Chapter 7 Rational Functions - College of the Redwoods

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630 Chapter 7 Rational Functions The Secant Line Consider the graph of the function f that we’ve drawn in Figure 3. Note that we’ve chosen two points on the graph of f, namely (a, f(a)) and (x, f(x)), and we’ve drawn a line L through them that mathematicians call the “secant line.” y (a, f(a)) (x, f(x)) L f x a x Figure 3. The secant line passes through (a, f(a)) and (x, f(x)). The slope of the secant line L is found by dividing the change in y by the change in x. Slope = ∆y f(x) − f(a) = ∆x x − a This slope provides the average rate of change of the variable y with respect to the variable x. Students in calculus use this “average rate of change” to develop the notion of “instantaneous rate of change.” However, we’ll leave that task for the calculus students and concentrate on the challenge of simplifying the expression equation (23) for the average rate of change. ◮ Example 24. Given the function f(x) = x 2 , simplify the expression for the average rate of change, namely f(x) − f(a) . x − a First, note that f(x) = x 2 and f(a) = a 2 , so we can write f(x) − f(a) x − a = x2 − a 2 x − a . We can now use the difference of two squares pattern to factor the numerator and cancel common factors. x 2 − a 2 x − a = (x + a)(x − a) x − a = x + a (23) Version: Fall 2007
Section 7.2 Reducing Rational Functions 631 Thus, provided, of course, that x ≠ a. f(x) − f(a) x − a = x + a, Let’s look at another example. ◮ Example 25. Consider the function f(x) = x 2 − 3x − 5. Simplify f(x) − f(2) . x − 2 First, f(x) = x 2 − 3x − 5 and therefore f(2) = (2) 2 − 3(2) − 5 = −7, so we can write f(x) − f(2) x − 2 = (x2 − 3x − 5) − (−7) x − 2 = x2 − 3x + 2 . x − 2 We can now factor the numerator and cancel common factors. Thus, x 2 − 3x + 2 x − 2 provided, of course, that x ≠ 2. = f(x) − f(2) x − 2 (x − 2)(x − 1) x − 2 = x − 1, = x − 1 Version: Fall 2007
Page 1 and 2: 7 Rational Functions In this chapte
Page 3 and 4: Section 7.1 Introducing Rational Fu
Page 19 and 20: Section 7.2 Reducing Rational Funct
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Page 39 and 40: Section 7.3 Graphing Rational Funct
Page 61 and 62: Section 7.4 Products and Quotients
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Section 7.6 Complex Fractions 695 7
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Section 7.9 Index 747 7.9 Index a a
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Chapter 7 Rational Functions - College of the Redwoods

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