This Write-Up will demonstrate the use of Graphing Calculator plus Algebra to discover how to take a parabola, switch the direction from up to down, move it to quadrant 2, and finally to make a parabola that is upside down with the same vertex as the original parabola.
First, we graph a parabola:
, which is shown in purple. Then we
try to turn it upside down, by changing the sign of each term.
Changing the sign of c from -4 to +4 is
shown in red. 
Changing the sign of b is shown in blue.
Finally, changing the sign of a works, and
the green parabola, upside down to our original purple parapola,
results. 
Now, from studying the graphs, we know that changing the b term from +3 to -3 moved the original parabola in a positive direction. We need to move our green parabola in a negative direction in terms of the x axis, and we also need to move it in a negative direction in terms of the y axis.
Let's deal with the x axis first, and try
changing b. Let's do a new graph so that it looks less confusing.
Now our original parabola 
is still purple, but the green parabola
in the above drawing which took us to the desired upside down
configuration 
is now red, and our "experiment"
parabola changing the b term, is blue 
.
Reducing the b term seems to be working, bringing our parabola in a more negative direction both in terms of the x axis and the y axis.
Next, we try changing the sign of a in the general quadratic
formula 
and discover that changing the sign
of a does turn the parabola upside down.
Next, I tried changing the size of the a:
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