332 CHAPTER 3 Polynomial and Rational Functions In the table on the preceding page, 3 represents 2003, 4 represents 2004, and so on, and the number of outpatient visits is given in millions. (a) Prepare a scatter diagram, and determine a quadratic model for these data. (b) Use the model from part (a) to predict the number of visits in 2020. SOLUTION (a) Linear regression is used to determine linear equations that model data. With a graphing calculator, we can use quadratic regression to find quadratic equations that model data. The scatter diagram in Figure 15(a) suggests that a quadratic function with a positive value of a (so the graph opens up) would be a reasonable model for the data. Using quadratic regression, the quadratic function ƒ1x2 = 0.2675x2 + 8.186x + 537.4 approximates the data well. See Figure 15(b). The quadratic regression values of a, b, and c are displayed in Figure 15(c). (b) The year 2020 corresponds to x = 20. ƒ1x2 = 0.2675x2 + 8.186x + 537.4 ƒ 1202 = 0.267512022 + 8.1861202 + 537.4 ƒ1202 ≈808 million The model predicts that there will be 808 million visits in 2020. S Now Try Exercise 77. f(x) = 0.2675x2 + 8.186x + 537.4 800 500 0 20 (b) (c) Figure 15 CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. A polynomial function with leading term 3x5 has degree . 2. The lowest point on the graph of a parabola that opens up is the of the parabola. 3. The highest point on the graph of a parabola that opens down is the of the parabola. 4. The axis of symmetry of the graph of ƒ1x2 = 21x + 422 - 6 has equation x = . 5. The vertex of the graph of ƒ1x2 = x2 + 2x + 4 has x-coordinate . 6. The graph of ƒ1x2 = -2x2 - 6x + 5 opens down with y-intercept 10, 2, so it has x-intercept(s). (no / one / two) 3.1 Exercises CONCEPT PREVIEW Match each equation in Column I with the description of the parabola that is its graph in Column II. I 7. y = 1x + 422 + 2 8. y = 1x + 222 + 4 9. y = -1x + 422 + 2 10. y = -1x + 222 + 4 II A. vertex 1-2, 42, opens up B. vertex 1-2, 42, opens down C. vertex 1-4, 22, opens up D. vertex 1-4, 22, opens down 800 500 0 20 (a)
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