241 2.4 Linear Functions The slope of a line can be found only if the line is nonvertical. This guarantees that x2 ≠x1 so that the denominator x2 - x1 ≠0. Slope Formula The slope m of the line through the points 1x1, y12 and 1x2, y22 is defined as follows. m= rise run = y x = y2 −y1 x2 −x1 , where x 30 That is, the slope of a line is the change in y divided by the corresponding change in x, where the change in x is not 0. LOOKING AHEAD TO CALCULUS The concept of slope of a line is extended in calculus to general curves. The slope of a curve at a point is understood to mean the slope of the line tangent to the curve at that point. The line in the figure is tangent to the curve at point P. P f(x) x 0 CAUTION When using the slope formula, it makes no difference which point is 1x1, y12 or 1x2, y22. However, be consistent. Start with the x- and y-values of one point (either one), and subtract the corresponding values of the other point. Use y2 - y1 x2 - x1 or y1 - y2 x1 - x2 , not y2 - y1 x1 - x2 or y1 - y2 x2 - x1 . Be sure to write the difference of the y-values in the numerator and the difference of the x-values in the denominator. EXAMPLE 5 Finding Slopes with the Slope Formula Find the slope of the line through the given points. (a) 1-4, 82, 12, -32 (b) 12, 72, 12, -42 (c) 15, -32, 1-2, -32 SOLUTION (a) Let x1 = -4, y1 = 8, and x2 = 2, y2 = -3. m= rise run = Δy Δx Definition of slope = -3 - 8 2 - 1-42 = -11 6 , or - 11 6 Subtract; -a b = - a b We can also subtract in the opposite order, letting x1 = 2, y1 = -3 and x2 = -4, y2 = 8. The same slope results. m= 8 - 1-32 -4 - 2 = 11 -6 , or - 11 6 (b) If we attempt to use the slope formula with the points 12, 72 and 12, -42, we obtain a 0 denominator—that is, Δx = x2 - x1 = 2 - 2 = 0. m= -4 - 7 2 - 2 = -11 0 Undefined A sketch would show that the line through 12, 72 and 12, -42 is vertical. The slope of a vertical line is undefined. Substitute carefully.
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