 Explore interactively the derivative of quadratic functions as defined in calculus.

A quadratic function of the form

f(x) = ax 2 + b x + c

and its first derivative are explored simultaneously in order to gain deep understanding of the concept of the derivative and also its graphical meaning.

#### Interactive Tutorial

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1 - Click on the button "click here to start" and maximize the window obtained. Three graphs are displayed: in blue the graph of f; in red the tangent line to the graph of f and in black the graph of the first derivative f ' of f.

2 - Use the left bottom slider ("Change Tangent Position") to change the position of the tangent line and observe as the graph of f ' is drawn. Why is the graph of f ' linear? Find the general formula for
f '(x).

3 - Set parameter a to a positive value and move the tangent line so that it is positioned at the local minimum. The slope of the tangent is approximately equal to zero. What is the value of
f'(x) at that point? What is the sign of f'(x) to the left of that point? What is the sign of f'(x) to the right of that point? What is the relationship between the sign of f'(x) and the fact that f is increasing or decreasing?

4 - Set parameter a to a negative value and move the tangent (red) so that it is positioned at the local maximum. Repeat the same activity as in part 3 above.

5 - The general formula for the derivative of
f(x) = ax 2 + b x + c is given by f '(x) = 2 ax + b. Show that the value of x that makes f '(x) = 0 is equal to - b / 2a which is the x coordinate of the vertex of a a parabola. Use different values of parameters a and b and compare the theoretical values to the graphically approximated values of the x coordinate of the vertex (maximum or minimum).
More on derivatives:

Derivatives of Polynomial Functions. The derivative of third order polynomial functions are explored interactively and graphically.

Derivatives of Sine (sin x) Functions. The derivative of sine functions are explored interactively.

Derivative of tan(x). The derivative of tan (x) is explored interactively to understand the behaviour of the tangent line close to a vertical asymptote.

Vertical Tangent. The derivative of f(x) = x 1 / 3 is explored interactively to understand the concept of vertical tangent.