Question

draw graph of function f:R-> defined byf(x)=3x²​

Answer 1

$\begin{gathered}\begin{gathered}\bf \:Given - \begin{cases} &\sf{f(x) \: = \: {3x}^{2} }\end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To\:find - \begin{cases} &\sf{sketch \: the \: graph \: of \: {3x}^{2} }  \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\bold{{\underline{Formula \: Used - }}}} \end{gathered}$

We know,

• For the parabola x² = 4ay,

• Vertex = (0, 0)

• Focus = (0, a)

• Equation of axis, x = 0

• Equation of directrix, y = - a

• Length of Latus Rectum = 4a

• Symmetric:- Symmetric along y - axis.

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

Given function is

• f(x) = 3x²

Let

• y = 3x²

$\rm :\implies\: {x}^{2} = \dfrac{1}{3} \: y$

• which is the equation of Parabola.

• On comparing with x² = 4ay,

we get

$\rm :\longmapsto\:4a = \dfrac{1}{3}$

$\rm :\implies\:a \: = \: \dfrac{1}{12}$

Now,

we have

• Vertex = (0, 0)

• Focus = (0, 1/12)

• Equation of axis, x = 0

• Equation of directrix, y = - 1/12

• Length of Latus Rectum = 1/3

• Symmetric:- Symmetric along y - axis.

Intercept on axis :-

To Find intercept on x - axis

• Put y = 0, we get x = 0

• It implies, curve has no intercept on x - axis.

To find intercept on y - axis,

• Put x = 0, we get y = 0.

• It implies, curve has no intercept on y - axis.

Points on curve :-

Now,

Let select few values of x, to get corresponding values of y

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 0 \\ \\ \sf 1 & \sf 3 \\ \\ \sf - 1 & \sf 3\\ \\ \sf 2 & \sf 12\\ \\ \sf - 2 & \sf 12\\ \\ \sf 3 & \sf 27\\ \\ \sf - 3 & \sf 27 \end{array}} \\ \end{gathered}$

➢ Now draw a graph using the points

➢ See the attachment graph.

Explore more :-

For the parabola x² = - 4ay,

• Vertex = (0, 0)

• Focus = (0, - a)

• Equation of axis, x = 0

• Equation of directrix, y = a

• Length of Latus Rectum = 4a

• Symmetric:- Symmetric along y - axis.

For the parabola y² = 4ax,

• Vertex = (0, 0)

• Focus = (a, 0)

• Equation of axis, y = 0

• Equation of directrix, x = - a

• Length of Latus Rectum = 4a

• Symmetric:- Symmetric along x - axis.

For the parabola y² = - 4ax,

• Vertex = (0, 0)

• Focus = (- a, 0)

• Equation of axis, y = 0

• Equation of directrix, x = a

• Length of Latus Rectum = 4a

• Symmetric:- Symmetric along x - axis.