SECTION 2.3 Properties of Functions 95 Point–slope form of a secant line THEOREM Slope of a Secant Line The average rate of change of a function from a to b equals the slope of the secant line containing the two points a f a , ( ) ( ) and b f b , ( ) ( ) on its graph. Finding an Equation of a Secant Line Suppose that ( ) = − + g x x x 3 2 3. 2 (a) Find the average rate of change of g from −2 to 1. (b) Find an equation of the secant line containing ( ) ( ) − − g 2, 2 and ( ) ( ) g 1, 1. (c) Using a graphing utility, draw the graph of g and the secant line obtained in part (b) on the same screen. EXAMPLE 8 Solution (a) The average rate of change of ( ) = − + g x x x 3 2 3 2 from −2 to 1 is g g Average rate of change 1 2 1 2 4 19 3 15 3 5 ( ) ( ) ( ) = − − − − = − = − = − (b) The slope of the secant line containing ( ) ( ) ( ) − − = − g 2, 2 2, 19 and ( ) ( ) ( ) = g 1, 1 1, 4 is = − m 5. sec Use the point–slope form to find an equation of the secant line. y y m x x y x y x y x 19 5 2 19 5 10 5 9 1 sec 1 ( ) ( ) ( ) − = − − = − − − − = − − = − + Figure 36 Graph of g and the secant line g 1 3 1 2 1 3 4 2 ( ) ( ) = − ⋅ + = g 2 3 2 2 2 3 19 2 ( ) ( ) ( ) − = − − − + = x y g m 2, 2 19, 5 1 1 sec ( ) =− = − = =− Distribute. Slope–intercept form of the secant line Now Work PROBLEM 71 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 2.3 Assess Your Understanding 1. The interval 2, 5 ( ) can be written as the inequality . (pp. A77–A78) 2. The slope of the line containing the points 2, 3 ( ) − and 3, 8 ( ) is . (pp. 32–34) 3. Test the equation y x5 1 2 = − for symmetry with respect to the x -axis, the y -axis, and the origin. (pp. 21–23) 4. Write the point-slope form of the line with slope 5 containing the point 3, 2 . ( ) − (p. 36) 5. The intercepts of the graph of y x 9 2 = − are . (pp. 20–21) Concepts and Vocabulary 6. A function f is on an interval I if, for any choice of x1 and x2 in I, with x x , 1 2 < then f x f x . 1 2 ( ) ( ) < 7. A(n) function f is one for which f x f x ( ) ( ) − = for every x and x− in the domain of f; a(n) function f is one for which f x f x ( ) ( ) − = − for every x and x− in the domain of f. 8. True or False A function f is decreasing on an interval I if, for any choice of x1 and x2 in I, with x x , 1 2 < then f x f x . 1 2 ( ) ( ) > 9. True or False A function f has a local minimum at c if there is an open interval I containing c so that f c f x ( ) ( ) ≤ for all x in this open interval. 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure (c) Figure 36 shows the graph of g along with the secant line = − + y x5 9 using Geogebra.
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