210 CHAPTER 4 Polynomial and Rational Functions SUMMARY Steps for Using a Graphing Utility to Analyze the Graph of a Polynomial Function Step 1 Determine the end behavior of the graph of the function. Step 2 Graph the function using a graphing utility. Step 3 Use the graphing utility to approximate the x- and y-intercepts of the graph. Step 4 Use the graphing utility to create a TABLE to find points on the graph around each x-intercept. Step 5 Approximate the turning points of the graph. Step 6 Use the information in Steps 1 through 5 to draw a complete graph of the function by hand. Step 7 From the graph, find the range of the polynomial function. Step 8 Use the graph to determine where the function is increasing and where it is decreasing. The range of f is the set of all real numbers. Step 7 From the graph, find the range of the polynomial function. Based on the graph, f is increasing on the intervals , 2.28 ( ] −∞ − and 0.63, . [ )∞ Also, f is decreasing on the interval 2.28, 0.63 . [ ] − Step 8 Use the graph to determine where the function is increasing and where it is decreasing. Now Work PROBLEM 23 2 Build Cubic Models from Data In Section 3.2 we found the line of best fit from data, and in Section 3.4 we found the quadratic function of best fit. It is also possible to find other polynomial functions of best fit. However, most statisticians do not recommend finding polynomials of best fit of degree higher than 3. Data that follow a cubic relation should look like Figure 21(a) or (b). Figure 21 Cubic relation y 5 ax3 1 bx2 1 cx 1 d, a . 0 (a) y 5 ax3 1 bx2 1 cx 1 d, a , 0 (b) A Cubic Function of Best Fit The data in Table 6 on the next page represent the weekly cost C (in thousands of dollars) of producing x thousand insulated tumblers. (a) Draw a scatter plot of the data using x as the independent variable and C as the dependent variable. Comment on the type of relation that may exist between the two variables x and C. (b) Using a graphing utility, find the cubic function of best fit C C x( ) = that models the relation between number of tumblers and cost. (c) Graph the cubic function of best fit on your scatter plot. (d) Use the function found in part (b) to predict the cost of producing 22 thousand tumblers per week. EXAMPLE 4
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