12 Analyzing Graphs of Functions and Relations


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1 Use the graph of each function to estimate the indicated function values. Then confirm the estimate algebraically. Round to the nearest hundredth, if necessary. The function value at x = 1 appears to be about 4. To confirm this estimate algebraically, find h( 1). 3. a. f ( 8) b. f ( 3) c. f (0) The function value at x = 8 appears to be about 10. To confirm this estimate algebraically, find f ( 8). The function value at x = 1.5 appears to be about. Find h(1.5). The function value at x = 3 appears to be about 5. Find f ( 3). The function value at x = 0 appears to be about 2. Find f (0). The function value at x = 2 appears to be about. Find h(2). 6. a. h( 1) b. h(1.5) c. h(2) esolutions Manual  Powered by Cognero Page 1
2 Use the graph of h to find the domain and range of each function. 9. The arrows on the left and right sides of the graph indicate that the graph will continue without bound in both directions. Therefore, the domain of h is (, ). The graph does not extend below h(0) or 2, but h(x) increases without bound for lesser and greater values of x. So, the range of h is [2, ). a. State the domain and range of each function. b. Use the graph to estimate the impact energy of each metal at 0 C. a. The arrows on the left and right sides of the graph that corresponds to the copper specimen indicate that the graph will continue without bound. So, the domain is all real numbers, or [, ]. The impact energy of the specimen for all temperatures from 150ºC to 150ºC appears to be about 1.75 joules, so the range is [1.75] The arrows on the left and right sides of the graph that corresponds to the aluminum specimen indicate that the graph will continue without bound. So, the domain is [, ]. The graph does not extend above f( 100) = 1.5 or below f (125) = 0.6. So, the range is [0.6, 1.5]. 12. The closed dot at (7, 1) indicates that x = 7 is in the domain of h. Although there is an open dot at (4, 1), the closed dot at (4, 1) indicates that x = 4 is in the domain of h. The arrow to the left indicates that the graph will continue without bound. Therefore, the domain of h is (, 7]. The closed dots at (4, 1) and (7, 1) indicate that y = 1 is in the range of h. The closed dot at (5, 1) indicates that y = 1 is not in the range of h. The arrow to the left indicates that the graph will continue without bound. Therefore, the range of h is [ 1] (1, ). 15. ENGINEERING Tests on the physical behavior of four metal specimens are performed at various temperatures in degrees Celsius. The impact energy, or energy absorbed by the sample during the test, is measured in Joules. The test results are shown. The arrows on the left and right sides of the graph that corresponds to the zinc specimen indicate that the graph will continue without bound. So, the domain is [, ]. The graph does not extend above f( 100) = 0.5 or below f (100) = 1.3. So, the range is [0.5, 1.3]. The arrows on the left and right sides of the graph that corresponds to the steel specimen indicate that the graph will continue without bound. So, the domain is [, ]. The graph does not extend above f( 125) = 0.2 or below f (100) = So, the range is [0.2, 1.75]. Note that absolute zero, the coldest temperature possible, is about C. Therefore, in the context of this problem, the true domain is [ , ]. b. Estimate the function value at x = 0 for each curve. esolutions Manual  Powered by Cognero Page 2
3 When x = 0, the copper, aluminum, zinc, and steel specimens appear to have impact energies of about 1.75 J, 1.2 J, 0.5 J, and 1.5 J, respectively. Use the graph of each function to find its y intercept and zero(s). Then find these values algebraically. 21. From the graph, it appears that f (x) intersects the y axis at approximately (0, 2), so the yintercept is 2. Find f (0). Therefore, the yintercept is 2. From the graph, the xintercepts appear to be at about and. Let f (x) = 0 and solve for x. 18. From the graph, it appears that f (x) intersects the y axis at approximately (0, 0), so the yintercept is 0. Find f (0). Therefore, the yintercept is 0. From the graph, it appears that there is an x intercept at 0. Let f (x) = 0 and solve for x. Therefore, the zeros of f are and. Use the graph of each equation to test for symmetry with respect to the xaxis, yaxis, and the origin. Support the answer numerically. Then confirm algebraically. Therefore, the zero of f is The graph appears to be symmetric with respect to the xaxis, yaxis, and origin because there appears to be mirror images about the axes and the origin. Also, for every point (x, y) on the graph, there is a point (x, y), a point ( x, y), and a point ( x, y), respectively. Make a table of values to support each part of this conjecture. esolutions Manual  Powered by Cognero Page 3
