Q. 24 - Calculus

Question

In Exercises 24–34 sketch the parametric curve by eliminating the parameter 

24. $x = 2t - 1, y = 3t + 5, t \in \mathrm{ℝ}$

Step-by-Step Solution

Answer

As a result, after removing the parameter, the needed equation is $\mathit{y}\mathbf{=}\mathbf{3}\mathbf{·}\frac{\mathbf{x}\mathbf{+}\mathbf{1}}{\mathbf{2}}\mathbf{+}\mathbf{5}$

Step 1 : Given information

Given : $x = 2t - 1, y = 3t + 5, t \in \mathrm{ℝ}$

Step 2 : Removing the parameter

Take the parametric curve  $x= 2t-1$

$1$ should be added to both sides of the equation:

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

Divide both sides by $2$ :

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

Substitute the parametric equation $t=\frac{x=1}{2}$ in place of both sides of the equation, $y=3t+5$

Then you've got :

$y=3·\frac{x+1}{2}+5$

As a result, after removing parameter $1$, the needed equation is $y=3·\frac{x+1}{2}+5$

Step 3 : Substituting different values for x

To create the equation's graph assume $x=-1,0,1,2$ :

1) Substitute $x=-1$ in the equation :

 $$\begin{gathered} y=5 \\ \Rightarrow (x,y)=(-1,5) \end{gathered}$$

2) Substitute $x = 0$:

 $$\begin{gathered} y = 1.5+5 \\ \Rightarrow y = 6.5 (0,6.5) \\ \Rightarrow (x,y)=(0,6.5) \end{gathered}$$

3) Replace $x = 1$:

$$\begin{gathered} y=3(1+1)+5 \\ = 8 \\ \Rightarrow (x,y)= (1,8) \end{gathered}$$

4) Substitute $x = 2$:

$$\begin{gathered} y=3(2+1) \\ =4.5+5 \\ =9.5 \\ \Rightarrow (x,y)= (2,9.5) \end{gathered}$$

Step 4 : Plotting the graph

The following is the graphical representation using the points $(-1,5) , (0,6.5) , (1,8) , (2,9.5)$ :

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