142 CHAPTER 3 Linear and Quadratic Functions Figure 3 Age Heartbeats x y 40 45 50 70 60 50 40 20 30 Age, x Maximum Number of Heartbeats, y y x = Δ Δ Average Rate of Change 20 50 47.5 50 30 20 0.25 − − = − 30 47.5 0.25 − 40 45 0.25 − 50 42.5 0.25 − 60 40 0.25 − 70 37.5 Table 5 Now Work PROBLEM 21 3 Determine Whether a Linear Function Is Increasing, Decreasing, or Constant When the slope m of a linear function is positive m 0 , ( ) > the line slants upward from left to right.When the slope m of a linear function is negative m 0 , ( ) < the line slants downward from left to right. When the slope m of a linear function is zero m 0 , ( ) = the line is horizontal. THEOREM Increasing, Decreasing, and Constant Linear Functions A linear function f x mx b ( ) = + is increasing over its domain if its slope, m, is positive. It is decreasing over its domain if its slope, m, is negative. It is constant over its domain if its slope, m, is zero. Now Work PROBLEM 13(d) Determining Whether a Linear Function Is Increasing, Decreasing, or Constant Determine whether the following linear functions are increasing, decreasing, or constant. (a) f x x5 2 ( ) = − (b) g x x2 8 ( ) = − + (c) s t t 3 4 4 ( ) = − (d) h z 7 ( ) = Solution EXAMPLE 3 (a) The linear function f x x5 2 ( ) = − has slope 5, which is positive. The function f is increasing on the interval , . ( ) −∞ ∞ (b) The linear function g x x2 8 ( ) = − + has slope 2, − which is negative. The function g is decreasing on the interval , . ( ) −∞ ∞ (c) The linear function s t t 3 4 4 ( ) = − has slope 3 4 , which is positive. The function s is increasing on the interval , . ( ) −∞ ∞ (d) The linear function h can be written as h z z0 7. ( ) = + Because the slope is 0, the function h is constant on the interval , . ( ) −∞ ∞
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