Some interesting equations for students to explore can be derived from quadratic functions of the form where the values of a, b, and c can vary to yield different graphs of a parabola. Students can examine these changing values to get a feel for just how the basic graph of can change. Of further interest is exploring how the graph is affected when an extra term is added to the equation, specifically the xy term.

A) Graph the quadratic function on Graphing Calculator 2.0 or other graphing program.

Describe the graph of this equation in terms of the opening of the parabola, horizontal or vertical shifts, vertex, and line of symmetry.

B) Now, on the same axes graph

Describe the new graph. Zoom out from the axes by changing the range on the y-axis from -25 to 25. Redraw and interpret the graph - what is it? Why do you think it became this type of graph? What do you think will happen when the coefficients of the xy term are changed? Click here to compare graphs.

C) Systematically try different coefficients for the xy term. Are they always the same types of curves? If not, what changes?

What happens when the xy coefficient is close to zero?

How does the sign of the coefficent change the graph?

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