Exploring families of functions

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.
  • If , 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 ? Does it depend on ? Does it depend on ?
  • How does the increasing/decreasing behavior of f(x) depend on the value of ?
  • Does the concavity of f(x) depend on any of the parameters?