260 CHAPTER 2 Graphs and Functions (a) Find an equation that models the data. (b) Use the equation from part (a) to estimate the cost of tuition and fees for in-state students at public four-year colleges in 2019. SOLUTION (a) The points in Figure 51 lie approximately on a straight line, so we can write a linear equation that models the relationship between year x and cost y. We choose two data points, 10, 80702 and 13, 87782, to find the slope of the line. m= 8778 - 8070 3 - 0 = 708 3 = 236 The slope 236 indicates that the cost of tuition and fees increased by about $236 per year from 2013 to 2017. We use this slope, the y-intercept 10, 80702, and the slope-intercept form to write an equation of the line. y = mx + b Slope-intercept form y = 236x + 8070 Substitute for m and b. (b) The value x = 6 corresponds to the year 2019, so we substitute 6 for x. y = 236x + 8070 Model from part (a) y = 236162 + 8070 Let x = 6. y = 9486 Multiply, and then add. The model estimates that average tuition and fees for in-state students at public four-year colleges in 2019 were about $9486. S Now Try Exercise 63(a) and (b). NOTE In Example 7, if we had chosen different data points, we would have obtained a slightly different equation. Guidelines for Modeling Step 1 Make a scatter diagram of the data. Step 2 Find an equation that models the data. For a line, this involves selecting two data points and finding the equation of the line through them. Linear regression is a technique from statistics that provides the line of “best fit.” Figure 52 shows how a TI-84 Plus calculator accepts the data points, calculates the equation of this line of best fit (in this case, y1 = 193.4x + 8114.6), and plots the data points and line on the same screen. Figure 52 (a) (b) −1 0 10,000 5 (c) 7
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