The concavity of the graph of a quadratic function of the form

*f(x) = a x *^{ 2} + bx + c

and its second derivative are explored simultaneously in order to develop deep understanding of the concept of the concavity of a graph and its graphical meaning.

__Interactive Tutorial__

1 - Click on the button "click here to start" and maximize the window obtained. Three graphs are shown: in blue color the graph of the quadratic function f. In red color the tangent line to the graph of f and in black color the graph of the second derivative f '' which is drawn as the position of the tangent line is changed using the bottom slider ("Change Tangent Position").

1 - Use the top slider to set coefficient a to a positive value (a = 1 for example).

2 - Use the slider at the bottom of the left panel to change the position of the tangent line (red). Starting from the left moving to the right, does the slope of the tangent (first derivative) increase or decrease? Is the second derivative f '' positive or negative? Is the graph concave up or down? Calculate the first and second derivatives of function f and use the theorem on concavity to explain the above observations.

3 - Use the top slider to set coefficient a to a negative value (a = -2 for example).

4 - Use the slider at the bottom of the left panel to change the position of the tangent line (red). Starting from the left moving to the right, does the slope of the tangent (first derivative) increase or decrease? Is the second derivative f '' positive or negative? Is the graph concave up or down? Calculate the first and second derivatives of function f and use the theorem on concavity to explain the above observations.

5 - Set a to a certain value. Now use the sliders to change coefficients b and c. Does the concavity of the graph change? Explain.

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