Exploring families of functions

Topic:
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?