This is the general understanding which we
have concluded from our discussions earlier: that b tells us where the axis will be and the relationship is b+Axis- & b-Axis+. I hope that is simple enough to reckon easily. And we knew that c gave us the y-intercept i.e. where the graph cuts the y-axis but additionally and very significantly, c gives us additional information about the sign of the roots, alpha & beta viz
c+both roots + or – (depending on axis i.e. depending on b)
c-roots of opposite signs
This is shown in the diagrams
b+ b – and
c+ c – getting us
b – c –
This can be worked out by the student on his own. However, all the possibilities are shown in the diagram above. Please redraw the graphs on your own and convince yourself why what is shown herein are correct. In these diagrams, instead of starting from b, we have started by assuming the axes to be in the positive or the negative direction and have deduced the sign of b given that a is negative.
Let us reiterate that you need not memorize anything but be able to discern the patterns and deduce the correct conclusions by a simple analysis of b and c being positive and negative and how it affects the roots and the axis.