Let y = x2 - 2x - 8.
The following table gives the values of y or f(x) for various values of x.
x |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
y=x2-2x-8 |
16 |
7 |
0 |
-5 |
-8 |
-9 |
-8 |
-5 |
0 |
7 |
16 |
Let us plot the points (-4, 16), (-3, 7), (-2, 0), (-1, -5), (0, - 8), (1, - 9), (2, - 8), (3, - 5), (4, 0), (5, 7) and (6, 16) on a graphs paper and draw a smooth free hand curve passing through these points. The curve thus obtained represents the graphs of the polynomial f(x) = x2 - 2x - 8. This is called a parabola. The lowest point P, called a minimum points, is the vertex of the parabola. Vertical line passing through P is called the axis of the parabola. Parabola is symmetric about the axis. So, it is also called the line of symmetry.
Observations :
Fro the graphs of the polynomial f(x) = x2 - 2x - 8, following observations can be drawn :
(i) The coefficient of x2 in f(x) = x2 - 2x - 8 is 1 (a positive real number) and so the parabola opens upwards.
(ii) D = b2 - 4ac = 4 + 32 = 36 > 0. So, the parabola cuts X-axis at two distinct points.
(iii) On comparing the polynomial x2 - 2x - 8 with ax2 + bx + c, we get a = 1, b = - 2 and c = - 8.
The vertex of the parabola has coordinates (1, -9) i.e. (-b/2a,D/4a), where D =b2 - 4ac.
(iv) The polynomial f(x) = x2 - 2x - 8 = (x - 4) (x + 2) is factorizable into two distinct linear factors (x - 4) and (x + 2). So, the parabola cuts X-axis at two distinct points (4, 0) and (-2, 0). the x-coordinates of these points are zeros of f(x).