Fundamentals: The shape of the curve
We all know that the graph of a quadratic equation is a parabola, i.e. a U-shaped curve. But how do we know whether it’s an upward facing U or a downward facing U? Now, some of you (the well-read types) may say that in the general equation of a quadratic, i.e.
y = Ax^2 + Bx + C
…the sign of the coefficient of x^2, i.e. A determines it. In other words, if A is positive, it is an upward parabola, if A is negative it is a downward parabola. You’re right, but I want to know why it is so?
Any ideas?
The reason is simple! As we consider larger and larger values of x, which of the three terms in Ax^2 + Bx + C is going to dominate? Obviously, it is Ax^2. Now as x^2 is always positive, whether y is positive or negative will depend only on the sign of A. In other words, y will be positive for positive A and negative for negative A (for very large values of x). Similarly, even when I take extremely large negative values of x, x^2 is still positive, hence y will still be positive for positive A and negative for negative A. Let us see this on a graph:
And now, let us assume that the quadratic equation has real roots. A root, by definition, is/are the value(s) of x for which y = 0. Now, y = 0 is nothing but the equation of the x-axis. Therefore, the curve has to meet the x-axis at the roots. Now, it should become absolutely clear that the shape of the curve will be upward parabola for positive A (yellow regions in graph) and downward parabola for negative A (green regions in graph).
The important extension of this concept is to any n-degree equation (curve). Let us see how. To determine whether y will be +ve or –ve for large +ve or –ve values of x, all we need to do is look at the sign of the coefficient of the highest power of x in the equation. For e.g. for…
y = 6x^3 + 4x^2 + 5x + 6
…we note that the coefficient of the highest power of x is +ve (6, i.e.). Hence, for large +ve values of x y is +ve and for large –ve values of x, x^3 is –ve and hence y will also be –ve. In graphical terms, the curve will lie somewherin the yellow regions for large +ve or –ve values of x:
Hence we can make out whether the curve will approach the x-axis from up or below from both left and right by simply seeing the sign of the coefficient of the highest power of x.
The Nature of the Roots
Another interesting point to note is that in a quadratic equation, depending on the value of the discriminant D, i.e.
D = (b^2-4ac)
…we get the following three cases on the nature of the roots:
1) D > 0 : 2 Distinct real roots, the graph will cut the x-axis at two distinct points
2) D = 0: A double root, the graph just touches the x-axis and bounces off.
3) D less than 0: No real roots, the graph doesn’t touch the x-axis at all, it is completely on one side of the x-axis only.
The important thing to note from here is that if the root is a single root (or any odd power like (x-1)^3 = 0) the graph cuts the x-axis at that root but if the root is a double root (for e.g. (x-2)^2 = 0 OR (x-3)^4 = 0 ) the curve simply touches the x-axis and bounces off.
Now, combining these 2 concepts of
1) From which side of x-axis will the curve approach the x-axis from both left and right. &
2) At which roots the curve will cut the x-axis and at which it will simply bounce...we can easily plot the curve of the algebraic equation.
Lets see how…
Supposing the question is
(x-1)(x^2-5x+6) less than 0
Now the coefficient of x^3(the highest power of x here) is 1 which is positive. Hence, the curve will approach the x-axis from above from the right side and from below from the left side. (Just like in the earlier example of y = 6x^3 + 4x^2 + 5x + 6
Secondly,
(x-1)(x^2-5x+6) less than 0
=> (x-1)(x-2)(x-3) less than 0
Hence the roots are x = 1, 2 & 3. Hence the curve has to cut the x-axis at these three points. Therefore the shape of the curve has to be like
Hence we can clearly see that y = (x-1)(x^2-5x+6) will be less than 0 when
x is less than 1 OR x lies between 2 and 3.
Similarly, we can solve any question based on inequalities.
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