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8-3 Graphing Reciprocal Functions Identify the asymptotes, domain, and range of each function. 5. ANSWER: 1. ANSWER: x = 1, f (x) = 0; D = {x | x 0} 1}; R = {f (x) | f (x) Graph each function. State the domain and range. Identify the asymptotes, domain, and range of each function. 3. ANSWER: 7. ANSWER: x = –4, f (x) = 0; D = {x | x 0} –4}; R = {f (x) | f (x) 5. ANSWER: 9. ANSWER: x = –6, f (x) = –2; D = {x | x –2} –6}; R = {f (x) | f (x) eSolutions Manual - Powered by Cognero Graph each function. State the domain and range. Page 1 ANSWER: x = –6, f (x) = –2; D = {x | x –6}; R = {f (x) | f (x) –2} 8-3 Graphing Reciprocal Functions D = {x | x 6}; R = {f (x) | f (x) 0} Graph each function. State the domain and range. 15. 11. ANSWER: ANSWER: D = {x | x 0}; R = {f (x) | f (x) 3} D = {x | x 0}; R = {f (x) | f (x) 0} 17. ANSWER: 13. ANSWER: D = {x | x 5}; R = {f (x) | f (x) 0} D = {x | x 6}; R = {f (x) | f (x) 0} 19. ANSWER: 15. ANSWER: D = {x | x eSolutions Manual - Powered by Cognero D = {x | x 0}; R = {f (x) | f (x) 3} –3}; R = {f (x) | f (x) 6} Page 2 D = {x | x Reciprocal (x) | f (x) 0} 5}; R = {fFunctions 8-3 Graphing D = {x | x –4}; R = {f (x) | f (x) –2} 23. CYCLING Marina’s New Year’s resolution is to ride her bike 5000 miles. 19. a. If m represents the mileage Marina rides each day and d represents the number of days, write an equation to represent the mileage each day as a function of the number of days that she rides. ANSWER: b. Graph the function. c. If she rides her bike every day of the year, how many miles should she ride each day to meet her goal? D = {x | x –3}; R = {f (x) | f (x) ANSWER: 6} a. 21. b. ANSWER: c. 13.7 mi D = {x | x –4}; R = {f (x) | f (x) –2} Graph each function. State the domain and range. 23. CYCLING Marina’s New Year’s resolution is to ride her bike 5000 miles. a. If m represents the mileage Marina rides each day and d represents the number of days, write an equation to represent the mileage each day as a function of the number of days that she rides. 25. ANSWER: b. Graph the function. c. If she rides her bike every day of the year, how many miles should she ride each day to meet her goal? ANSWER: eSolutions Manual - Powered by Cognero a. D = {x | x 2}; R = {f (x) | f (x) 0} Page 3 D= c. 13.7 mi Reciprocal Functions 8-3 Graphing R = {f (x) | f (x) 0} Graph each function. State the domain and range. 29. BASEBALL The distance from the pitcher’s mound to home plate is 60.5 feet. a. If r represents the speed of the pitch and t represents the time it takes the ball to get to the plate, write an equation to represent the speed as a function of time. 25. ANSWER: b. Graph the function. c. If a two-seam fastball reaches the plate in 0.48 second, what was its speed? ANSWER: a. D = {x | x 2}; R = {f (x) | f (x) 0} b. 27. ANSWER: c. 126 ft/s Graph each function. State the domain and range, and identify the asymptotes. D= 31. R = {f (x) | f (x) 0} 29. BASEBALL The distance from the pitcher’s mound to home plate is 60.5 feet. ANSWER: D = {x | x –2}; R = {f (x) | f (x) (x) = –5 –5}; x = –2, f a. If r represents the speed of the pitch and t represents the time it takes the ball to get to the plate, write an equation to represent the speed as a function of time. b. Graph the function. eSolutions - Powered by Cognero c. IfManual a two-seam fastball reaches second, what was its speed? the plate in 0.48 Page 4 c. 126 ft/s Reciprocal Functions 8-3 Graphing Graph each function. State the domain and range, and identify the asymptotes. 35. ANSWER: D = {x | x 7}; R = {f (x) | f (x) –8 31. ANSWER: D = {x | x –2}; R = {f (x) | f (x) (x) = –5 –8}; x = 7, f (x) = –5}; x = –2, f 37. MULTIPLE REPRESENTATIONS Consider the functions and 33. a. TABULAR Make a table of values comparing the two functions. ANSWER: D = {x | x 4}; R = {f (x) | f (x) 3 3}; x = 4, f (x) = b. GRAPHICAL Use the table of values to graph both functions. c. VERBAL Compare and contrast the two graphs. d. ANALYTICAL Make a conjecture about the difference between the graphs of functions of the form with an even exponent in the denominator and those with an odd exponent in the denominator. ANSWER: a. 35. ANSWER: D = {x | x 7}; R = {f (x) | f (x) –8 –8}; x = 7, f (x) = eSolutions Manual - Powered by Cognero Page 5 denominator and those with an odd exponent in the denominator. 