Q. 25 - Calculus

Question

Sketch the parametric curve by eliminating the parameter 

$x=2t-1,y=3t^2+5,t\in \mathrm{ℝ}$

Step-by-Step Solution

Answer

The required graph is 

Step 1: Given information

The parametric curve is

$x=2t-1,y=3t^2+5,t\in \mathrm{ℝ}$

Step 2: Calculation

Consider the parametric equations $x=2t-1,y=3t^2+5,t\in \mathrm{ℝ}$

The objective is to sketch the parametric curve by eliminating the parameters.

Take the parametric curve $x=2 t-1$

Add I to both sides of the equation.

$$\begin{gathered} x+1=2t-1+1 \\ x+1=2t \end{gathered}$$

Now divide the equation on both sides by 2 .

$$\begin{gathered} \frac{x+1}{2}=\frac{2t}{2} \\ \frac{x+1}{2}=\frac{2t}{2} \\ \frac{x+1}{2}=t \end{gathered}$$

Now substitute $t=\frac{x+1}{2}$in the parametric equation $y=3t^2+5$.

Then,

$$\begin{gathered} y=3·{\left(\frac{x+1}{2}\right)}^2+5\left[sincet=\frac{x+1}{2}\right] \\ y=3·\frac{(x+1{)}^2}{2^2}+5 \\ y=\frac{3(x+1{)}^2}{4}+5 \end{gathered}$$

Thus, the required equation after eliminating the parameter $t$ is $y=\frac{3(x+1{)}^2}{4}+5$.

To draw the graph of the equation, assume$x=-2,-1,0,1,2$.

Substitute $x=-2$in the equation $y=\frac{3(x+1{)}^2}{4}+5$.

Then,

$$\begin{gathered} y=3\frac{(-2+1{)}^2}{4}+5 \\ y=.75+5(x,y) \\ =(-2,5.75) \end{gathered}$$

Step 3: Further calculation

Substitute $x=-1$in the equation $y=\frac{3(x+1{)}^2}{4}+5.$

Then,

$$\begin{gathered} y=3\frac{(-1+1{)}^2}{4}+5 \\ y=3\frac{\left(0\right)}{4}+5 \\ (x,y)=(-1,5) \end{gathered}$$

Substitute $x=0$ in the equation $y=\frac{3(x+1{)}^2}{4}+5$.

Then,

$$\begin{gathered} y=3\frac{(0+1{)}^2}{4}+5 \\ y=.75+5 \\ y=5.75 \\ (x,y)=(0,5.75) \end{gathered}$$

Substitute $x=1$ in the equation $y=\frac{3(x+1{)}^2}{4}+5$Then,

$$\begin{gathered} y=3\frac{(1+1{)}^2}{4}+5 \\ y=3\frac{(2{)}^2}{4}+5 \\ y=3\frac{A}{A}+5 \\ (x,y)=(1,8) \end{gathered}$$

Substitute $x=2$in the equation $y=\frac{3(x+1{)}^2}{4}+5^{\text{. Then, }}$

$$\begin{gathered} y=3\frac{(2+1{)}^2}{4}+5 \\ y=6.75+5 \\ y=11.75 \\ (x,y)=(2,11.75) \end{gathered}$$

Step 4: Sketch the graph.

The graphical representation using the points $(-2,5.75)(-1,5)(0,5.75)(1,8)(2,11.75)$is as follows,

Therefore, the required equation after eliminating the parameter is $y=\frac{3(x+1{)}^2}{4}+5$