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:
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.
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)]
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)]
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)
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.