This diagram we have seen in the previous lesson. We have understood that alpha and beta, the two roots lie on the x axis and the point (0,c) lies on the y axis. That is fine. But as shown in this diagram, must the y axis be exactly where it is shown? Couldn’t it be elsewhere? Sure it can be. We can move the y axis left or right and see what more information we can eke out of this graph of the quadratic function.
Instead of moving the y axis we have taken three cases of the U shaped quadratic graph. What we see are two things:
1) the roots are either
a) both negative
b) one – ve & one +ve
c) both positive
& 2) c is either +ve or – ve
ALGEBRA to GEOMETRY: change a, b & c : the graph changes too
Remember that the axis of the quadratic is given by
x = – b/2a
Thus, if both a and b are positive, then x will be negative as shown. And vice versa if b is negative while a is positive
Remember also that a is always positive for the U shaped quadratic graph. When a < 0, the U upturns and becomes inverted.
In the above diagram also note that when b is negative the roots are both positive and vice versa: i.e. when b is positive, the roots are both negative. This is an important observation. You do not have to memorize it but must know how to arrive at the conclusion on your own. So, it is advised that you must draw these diagrams by yourself.
But if you want to memorize you can use sequences like this: a+b+c+roots- & a+b-c+roots+
The other two cases are shown here when the roots are of opposite signs.
If you have understood the previous diagram properly you will notice that the only difference here is in the value of c which is negative. In the previous diagram c is +ve while here it is – ve. And just that change alters the position of the roots. Now the roots are on the two sides of the y axis making them of the opposite signs.
The memorization sequence can be a+b+c- and a+b-c- when roots are of opposite signs.
But seriously, we do not advocate any kind of memorization as of now. Things will start to gel as you get more and more familiar with the graph and the algebra and their connections.
[c+ b+] implies both roots –
[c+ b-] implies both roots +
[c -] implies roots of opp signs
For each of the four graphs in the two diagrams above we can make certain general observations:
1) all the graphs shown have a>0
2) if b>0, axis is on the left of y-axis i.e. it is on the negative side or we can even say: b is positive, axis is negative and vice versa
3) if b<0, axis is on the right i.e. the axis is on the positive part of the x axis i.e. b is negative, axis is positive
4) c gives us the y-intercept: this we already knew. But if you look closely there are two more observations
5) if c is positive, the roots are either both positive or both negative depending on b being negative or positive. We will generalize this shortly
6) if c is negative, the roots are of opposite signs and the value of b does not matter so much in determining the roots.
As an example, in this diagram notice when b>0 and c>0 both roots are negative and in the graph below, when c<0, we need not know how b is: the roots have to be of opposite signs.