To explore the graphs of the parabola, I began by graphing the equation

for different values of a, b, and c. Approaching this enterprise in 3
steps, I fixed two of the values for a, b, and c, and varied the third,
making at least 5 graphs on the same axes. See the examples below.

By examining several "families" of such equations for which
a and b are constant in each such family, while c varies, we notice that
the curves all are alined on a single vertical axis. The value for c always
is the y coordinate for the y-intercept since evaluating for x=0 will give
y=c as the result. The effects of varyining the coefficients a and b are
demonstrated and discussed in the following examples. (This exploration
was actually best accomplished on the graphing calculator; the same was
true for the two which follow by allowing either a or b to vary and observing
the results.)

Note that the line is the graph for the equation when a=0. The graphs for the equations when a< 0 are concave upward and lie completely above the line where a=0. When a>0, they are concave downward and lie completely below that line. Again, each graph interscects the y-axis only at the point (0,c), that is, (0,2) in this example.

This is , perhaps, the most difficult effect describe; I will explore
it further in assignment 3! when b = 0, the curve passes through the y-axis
at its vertex. For positive values of b, the curve moves to the right and
downward; for negative values of b, to the left and downward. As shown below,
for a<0, varying the values of b move the curve up and right for positive
values and up and left for negative values.

For demonstrations in a classroom setting, I would prefer to graph each
of these families one graph at a time, adding the new graphs to the others
until the pattern is perceived. Otherwise, it is difficult to see what the
impact of each change is. I would definitlely like to demonstrate these
concepts on the graphing calculator. In fact, I would also use the graphing
calculator to "back-up" and display several graphs of the following.

In the first case, we would again observe that when a (now n) is 0, the graph is a horizontal line, when a (now n) is positive the curve is concave upward, and when negative, downward. The larger the coefficient, in absolute value, the tighter the curve; the smaller, the flatter. In the second demonstration, we observe a more predictable way to shift the curves to right and left with positive and negative, respectively, values of n.

Note that both graphs pass through the x and y axes at the same points, as would be expected from the algebra of replacing either x or y with 0. This will remain true for all the variations explored below.

Note that as the coefficient goes from 1 to 3 1/2, the two "halves" of the curve appear more pointed and the part in the 3rd quadrant appears to approach the y-axis (vertical) with the point close to the y-intercept. In the next graph, the effect of letting the coefficient increase from 3 3/4. (I wonder exactly where the boundary is? Somewhere between 3 1/2 and 3 3/4?)

Note that as the coefficient increases from 3 3/4, again the two "halves" of the curve appear more pointed . But here the curves have flipped to an upper right and lower left part.

Here, as n gets closer to 0 the top part of the curve gets close to the parabloa where n=0 while the bottom part appears to slide away into the 4th quadrant.

That is, values of n between 0 (where the graph is a circle) and 2 (but not including 2)

These are the graphs of the same equations as above, except that the coefficient, n, of the xy terms are negative instead of positive:

I hope that you have enjoyed this exploration into a small corner of the world of quadratic equations and their graphs.

by Jackie Cohen