8-3 Graphing Reciprocal Functions ANSWER: a. d. Sample answer: When n is even, the graph will show symmetry with respect to the y-axis. When the n is odd, the graph will show symmetry with respect to the origin. 39. REASONING Compare and contrast the graphs of each pair of equations. a. b. c. d. Without making a table of values, use what you observed in parts a-c to sketch a graph of b. ANSWER: a. The first graph has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The second graph is translated 7 units up and has a vertical asymptote at x = 0 and a horizontal asymptote at y = 7. b. Both graphs have a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The second graph is stretched by a factor of 4. c. The positive portion of graph of is similar to the Positive values of x produce positive values of f (x). The negative portion of appears to be a reflection of c. The first graph has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The second graph is translated 5 units to the left and has a vertical asymptote at x = –5 and a horizontal asymptote at y = 0. d. over the x-axis. Negative values of x produce positive values of g(x). d. Sample answer: When n is even, the graph will show symmetry with respect to the y-axis. When the n is odd, the graph will show symmetry with respect to the origin. 39. REASONING Compare and contrast the graphs of each pair of equations. a. eSolutions Manual - Powered by Cognero 41. CHALLENGE Write two different reciprocal Page 6 functions with graphs having the same vertical and horizontal asymptotes. Then graph the functions. ANSWER: B 8-3 Graphing Reciprocal Functions 41. CHALLENGE Write two different reciprocal functions with graphs having the same vertical and horizontal asymptotes. Then graph the functions. 47. If –1 < a < b < 0, then which of the following has the greatest value? Aa–b ANSWER: Bb–a and Sample answer: Ca+b D 2b – a ANSWER: B Simplifying each expression. 49. 43. SHORT RESPONSE What is the value of (x + y) 2 2 (x + y) if xy = –3 and x + y = 10? ANSWER: –2p ANSWER: 4 45. If and d > 1, then c could equal ___. 51. A ANSWER: B C Graph each function. State the domain and range. x D 53. y = 5(2) ANSWER: D = {x | x is all real numbers}, R = {y | y > 0} ANSWER: B eSolutions by Cognero < a <- bPowered < 0, then which 47. If –1Manual greatest value? of the following has the Page 7 ANSWER: 8-3 Graphing Reciprocal Functions Graph each function. State the domain and range. Find (f + g)(x), (f - g)(x), (f · g)(x), and for each f (x) and g(x). x 53. y = 5(2) ANSWER: D = {x | x is all real numbers}, R = {y | y > 0} 57. ANSWER: 55. ANSWER: D = {x | x is all real numbers}, R = {y | y > 0} 59. GEOMETRY The width of a rectangular prism is w centimeters. The height is 2 centimeters less than the width. The length is 4 centimeters more than the width. If the volume of the prism is 8 times the measure of the length, find the dimensions of the prism. ANSWER: Find (f + g)(x), (f - g)(x), (f · g)(x), and for each f (x) and g(x). Graph each polynomial function. Estimate the xcoordinates at which the relative maxima and relative minima occur. State the domain and range for each function. 57. 4 2 61. f (x) = x – 8x + 10 ANSWER: ANSWER: eSolutions Manual - Powered by Cognero Page 8 ANSWER: 8-3 Graphing Reciprocal Functions Graph each polynomial function. Estimate the xcoordinates at which the relative maxima and relative minima occur. State the domain and range for each function. 4 2 61. f (x) = x – 8x + 10 ANSWER: Sample answer = rel. max. at x = 0, rel. min. at x = – 2 and at x = 2; D = {all real numbers}, R = {f (x) | f (x) –6} eSolutions Manual - Powered by Cognero Page 9