339 3.1 Quadratic Functions and Models 78. Private College Enrollment The table lists total fall enrollments in private colleges in the United States for selected years. (a) Plot the data. Let x = 0 correspond to the year 2000. (b) Find a quadratic function g1x2 = ax2 + bx + c that models the data. (c) Plot the data together with g in the same window. How well does g model enrollment? (d) Use g to estimate total enrollment in 2020 to the nearest tenth of a million. 79. Foreign-Born Americans The table lists the percent of the U.S. population that was foreign-born for selected years. (a) Plot the data. Let x = 0 correspond to the year 1930, x = 10 correspond to 1940, and so on. (b) Find a quadratic function ƒ1x2 = a1x - h22 + k that models the data. Use 140, 4.72 as the vertex and 120, 6.92 as the other point to determine a. (c) Plot the data together with ƒ in the same window. How well does ƒ model the percent of the U.S. population that is foreign-born? (d) Use the quadratic regression feature of a graphing calculator to determine the quadratic function g that provides the best fit for the data. (e) Use functions ƒ and g to predict the percent, to the nearest tenth, of the U.S. population in 2022 that will be foreign-born. 80. Automobile Stopping Distance Selected values of the stopping distance y, in feet, of a car traveling x miles per hour are given in the table. (a) Plot the data. (b) The quadratic function ƒ1x2 = 0.056057x2 + 1.06657x is one model that has been used to approximate stopping distances. Find ƒ1452 to the nearest foot, and interpret this result. (c) How well does ƒ model the car’s stopping distance? Year Enrollment (in millions) 2000 3.6 2004 4.3 2008 5.1 2012 5.8 2016 5.3 Data from National Center for Education Statistics. Year Percent 1930 11.6 1940 8.8 1950 6.9 1960 5.4 1970 4.7 1980 6.2 1990 7.9 2000 10.4 2010 12.9 2017 13.7 Data from U.S. Census Bureau. Speed (in mph) Stopping Distance (in feet) 20 46 30 87 40 140 50 240 60 282 70 371 Data from National Safety Institute Student Workbook. Concept Check Work each problem. 81. Find a value of c so that y = x2 - 10x + c has exactly one x-intercept. 82. For what values of a does y = ax2 - 8x + 4 have no x-intercepts? 83. Define the quadratic function ƒ having x-intercepts 12, 02 and 15, 02 and y-intercept 10, 52. 84. Define the quadratic function ƒ having x-intercepts 11, 02 and 1-2, 02 and y-intercept 10, 42. 85. The distance between the two points P1x1, y12 and R1x2, y22 is d1P, R2 = 21x1 - x222 + 1y 1 - y222. Distance formula Find the closest point on the line y = 2x to the point 11, 72. (Hint: Every point on y = 2x has the form 1x, 2x2, and the closest point has the minimum distance.)
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