140 CHAPTER 3 Linear and Quadratic Functions x ( ) = = − + 3 7 y f x x Average Rate of Change y x = Δ Δ 2− 13 10 13 1 2 3 1 3 ( ) − − − − = − = − 1− 10 7 10 0 1 3 1 3 ( ) − − − = − = − 0 7 3− 1 4 3− 2 1 3− 3 −2 Table 1 It is not a coincidence that the average rate of change of the linear function f x x3 7 ( ) = − + is the slope of the linear function. That is, y x m 3. Δ Δ = = − The following theorem states this fact. In Words The average rate of change of a linear function, whose graph is a line, equals the slope of the line. THEOREM Average Rate of Change of a Linear Function Linear functions have a constant average rate of change. The average rate of change of a linear function f x mx b ( ) = + is y x m Δ Δ = Proof The average rate of change of f x mx b ( ) = + from x1 to x x x , , 2 1 2 ≠ is y x f x f x x x mx b mx b x x mx mx x x m x x x x m 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 ( ) ( ) ( ) ( ) ( ) Δ Δ = − − = + − + − = − − = − − = ■ Based on the theorem just proved, the average rate of change of the function g x x 2 5 5 ( ) = − + is 2 5 . − Now Work PROBLEM 13(C) As it turns out, only linear functions have a constant average rate of change. Because of this, the average rate of change can be used to determine whether a function is linear. This is especially useful if the function is defined by a data set. Using the Average Rate of Change to Identify Linear Functions (a) A strain of E. coli known as Beu 397-recA441 is placed into a Petri dish at 30° Celsius and allowed to grow. The data shown in Table 2 are collected. The population is measured in grams and the time in hours. Plot the ordered pairs x y , ( ) in the Cartesian plane, and use the average rate of change to determine whether the function is linear. EXAMPLE 2
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