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