Match the parametric equations with the graphs labeled IVL G

Match the parametric equations with the graphs labeled I-VL Give reasons for your choices. (Do not use a graphing device.)

look at (b) ,
x = t^2 - 2t , y = t

x = y^4 - 2y^2 = y^2(y^2 - 2)

this curve is symmetric about x-axis as y = +p or -p will yield same x value.

this reduces us to two possibility, II and VI.

To decide among these, lets check y at x = 0,

y^2(y^2 - 2) = 0 => y^2 = 0 or y^2 = 2 => y = 0 or y = ±2

so there are three values of y for x = 0, which is only in curve 2.

(b) matches with II.

Now look at (f),

x^2 + y^2 = [(sin 2t)^2 + (cos 2t)^2]/(4+t^2)^2 = (1/4+t^2)^2

now observe this eq is symmetric to y and x axis.

only curve left which is symmetric to x-axis is VI

hence (f) matches with VI

Now look at (e), y = t^2 + cos 3t,

t^2 >= 0 , -1 <= cos 3t <= 1 , so y is never 0

which is only in (I)

so (e) matches with (I)

(a) matches with (V) as, y = t^2, y is always positive as t^2 is always positive.

y >= 0

y = 0 only at t = 0

where x = 1

so only curve matches is V

(d) is (III) as at y = 0, there are infinite t\'s possible corresponding to which there are infinite x\'s for y = 0

hence only curve matches is (III)