Q4: Lowest Pt. on Axis

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This diagram has been explained in the previous lesson and is being shown to remind you of it and to impress upon you its importance 

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this diagram is a repeat of the earlier graph showing it more simply the two roots alpha and beta and their placement. the central line can also be written as (alpha+beta)/2. Check it out for yourself.

The additional point here in this diagram is (0,c) which is very simple and easy to remember and there is no complication.
Whenever a quadratic cuts the y-axis it is always at this point (0,c)


The Minimum point in a quadratic OR its minimum value 

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It is a question of symmetry. The U shaped graph of a quadratic function has a minimum point as you can see.

Since the U is a symmetric diagram, it has an axis about which it is symmetric and this axis lies between the two roots alpha and beta and as we have seen earlier the axis is given as x= – b/2a.
So we can find any y point in a graph by putting the x value into the expression. That is precisely what has been done and shown in the diagram above and we get the result as (- D/4a).


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Finally, a snapshot. This shows the quadratic function with the two roots i.e. the two points where the graph cuts the x axis and it also shows the axis and the lowest point of the quadratic lying on this axis. The graph also shows where the curve cuts the y-axis. 


A Diagram to embed in your mind

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Commit this diagram into your mind, embed it graphically, understand it algebraically, see its constituent parts, see the Axis, the lowest point the distance between the mid point and alpha & beta, the minimum value of the quadratic. Do you see how many different facets of the quadratic now becomes clear?