Draw the graphs of the following polynomials and write the observations.
i) f(x) = 3x + 8
ii) f(x) = 3 – 2x - x^2
iii) f (x) = x^3- 4x
Answers
(i) Graph :- In the attachment no. 1
(i) Graph :- In the attachment no. 1(ii) Let y = f(x) or, y = 3 - 2x - x².
Let us list a few values of y = 3 - 2x - x² corresponding to a few values of x as follows :-
In the attachment no. 2
Thus, the following points lie on the graph of the polynomial y = 2 - 2x - x² :-
(-5, -12), (-4, -5), (-3, 0), (-2, 4), (-1, 4), (0, 3), (1, 0), (2, - 5), (3, -12) and (4, - 21).
Let plot these points on a graph paper and draw a smooth free hand curve passing through these points to obtain the graphs of y = 3 - 2x - x². The curve thus obtained represents a parabola, as shown in figure. The highest point P(-1, 4), is called a maximum points, is the vertex of the parabola. Vertical line through P is the axis of the parabola. Clearly, parabola is symmetric about the axis.
Observations :-
Following observations from the graph of the polynomial f(x) = 3 - 2x - x2 is as follows :-
(i) The coefficient of x2 in f(x) = 3 - 2x - x² is - 1 i.e. a negative real number and so the parabola opens downwards.
(ii) D = b² - 4ax = 4 + 12 = 16 > 0. So, the parabola cuts x-axis two distinct points.
(iii) On comparing the polynomial 3 - 2x - x² with ax² + bc + c, we have a = - 1, b = - 2 and c = 3. The vertex of the parabola is at the point (-1, 4) i.e. at (-b/2a,-D/4a), where D = b² - 4ac.
(iv) The polynomial f(x) = 3 - 2x - x² = (1 - x) (x + 3) is factorizable into two distinct linear factors (1 - x) and (x + 3). So, the parabola cuts X-axis at two distinct points (1, 0) and (-3, 0). The co-ordinates of these points are zeros of f(x).
Graph :- In the attachment no. 3
(iii) Let y = f(x) or, y = x² - 4x.
The values of y for variable value of x are listed in the following table :-
In the attachment no. 4
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).
Graph :- In the attachment no. 5
