C 2 Before drawing any more graphs let s see what members of this family have in common Since 3 lim 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 lim x 1 x 2x 1 0 f x Now we compute the derivative When c 1 there are two vertical asymptotes x 1 3 lim x 1 x 1 x 2x c This shows that f x 0 when x maximum value f 1 0 For c 1 this means that f increases on the interval in interval notation if c 1 f x 0 when x 1 and f x 0 when x 1 the bottom graph above For c 1 f 1 0 Solving this equation we get x 1 and decreases on the interval in interval notation For c 1 there is an absolute is a local maximum value and the intervals of increase and decrease are interrupted at the vertical asymptotes When 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 0 as c 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 c 1 1