Similar Derivatives
This construction explores functions f(x) and their antiderivatives F(x) for which f(x) may be expressed as the reciprocal of a quadratic function.
Here, f(x) = 1/(x² – 4x + c), where the value of c may be controlled by a slider.
Suggested exploration:
- Observe the effect on the graph of f(x) for different values of c. 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 c values? How would you generalize your observation for other quadratic denominators of f(x)?
- Display the graph (parabola) of 1/f(x), the quadratic denominator. Make observations of how the graph of 1/f(x) varies with c that support your observations about f(x).
- Display the graph/equation for the function F(x). Observe how the type of function(s) that appear in the F(x) equation varies with different c values.
- If you haven't already done so, make sure you can analytically determine the different types of displayed F(x) equations for yourself. What Calculus techniques did each type of equation require?
- For some c values, an unlabeled slider appears under the "Antiderivative F" checkbox. Split that F(x) 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 c slider and once again observe the effects of varying the c values.
- Even though various types of function make up F(x) depending on the c value, do you observe the strong visual similarities in their graphs?