Derivatives of Quadratic Functions

Derivative Rules for Quadratic Functions

In the App Adjust the values of the coefficients a, b, and c via the sliders or input boxes. Notice that when the graph of f(x) is increasing its derivative will be positive (above the x-axis), and when the graph of f(x) is decreasing its derivative will be negative (below the x-axis). When the graph of f(x) is at the extremum (maximum if a < 0 and minimum if a > 0), then the derivative will be 0 (on the x-axis). Show the graph and formula for the derivative by checking its checkbox. Can you see a general pattern for the formula of the derivative? Test out your conjecture by trying different values of a, b, and c. The graph and formula for f "(x) can also be displayed. Note that if a > 0 then the graph of the second derivative is always above the x-axis (positive valued) and the graph of the first derivative is increasing and the graph of the original function is concave up. If a < 0 then the second derivative is negative, the first derivative is decreasing, and the original function is concave down. Check Derivative Rule to see the shortcut rule for derivatives of quadratic functions. Check Proof to see an algebraic proof of this derivative rule.