Parametric Equations?

Question

Graph the curve whose parametric equations are given and show its orientation. Also, find the rectangular equation of each curve.

(1) x = t - 3, y = 2t + 4, where t lies in the interval [0, 2].

(2) x = sqrt{t} + 4, y = sqrt{t} - 4, where t is greater than or equal to 0.

Answer 1

(1)
x = t - 3, y = 2t + 4
so
t = x+3
then
y = 2t + 4 = 2(x+3) + 4 = 2x +10
straight line gradient = 2, y-intercept = 10
graph
from x = 0 - 3 = -3
to x = 2 - 3 = -1
ie graph for -3 ≤ x ≤ -1

(2) this would be several functions if it wasn't that t ≥ 0
In this case it is simple
x = √t + 4, y = √t - 4
so
√t = x - 4
since t ≥ 0 substituting √t = x - 4 in y is fine, you would have to find t and substitute this instead so it would be more complicated.
then
y = √t - 4 = x - 4 - 4 = x - 8
straight line gradient = 1, y-intercept = -8
graph
from x = √0 - 4 = -4
to x = √∞ - 3 = ∞
(note: I should have used limits (as t --> ∞) here)
ie graph for -4 ≤ x ≤ ∞

the end
.

Answer 2

1) It is a straight line

2) Pair of Straight lines or rectangular hyperbola.