Question

Draw the graphs of the polynomial f(x) = x³ - 4x.

Answer 1

Let y = f(x) or, y = x² - 4x.

The values of y for variable value of x are listed in the following table :

x	-3	-2	-1	0	1	2	3
y=x3-4x	-15	0	3	0	-3	0	15

Thus, the curve y = x3 - 4x passes through the points (-3, -15), (-2, 0), (-1, 3), (0 ,0), (1, -3), (2, 0), (3, 15), (4,48) etc. Plotting these points on a graph paper and drawing a free hand smooth curve through these points, we obtain the graph of the given polynomial as shown figure.

Observations :

For the graphs of the polynomial f(x) = x³ - 4x, following observations are as follows :-

(i) The polynomial f(x) = x³ - 4x = x(x² - 4) = x(x - 2) (x + 2) is factorizable into three distinct linear factors. The curve y = f(x) also cuts X-axis at three distinct points.

(ii) We have, f(x) = x (x - 2) (x + 2) Therefore 0, 2 and -2 are three zeros of f(x). The curve y = f(x) cuts X-axis at three points O (0, 0), P(2, 0) and Q (-2, 0).