of intersection of the two graphs are visible in our viewing window. ii. Cop the image in our viewing window onto our homework paper. Label and scale each ais with min, ma, min, and ma. Label the graphs with their equations. Remember to draw all lines with a ruler. iii. Use the intersect utilit to determine the coordinates of the point(s) of intersection. Plot the point of intersection on our homework paper and label it with its coordinates. iv. Solve the equation f() = k algebraicall. Place our work and solution net to our graph. Do the solutions agree?. f() = + 3, k =. f() = 4, k = 3 3. f() = 7, k = 4 4. f() = 3 + 5, k = 5 5. f() = 5 +, k = 4 6. f() = 4, k = 5 In Eercises 7-, use an algebraic technique to solve the given equation. Check our solutions = = = 3 + = = 3 For the rational functions in Eercises 3-6, perform each of the following tasks. i. Load the function f and the line = k into our graphing calculator. Adjust the viewing window so that all point(s) of intersection of the two graphs are visible in our viewing window. ii. Cop the image in our viewing window onto our homework paper. Label and scale each ais with min, ma, min, and ma. Label the graphs with their equations. Remember to draw all lines with a ruler. iii. Use the intersect utilit to determine the coordinates of the point(s) of intersection. Plot the point of intersection on our homework paper and label it with its coordinates. iv. Solve the equation f() = k algebraicall. Place our work and solution net to our graph. Do the solutions agree? 3. f() = + 3 +, k = 9 4. f() = + 6, k = 4 5. f() = 5, k = = 6. f() = + 5 +, k = 7 Coprighted material. See:

22 956 Chapter 9 Radical Functions In Eercises 7-4, use an algebraic technique to solve the given equation. Check our solutions = = = = + 5 = = = = 5 For the rational functions in Eercises 5-8, perform each of the following tasks. i. Load the function f and the line = k into our graphing calculator. Adjust the viewing window so that all point(s) of intersection of the two graphs are visible in our viewing window. ii. Cop the image in our viewing window onto our homework paper. Label and scale each ais with min, ma, min, and ma. Label the graphs with their equations. Remember to draw all lines with a ruler. iii. Use the intersect utilit to determine the coordinates of the point(s) of intersection. Plot the point of intersection on our homework paper and label it with its coordinates. iv. Solve the equation f() = k algebraicall. Place our work and solution net to our graph. Do the solutions agree? 7. f() = , k = 8. f() = , k = In Eercises 9-40, use an algebraic technique to solve the given equation. Check our solutions = = = = 49 = = = = = = = = 5 5. f() = + + 6, k = 7 6. f() = , k = 7

23 Section 9.5 Radical Equations Answers. = (,) f()= +3 = = 6 0 f()= +3+ (6,9) =9 3. = 9/ 0 f()= 7 ( 4.5,4) =4 5. = = 0 (,4) f()= +5 =4 (9, 7) = 7 0 f()=

24 958 Chapter 9 Radical Functions 3. 8, 9 5. = 0 f()= + +6 (,7) = = 0 f()= (,) =

25 Section 9.6 The Pthagorean Theorem Eercises In Eercises -8, state whether or not the given triple is a Pthagorean Triple. Give a reason for our answer... (8, 5, 7). (7, 4, 5) 3. (8, 9, 7) 4. (4, 9, 3). 5. (, 35, 37) 6. (, 7, 9) 7. (, 7, 8) (, 60, 6) In Eercises 9-6, set up an equation to model the problem constraints and solve. Use our answer to find the missing side of the given right triangle. Include a sketch with our solution and check our result Coprighted material. See:

26 97 Chapter 9 Radical Functions 4. legs One leg of a right triangle is 3 feet longer than 3 times the length of the first leg. The length of the hpotenuse is 5 feet. Find the lengths of the legs. 5.. Pthagoras is credited with the following formulae that can be used to generate Pthagorean Triples a = m b = m, c = m Use the technique of Eample 6 to demonstrate that the formulae given above will generate Pthagorean Triples, provided that m is an odd positive integer larger than one. Secondl, generate at least 3 instances of Pthagorean Triples with Pthagoras s formula. In Eercises 7-0, set up an equation that models the problem constraints. Solve the equation and use the result to answer the question. Look back and check our result. 7. The legs of a right triangle are consecutive positive integers. The hpotenuse has length 5. What are the lengths of the legs? 8. The legs of a right triangle are consecutive even integers. The hpotenuse has length. What are the lengths of the legs?. Plato (380 BC) is credited with the following formulae that can be used to generate Pthagorean Triples. a = m b = m, c = m + Use the technique of Eample 6 to demonstrate that the formulae given above will generate Pthagorean Triples, provided that m is a positive integer larger than. Secondl, generate at least 3 instances of Pthagorean Triples with Plato s formula. 9. One leg of a right triangle is centimeter less than twice the length of the first leg. If the length of the hpotenuse is 7 centimeters, find the lengths of the

