By Bell E.T.
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Additional resources for Periodic recurring series
1 Lines y L P2(x2, y2) P1(x1, y1) ⌬y ϭ y2 Ϫ y1 (rise) ⌬x ϭ x2 Ϫ x1 (run) The quantity ⌬y ϭ y2 Ϫ y1 (⌬y is read “delta y”) measures the change in y from P1 to P2 and is called the rise; the quantity ⌬x ϭ x 2 Ϫ x 1 measures the change in x from P1 to P2 and is called the run. Thus, the slope of a line is the ratio of its rise to its run. Since the ratios of corresponding sides of similar triangles are equal, we see from Figure 3 that the slope of a line is independent of the two distinct points that are used to compute it; that is, x 0 3 y2 Ϫ y1 y 2œ Ϫ y 1œ ϭ œ x2 Ϫ x1 x 2 Ϫ x 1œ mϭ FIGURE 2 The slope of the line L is ⌬y y2 Ϫ y1 rise mϭ .
In this case we refer to the function f as a real-valued function of a real variable. We can think of a function f as a machine or processor. In this analogy the domain of f consists of the set of “inputs,” the rule describes how the “inputs” are to be processed by the machine, and the range is made up of the set of “outputs” (see Figure 1). FIGURE 1 A function machine x Input f f(x) Output Processor As an example, consider the function that associates with each nonnegative number x its square root, 1x.
60. 3x Ϫ 4y ϭ 8 61. x Ϫ 3 ϭ 0 and 6x Ϫ 8y ϭ 10 and y Ϫ 5 ϭ 0 62. 2x Ϫ 3y Ϫ 12 ϭ 0 63. y x ϩ ϭ1 a b and and 3x ϩ 2y Ϫ 6 ϭ 0 y x Ϫ ϭ1 a b In Exercises 64–65, find the point of intersection of the lines with the given equations. 35. A(Ϫ3, 6), B(3, 3), and C(6, 0) 36. 2x Ϫ 3y Ϫ 12 ϭ 0 58. Passes through (3, Ϫ4) and is perpendicular to the line through (Ϫ1, 2) and (3, 6) In Exercises 46–59, find an equation of the line satisfying the conditions. Write your answer in the slope-intercept form. 46. Is perpendicular to the x-axis and passes through the point (p, p2) 47.