Q9: Inequality

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If we know the equation of the line shown viz. y=(sq rt 3)x

As we take different values of x = 1,2,3,…
we can immediately find the corresponding point on the line for the given x.

See the three points based on x=-1,1 & 3

Since y is a function of x which means that the value of y depends on what x we have chosen. Thus y is called dependent variable since it is dependent on x: as we change x, the value of y automatically change as per the equation we are dealing with. In this case, we are dealing with the linear function, y=(sq rt 3) x.

Another way of writing the same thing is y=f(x). 
When x = 2, y = f(2) & x=a, y=f(a) where ‘a’ is any value of x.

We will extrapolate this understanding onto the quadratic function as shown below:


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For different values of x we have shown the calculation of the y values for the corresponding x values. 

A few things to note in this diagram:

1. f(0), f(1), f(2) as well as f(-3) are all positive and when y is positive the point is above the x axis.

2. f(-1)=f(-2)=0 and y value is zero means the curve is cutting the x axis at these points. Thus, -1 & -2 are the two roots of the quadratic

3. f(0) means when x = 0 which means f(0) is the value of y where the curve cuts the y axis. 

4. this is not shown in the diagram, but you ought to be able to see that for any value of x between -2 & -1, the y value will be negative. This is explained in detail in the next diagrams. Go for it. 


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The quadratic function is shown here with various y values (vertical lines)

So, we can write the point on the curve where x=1 as [1, f(1)] and 
when x=5, the point on the curve will be [5, f(5)]


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This diagram is really very very important and thus permit us to reiterate that by showing you more of the same thing. This has to get embedded in your mind as the questions of inequality of a quadratic function can easily be understood if you comprehend this concept of function thoroughly. 

A few points are shown

x=a,b,c,d & x= alpha, beta

and in the diagram below the corresponding points on the curve are shown:

[a, f(a)] to [d, f(d)]


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This is an important diagram, the y values are shown by vertical dotted lines for specific values of x, viz x=p, alpha, q, beta, r. 

At x=alpha or beta, the y values are zero

The y values are positive when p & q are outside the range of values between alpha and beta. 

So, as it turns out f(alpha)=f(beta)=0 while f(p) & f(r) are positive and f(q) is less than zero or negative (q lies between alpha & beta)


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We can generally conclude that when a point lies between the roots on the x-axis, its corresponding y values are negative
and
when a point on the x-axis lies beyond alpha and beta i.e. those x points which are less than alpha and which are larger than beta, for those x points, the y values are seen to be positive. 

And when we say y is positive, we mean that the Quadratic, let’s call it Q, is positive i.e. Q>0 for all x< alpha or x>beta. 
and conversely, as in the diagram, Q<0 when alpha<x<beta i.e when x lies between alpha and beta. 

Note that we are using Q or y or f(x) or Curve or the quadratic expression or the quadratic function, interchangeably. All of these words mean the same thing, the diagram you see above.


Look at these diagram once more. Once you become absolutely confidently familiar with these diagrams and understand the inequality that comes out from this diagram, you are well and truly into cracking the tough problems related to quadratic function.