Question

In problem graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility.

$x=3t+2,y=t+1;0\le t\le 4$

Short answer

The answer is$x-3y+1=0.$

Step-by-step solution

Step 1. Given information.

$$\begin{gathered} x=3t+2 \\ y=t+1 \end{gathered}$$

is given.

Step 2. Calculation.

Graph the curve defined by the parametric equations -

$x=3t+2,y=t+1,0\le t\le 4$

t	x	y	(x, y)
0	2	1	(2,1)
1	5	2	(5,2)
2	8	3	(8,3)
3	11	4	(11,4)
4	14	5	(14,5)

We find the rectangular equation by eliminating the parameter t from the parametric equations.

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

Put the value of t -

$$\begin{gathered} x=3t+2 \\ x=3(y-1)+2 \\ x=3y-3+2 \\ x=3y-1 \\ x-3y+1=0 \end{gathered}$$

Thus, the equation is$x-3y+1=0.$