Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction as t increases. Eliminate parameter to find a Cartesian equation of the curve.

Question

1) x= 2t-1 y=1/2t+1

2) x=3t+2 y=2t+3

3)x=t^3 -3 y=t+2 -3≤t≤3

Answer 1

#1. x = 2t - 1 and y = 1 / (2t + 1)

From the first equation, 2t = x + 1. So, y = 1 / (x+1)+1.

Therefore, y = 1 / (x+2) (graph is a hyperbola).

#2. x = 3t + 2 and y = 2t + 3

From the first equation, t = (x - 2) / 3

So, y = 2 [ (x-2)/3 ] + 3

Therefore, y = (2/3)x + (5/3) (graph is a line)

#3 x = t³ - 3 and y = t + 2

From the first equation, t = (x+3)^1/3. So, y = (x+3)^1/3 + 2.

Since -3 ≤ t ≤ 3, -30 ≤ x ≤ 24.