Applet for exploring families of functions, Section 3.2 in Active Calculus.
Drag the sliders back and forth to watch the parabola change. Pay close attention to the increasing/decreasing behavior, concavity, and critical values of the function. [list] [*] If [math]a \neq 0[/math], does f(x) always have a critical value? If so, what does its x-coordinate always equal? [*] Does the location of the critical value depend on the value of the parameter [math]a[/math]? Does it depend on [math]k[/math]? Does it depend on [math]h[/math]? [*] How does the increasing/decreasing behavior of f(x) depend on the value of [math]a[/math]? [*] Does the concavity of f(x) depend on any of the parameters? [/list]