Question

Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve. $x=t^{2}y=t^3,t\in [-2,2]$

Short answer

The Curve plot

Step-by-step solution

Given information

The parametric curves, $x=t^{2}y=t^3,t\in [-2,2]$

Calculation

The goal is to draw the parametric curve.

Assume$-2,0,1,2$when drawing the graph for the parametric equations.

Find the values of $x,y$by substituting different $t$values in the parametric equations.

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

$(x,y)=\left(t^2,t^3\right)$

$(x,y)=\left((-2{)}^2,(-2{)}^3\right)$[since by substituting $\left.t=-2\right]$

$(x,y)=(4,-8)$simplify

The point $(x,y)$When $t=-1$is,

$$\begin{gathered} (x,y)=\left(t^2,t^3\right) \\ (x,y)=\left((-1{)}^2,(-1{)}^3\right)[\text{since by substituting}t=-1] \\ (x,y)=(1,-1)\text{simplify} \end{gathered}$$

The point $(x,y)$When $t=0$is,

$(x,y)=\left(t^2,t^3\right)$

$(x,y)=\left((0{)}^2,(0{)}^3\right)[$since by substituting $t=0]$

$(x,y)=(0,0)$simplify

The point $(x,y)$ When $t=1$ is,

$$\begin{gathered} (x,y)=\left(t^2,t^3\right) \\ (x,y)=\left(1^2,1^3\right)[\text{since by substituting}t=1] \\ (x,y)=(1,1)\text{simplify} \end{gathered}$$

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

$$\begin{gathered} (x,y)=\left(t^2,t^3\right) \\ (x,y)=\left(2^2,2^3\right)[\text{since by substituting}t=2] \\ (x,y)=(4,8)\text{simplify} \end{gathered}$$

Calculation

The tabular representation of the points is as follows,

The graphical representation is shown below,

Therefore, the solution is the required graph.