369 3.4 Polynomial Functions: Graphs, Applications, and Models Approximations of Real Zeros EXAMPLE 7 Approximating Real Zeros of a Polynomial Function Approximate the real zeros of ƒ1x2 = x4 - 6x3 + 8x2 + 2x - 1. SOLUTION The dominating term is x4, so the graph will have end behavior similar to the graph of ƒ1x2 = x4, which is positive for all values of x with large absolute values. That is, the end behavior is up at the left and the right, . There are at most four real zeros because the polynomial is fourth-degree. Since ƒ102 = -1, the y-intercept is 10, -12. Because the end behavior is positive on the left and the right, by the intermediate value theorem ƒ has at least one real zero on either side of x = 0. The graph in Figure 36 shows that there are four real zeros, and the table indicates that they are between -1 and 0, 0 and 1, 2 and 3, and 3 and 4 because there is a sign change in ƒ1x2 = y1 in each case. Figure 37 shows that the negative zero is approximately -0.4142136. Similarly, we find that the other three zeros are approximately 0.26794919, 2.4142136, and 3.7320508. S Now Try Exercise 77. Polynomial Models EXAMPLE 8 Examining a Polynomial Model The table shows the average price, in dollars, of a pound of chocolate chip cookies from 2007 to 2018. (a) Using x = 0 to represent 2007, x = 1 to represent 2008, and so on, use the regression feature of a calculator to determine the quadratic function that best fits the data. Plot the data and the graph. (b) Repeat part (a) for a cubic function (degree 3). (c) Repeat part (a) for a quartic function (degree 4). (d) The correlation coefficient, R, is a measure of the strength of the relationship between two variables. The values of R and R2 are used to determine how well a regression model fits a set of data. The closer the value of R2 is to 1, the better the fit. Compare R2 for the three functions found in parts (a)–(c) to decide which function best fits the data. Year Price (in dollars) 2007 2.70 2008 2.88 2009 3.17 2010 3.25 2011 3.35 2012 3.62 2013 3.64 2014 3.41 2015 3.35 2016 3.35 2017 3.49 2018 3.50 Data from Consumer Price Index. −6.2 −4.7 6.2 4.7 Figure 36 Figure 37 −6.2 −4.7 6.2 4.7
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