This construction explores functions [color=#ff0000]f(x)[/color] and their antiderivatives [color=#0000ff]F(x)[/color] for which [color=#ff0000]f(x)[/color] may be expressed as the reciprocal of a quadratic function.[br][br]Here, [color=#ff0000]f(x)[/color] = 1/(x² – 4x + [b]c[/b]), where the value of [b]c[/b] may be controlled by a slider.[br][br]Suggested exploration:[br][list][*]Observe the effect on the graph of [color=#ff0000]f(x)[/color] for different values of [b]c[/b]. Which values cause there to be two vertical asymptotes? One vertical asymptote? No vertical asymptotes?[/*][*]Why does the number of vertical asymptotes vary for these [b]c[/b] values? How would you generalize your observation for other quadratic denominators of [color=#ff0000]f(x)[/color]?[/*][*]Display the graph (parabola) of [color=#e69138]1/f(x)[/color], the quadratic denominator. Make observations of how the graph of [color=#e69138]1/f(x)[/color] varies with [b]c[/b] that support your observations about [color=#ff0000]f(x)[/color].[/*][*]Display the graph/equation for the function [color=#0000ff]F(x)[/color]. Observe how the type of function(s) that appear in the [color=#0000ff]F(x)[/color] equation varies with different [b]c[/b] values.[/*][*]If you haven't already done so, make sure you can analytically determine the different types of displayed [color=#0000ff]F(x)[/color] equations for yourself. What Calculus techniques did each type of equation require?[/*][*]For some [b]c[/b] values, an unlabeled slider appears under the "Antiderivative F" checkbox. Split that [color=#0000ff]F(x)[/color] graph into two separate branches by sliding that slider to the right. Is the derivative/antiderivative relationship maintained throughout the range of that slider?[/*][*]Return to the [b]c[/b] slider and once again observe the effects of varying the [b]c[/b] values.[/*][*]Even though various types of function make up [color=#0000ff]F(x)[/color] depending on the c value, do you observe the strong visual similarities in their graphs?[/*][/list]