Question

Draw Graph and polynomial
for 6-x-x²

Answer 1

Step-by-step explanation:

The resultant graph is parabola. The graph cuts x-axis at (-3,0) and (2,0).

Answer 2

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

Given quadratic polynomial is

$\rm \: f(x) = y = 6 - x - {x}^{2}$

can be rewritten as

$\rm \: f(x) = y = - {x}^{2} - x + 6$

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,

$\rm \: a = - 1$

$\rm \: b = - 1$

$\rm \: c = 6$

So, on substituting the values, we get

$\rm \: \:Vertex = \bigg( - \dfrac{ ( - 1)}{2( - 1)} , \: \dfrac{4( - 1)(6) - {( - 1)}^{2} }{4( - 1)} \bigg)$

$\rm \: \:Vertex = \bigg( - \dfrac{1}{2} , \: \dfrac{ - 24 - 1 }{ - 4} \bigg)$

$\rm \: \:Vertex = \bigg( - \dfrac{1}{2} , \: \dfrac{ - 25 }{ - 4} \bigg)$

$\rm \: \:Vertex = \bigg( - \dfrac{1}{2} , \: \dfrac{25 }{4} \bigg)$

Step :- 2 Point of intersection with x - axis

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

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

$\rm \: - {x}^{2} - x + 6 = 0$

$\rm \: - ({x}^{2} + x - 6) = 0$

$\rm \: {x}^{2} + x - 6 = 0$

$\rm \: {x}^{2} + 3x - 2x - 6 = 0$

$\rm \: x(x + 3) - 2(x + 3) = 0$

$\rm \: (x + 3)(x - 2) = 0$

$\rm\implies \:x = - 3 \: \: or \: \: x = 2$

Hence, the point of intersection with x- axis is (2, 0) and ( - 3, 0).

Now, Point of intersection with y - axis.

We know, on y - axis, x = 0

So, on Substituting the value of x = 0 in given curve, we get

$\rm\implies \:\rm \: y = 6$

Hence, the point of intersection with y- axis is (0, 6).

Hᴇɴᴄᴇ,

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

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 6 \\ \\ \sf 2 & \sf 0\\ \\ \sf - 3 & \sf 0 \\ \\ \sf - 0.5 & \sf 6.25 \end{array}} \\ \end{gathered}$

➢ Now draw a graph using the points.

➢ See the attachment graph.