4 The positive yvalues produce the same xvalues as their corresponding yvalues, so x 2 + 4( y) 2 = 16 is equivalent to x 2 + 4y 2 = 16 and the graph is symmetric with respect to the xaxis. Because ( x) 2 + 4( y) 2 = 16 is equivalent to x 2 + 4y 2 = 16, the graph is symmetric with respect to the origin. ( y) 2 is equidistant to y 2, so x 2 + 4( y) 2 = 16 is equivalent to x 2 + 4y 2 = The graph appears to be symmetric with respect to the xaxis, yaxis, and origin because there appears to be mirror images about all three. Also, for every point (x, y) on the graph, there is a point (x, y), a point ( x, y), and a point ( x, y), respectively. Make a table of values for each part of this conjecture. The positive xvalues produce the same yvalues as their corresponding xvalues, so ( x) 2 + 4y 2 = 16 is equivalent to x 2 + 4y 2 = 16 and the graph is symmetric with respect to the yaxis. ( x) 2 is equidistant to x 2, so ( x) 2 + 4y 2 = 16 is equivalent to x 2 + 4y 2 = 16. esolutions Manual  Powered by Cognero Page 4
5 The positive yvalues produce the same xvalues as their corresponding yvalues, so 9x 2 25( y) 2 = 1 is equivalent to 9x 2 25y 2 = 1 and the graph is symmetric with respect to the xaxis. 25y 2 =1, the graph is symmetric with respect to the origin. ( y) 2 is equivalent to y 2, so 9x 2 25( y) 2 = 1 is equivalent to 9x 2 25y 2 = The graph does not appear to be symmetric with respect to the xaxis, yaxis, or origin because there are no mirror images with respect to these areas of the graph. Because neither y = x 3 2x 2 + 3x 4, y = x 3 2x 2 3x 4, nor y = x 3 2x 2 3x 4 are equivalent to y = x 3 2x 2 + 3x 4, the graph is not symmetric with respect to the xaxis, yaxis, or origin, respectively. The positive xvalues produce the same yvalues as their corresponding xvalues, so 9( x) 2 25y 2 = 1 is equivalent to 9x 2 25y 2 =1 and the graph is symmetric with respect to the yaxis. ( x) 2 is equivalent to x 2, so 9( x) 2 25y 2 = 1 is equivalent to 9x 2 25y 2 =1. Because 9( x) 2 25( y) 2 = 1 is equivalent to 9x 2 esolutions Manual  Powered by Cognero Page 5
6 GRAPHING CALCULATOR Graph each function. Analyze the graph to determine whether each function is even, odd, or neither. If odd or even, describe the symmetry of the graph of the function. 36. g(x) = 33. The graph appears to be symmetric with respect to the yaxis because there appears to be a mirror image about the yaxis. Also, for every point (x, y) on the graph, there is a point ( x, y). Make a table of values to support this conjecture. It does not appear that the graph of the function is symmetric with respect to the xaxis, yaxis, or origin. Test this conjecture. The function is neither even nor odd because g(x) g(x) and g( x) g(x). The positive xvalues produce the same yvalues as their corresponding xvalues, so (y 6) 2 + 8( x) 2 = 64 is equivalent to (y 6) 2 + 8x 2 = 64 and the graph is symmetric with respect to the yaxis. ( x) 2 is equivalent to x 2, so (y 6) 2 + 8( x) 2 = 64 is equivalent to (y 6) 2 + 8x 2 = 64. esolutions Manual  Powered by Cognero Page 6
7 39. f (x) = x 3 Use the graph of each function to estimate the indicated function values. It appears that the graph of the function is symmetric with respect to the yaxis. Test this conjecture. 42. a. f ( 2) b. f ( 6) c. f (0) a. The function value at x = 2 appears to be 2. b. The circle at ( 6, 8) indicates that 6 is not part of the domain of the function. Therefore, the function value at x = 6 is undefined. The function is even because f ( x) = f (x). Therefore, the graph of the function is symmetric with respect to the yaxis. c. The domain of the function is ( 6, 2] [2, ). Because 0 is not in the domain of the function, the function value at x = 0 is undefined. 45. PHONES The number of households h in millions with only wireless phone service from 2001 to 2005 can be modeled by h(x) = 0.5x x + 1.2, where x represents the number of years after a. State the relevant domain and approximate the range. b. Use the graph to estimate the number of households with only wireless phone service in Then find it algebraically. c. Use the graph to approximate the yintercept of the function. Then find it algebraically. What does the yintercept represent? d. Does this function have any zeros? If so, estimate esolutions Manual  Powered by Cognero Page 7
8 them and explain their meaning. If not, explain why. a. Because x represents years since 2001 and the model is only valid from 2001 to 2005, the relevant domain of the function is {x 0 x 4, x } or [0, 4]. The graph does not appear to extend below f (0) = 1.2 or above f (4) Therefore, the range of the function is approximately R = {y 1.2 million y 11.2 million, y } or [1.2, 11.2]. b. Because 2003 is 2 years after 2001, x = 2. Sample answer: From the graph, it appears that the function value when x = 2 is about 4.1. Find h(2). c. Sample answer: From the graph, it appears that the yintercept is about 1.1. Find h(0). GRAPHING CALCULATOR Graph each function and locate the zeros. Confirm your answers algebraically. 48. From the graph, there appears to be an xintercept at about 4. Let h(x) = 0 and solve for x. Isolate the radical. The yintercept represents the number of millions of households with only wireless service in d. Sample answer: There are no zeros because there were more than zero households with only wireless phone service for all of the years in the domain. 51. From the graph, the xintercepts appear to be at about 1 and 3. Let h(x) = 0 and solve for x. esolutions Manual  Powered by Cognero Page 8
13 Continuity, End Behavior, and Limits
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