Q 16. - Calculus

Question

sketch the parametric curve by plotting points.

$x = t, y = t^2, t \in R$

Step-by-Step Solution

Answer

The graph 

Step 1: Given information

$x = t, y = t^2, t \in R$

Step 2: Concept

A collection of quantities is defined as a function of one or more independent variables called parameters in a parametric equation. 

Step 3: Calculation

Consider the parametric curve $x=t, y=t^2$ at $t\in \mathrm{ℝ}$

The goal is to draw the parametric curve.

To draw the graph for the parametric equations assume $t=-2,-1,0,1,2$

Substitute different $t$ values in the parametric equations and find the values of $x, y$

The point $(x, y)$ When $t=2$ is,

$(x,y)=\left(t,t^2\right)$

$(x,y)=\left(-2,(-2{)}^2\right)[$ since by substituting $t=-2]$

$(x, y)=(-2,4)$ simplify

The point $(x, y)$ When $t=-1$ is,

$$\begin{gathered} (x,y)=\left(t,t^2\right) \\ (x,y)=\left(-1,(-1{)}^2\right)[\text{ since by substituting }t=-1] \\ (x,y)=(-1,1)\text{ simplify } \end{gathered}$$

The point $(x, y)$ When $t=0$ is,

$$\begin{gathered} (x,y)=\left(t,t^2\right) \\ (x,y)=\left(0,(0{)}^2\right) \text{ [since by substituting }t=0\left] \\ \right(x,y)=(0,0)\text{ simplify } \end{gathered}$$

The point $(x, y)$ When $t=1$ is,

$$\begin{gathered} (x,y)=\left(t,t^2\right) \\ \left.(x,y)=\left(1,(1{)}^2\right)\text{ [since by substituting }t=1\right] \\ (x,y)=(1,1)\text{ simplify } \end{gathered}$$

The point $(x,y)$ When $t=2$ is,

$$\begin{gathered} (x,y)=\left(t,t^2\right) \\ (x,y)=\left(2,(2{)}^2\right) [\text{ since by substituting }t=1] \\ (x,y)=(2,4)\text{ simplify } \end{gathered}$$

Step 4: Calculation

The tabular representation of the points is as follows,

The graphical representation is shown below, 

Therefore, the solution is a required graph.