Question

draw the polynomial p(x) = x to the power 2 -3x-4 on the graph and find the CO ordinate where it intersect at x axis??​

Answer 1

$\large\underline{\sf{Solution-}}$

Given polynomial is

$\red{ \boxed{ \rm{ \: \rm :\longmapsto\:p(x) = {x}^{2} - 3x - 4}}}$

Let assume that,

$\rm :\longmapsto\:y = {x}^{2} - 3x - 4$

To plot the graph of the quadratic polynomial which is always parabola, the following steps have to be followed :-

Step :- 1 Vertex of parabola

We know, vertex of parabola of quadratic polynomial ax² + bx + c is given by

$\blue{ \boxed{\bf \:Vertex = \bigg( - \dfrac{ b}{2a} , \: \dfrac{4ac - {b}^{2} }{4a} \bigg)}}$

Here,

$\rm :\longmapsto\:a = 1$

$\rm :\longmapsto\:b = - 3$

$\rm :\longmapsto\:c = - 4$

So, vertex of quadratic polynomial is

$\rm :\longmapsto\: \:Vertex = \bigg( - \dfrac{ ( - 3)}{2} , \: \dfrac{4(1)( - 4) - {( - 3)}^{2} }{4} \bigg)$

$\rm :\longmapsto\: \:Vertex = \bigg( \dfrac{3}{2} , \: \dfrac{ - 16 - {9}}{4} \bigg)$

$\rm :\longmapsto\: \:Vertex = \bigg( \dfrac{3}{2} , \: \dfrac{ - 25}{4} \bigg)$

Step :- 2

Point of intersection with x - axis

We know, on x - axis, y = 0.

So, on substituting the value of y = 0, we get

$\rm :\longmapsto\: {x}^{2} - 3x - 4 = 0$

$\rm :\longmapsto\: {x}^{2} - 4x + x- 4 = 0$

$\rm :\longmapsto\:x(x - 4) + 1(x - 4) = 0$

$\rm :\longmapsto\:(x - 4)(x + 1) = 0$

$\rm :\implies\:x = 4 \: \: \: or \: \: \: x = - \: 1$

Hence, the point of intersection with x- axis is (- 1, 0) and ( 4, 0).

Now,

Point of intersection with y - axis.

We know, on y - axis, x = 0

So, on Substituting the value in given curve, we get

$\rm :\longmapsto\: y = {0}^{2} - 3(0)- 4$

$\rm :\implies\:y = - 4$

Hence, the point of intersection with y- axis is (0, - 4).

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf - 4 \\ \\ \sf - 1 & \sf 0 \\ \\ \sf 4 & \sf 0\\ \\ \sf 1.5 & \sf - 6.25 \end{array}} \\ \end{gathered}$

➢ Now draw a graph using the points.

➢ See the attachment graph.