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 & bAxis+. I hope that is simple enough to reckon easily. And we knew that c gave us the yintercept i.e. where the graph cuts the yaxis 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)
croots of opposite signs
This is shown in the diagrams
below
The reader may think that we are obsessed with these characteristics to be showing these diagrams again and again and in every which way. Let us then reiterate the importance of aesthetics in mathematics. If you are able to see the beauty in these graphical shapes, the elegance of its dimensions, the symmetry of its shape, the symmetrical position of alpha and beta about the axis of the quadratic curve, if you see it not merely as mathematics but also as a work of art drawn by Nature itself because if you realize, we have not constructed anything here of our own. We have simply taken a second degree algebraic expression and turned it into a graph and are merely observing what comes up. It is a discovery, a voyage into a mathematical territory which has geometry and beauty and elegance and symmetry. In short, it is fascinating journey into what Nature has drawn out for us to discover.
A summary: Once Again 🙂
See the elegant U shaped curve once more showing the four possibilities out of
b+ b – and
c+ c – getting us
4 cases:
b+c+
b+c –
b c+
b – c –
b+ b – and
c+ c – getting us
4 cases:
b+c+
b+c –
b c+
b – c –
The other side of the story: when a is NEGATIVE and the graph is an INVERTED U
Having covered all the cases when a is greater than zero, we now proceed to reckon the case when a is negavite i.e. a < 0
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.
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.