27 Section 9.6 The Pthagorean Theorem 973 In Eercises 3-8, set up an equation that models the problem constraints. Solve the equation and use the result to answer the question. Look back and check our result. 3. Fritz and Greta are planting a - foot b 8-foot rectangular garden, and are laing it out using string. The would like to know the length of a diagonal to make sure that right angles are formed. Find the length of a diagonal. Approimate our answer to within 0. feet. 4. Angelina and Markos are planting a 0-foot b 8-foot rectangular garden, and are laing it out using string. The would like to know the length of a diagonal to make sure that right angles are formed. Find the length of a diagonal. Approimate our answer to within 0. feet. 5. The base of a 36-foot long gu wire is located 6 feet from the base of the telephone pole that it is anchoring. How high up the pole does the gu wire reach? Approimate our answer to within 0. feet. 6. The base of a 35-foot long gu wire is located feet from the base of the telephone pole that it is anchoring. How high up the pole does the gu wire reach? Approimate our answer to within 0. feet. 7. A stereo receiver is in a corner of a 3-foot b 6-foot rectangular room. Speaker wire will run under a rug, diagonall, to a speaker in the far corner. If 3 feet of slack is required on each end, how long a piece of wire should be purchased? Approimate our answer to within 0. feet. 8. A stereo receiver is in a corner of a -foot b 5-foot rectangular room. Speaker wire will run under a rug, diagonall, to a speaker in the far corner. If 4 feet of slack is required on each end, how long a piece of wire should be purchased? Approimate our answer to within 0. feet. In Eercises 9-38, use the distance formula to find the eact distance between the given points. 9. ( 8, 9) and (6, 6) 30. (, 0) and ( 9, ) 3. ( 9, ) and ( 8, 7) 3. (0, 9) and (3, ) 33. (6, 5) and ( 9, ) 34. ( 9, 6) and (, 4) 35. ( 7, 7) and ( 3, 6) 36. ( 7, 6) and (, 4) 37. (4, 3) and ( 9, 6) 38. ( 7, ) and (4, 5) In Eercises 39-4, set up an equation that models the problem constraints. Solve the equation and use the result to answer the question. Look back and check our result. 39. Find k so that the point (4, k) is units awa from the point (, ). 40. Find k so hat the point (k, ) is units awa from the point (0, ).

28 974 Chapter 9 Radical Functions 4. Find k so that the point (k, ) is 7 units awa from the point (, 3). 4. Find k so that the point (, k) is 3 units awa from the point ( 4, 3). 43. Set up a coordinate sstem on a sheet of graph paper. Label and scale each ais. Plot the points P (0, 5) and Q(4, 3) on our coordinate sstem. a) Plot several points that are equidistant from the points P and Q on our coordinate sstem. What graph do ou get if ou plot all points that are equidistant from the points P and Q? Determine the equation of the graph b eamining the resulting image on our coordinate sstem. b) Use the distance formula to find the equation of the graph of all points that are equidistant from the points P and Q. Hint: Let (, ) represent an arbitrar point on the graph of all points equidistant from points P and Q. Calculate the distances from the point (, ) to the points P and Q separatel, then set them equal and simplif the resulting equation. Note that this analtical approach should provide an equation that matches that found b the graphical approach in part (a). 44. Set up a coordinate sstem on a sheet of graph paper. Label and scale each ais. Plot the point P (0, ) and label it with its coordinates. Draw the line = and label it with its equation. a) Plot several points that are equidistant from the point P and the line = on our coordinate sstem. What graph do ou get if ou plot all points that are equidistant from the points P and the line =. b) Use the distance formula to find the equation of the graph of all points that are equidistant from the points P and the line =. Hint: Let (, ) represent an arbitrar point on the graph of all points equidistant from points P and the line =. Calculate the distances from the point (, ) to the points P and the line = separatel, then set them equal and simplif the resulting equation.

29 Section 9.6 The Pthagorean Theorem Cop the following figure onto a sheet of graph paper. Cut the pieces of the first figure out with a pair of scissors, then rearrange them to form the second figure. Eplain how this proves the Pthagorean Theorem. 46. Compare this image to the one that follows and eplain how this proves the Pthagorean Theorem. b a a c c b b c c a a b b a b c b a c a b a

30 976 Chapter 9 Radical Functions 9.6 Answers. Yes, because = 7 3. No, because Yes, because + 35 = a) In the figure that follows, XP = XQ. 7. No, because P (0,5) X(,) =(/) The legs have lengths 3 and The legs have lengths 8 and 5 centimeters.. (3, 4, 5), (5,, 3), and (7, 4, 5), with m = 3, 5, and 7, respectivel ft ft ft 5 b) = (/) Q(4, 3) = = k = 3,. 4. k =, 3.