C=-2 Before drawing any more graphs, let's see what members of this family have in common. Since 3 lim = 0 x x + 2x + c for any value of c, they all have the x-axis as a horizontal asymptote. A vertical asymptote will occur when c> 1, there is no vertical asymptote (the top graph above). When c = 1, the graph has a single vertical asymptote x = -1 because 3 x-1 (x + 1)2 1 (x + 1) = 80. 3 lim lim x-1 x + 2x + 1 When c < 1, there are two vertical asymptotes: x = -1 (the bottom graph above). Now we compute the derivative. f'(x) = (x2+2x+ + 2x + c) = 0. Solving this equation, we get x = -1 ]). When This shows that f'(x) = 0 when x = (if c-1), f'(x) > 0 when x < -1, and f'(x) < 0 when x > -1. For c 1, this means that f increases on the interval, in interval notation, and decreases on the interval, in interval notation, . For c> 1, there is an absolute maximum value f(-1) = . For c < 1, f(-1) = is a local maximum value and the intervals of increase and decrease are interrupted at the vertical asymptotes. The figures below are a "slide show" displaying five members of the family, all graphed in the viewing rectangle [-5, 4] by [-6, 6]. As predicted, c = 1 is a value at which a transition takes place from co. As c decreases two vertical asymptotes to one and then to none. As c increases from 1, we see that the maximum point becomes lower; this is explained by the fact that. 1 - 0 as c-> (c- 1)