SECTION 3.1 Properties of Linear Functions and Linear Models 141 (b) The data in Table 3 represent the maximum number of heartbeats that a healthy individual of different ages should have during a 15-second interval of time while exercising. Plot the ordered pairs x y , ( ) in the Cartesian plane, and use the average rate of change to determine whether the function is linear. 0 1 2 3 4 5 (0, 0.09) (1, 0.12) (2, 0.16) (3, 0.22) (4, 0.29) (5, 0.39) 0.09 0.12 0.16 0.22 0.29 0.39 Time (hours), x Population (grams), y (x, y) Table 2 Age, x (x, y) Maximum Number of Heartbeats, y 20 30 40 50 60 70 50 47.5 45 42.5 40 37.5 (20, 50) (30, 47.5) (40, 45) (50, 42.5) (60, 40) (70, 37.5) Source: American Heart Association. Table 3 Solution Compute the average rate of change of each function. If the average rate of change is constant, the function is linear. If the average rate of change is not constant, the function is nonlinear. (a) Figure 2 shows the points listed in Table 2 plotted in the Cartesian plane. Note that it is impossible to draw a straight line that contains all the points. Table 4 displays the average rate of change of the population. Figure 2 Time (hours) Population (grams) x y 0.1 0.2 0.3 0.4 2 3 4 5 0 1 Time (hours), x Population (grams), y y x = Δ Δ Average Rate of Change 0 0.09 0.12 0.09 1 0 0.03 − − = 1 0.12 0.04 2 0.16 0.06 3 0.22 0.07 4 0.29 0.10 5 0.39 Table 4 Because the average rate of change is not constant, the function is not linear. In fact, because the average rate of change is increasing as the value of the independent variable increases, the function is increasing at an increasing rate. So not only is the population increasing over time, but it is also growing more rapidly as time passes. (b) Figure 3 on the next page shows the points listed in Table 3 plotted in the Cartesian plane. Table 5 on the next page displays the average rate of change of the maximum number of heartbeats.The average rate of change of the heartbeat data is constant, 0.25 − beat per year, so the function is linear, and the points in Figure 3 lie on a line. (continued)
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