[SOLVED] Eliminating more trig parameters (sin & cos)

Ok so I thought I'd post this one now that I've been able to work through it a little farther because it didn't quite turn out how I expected graphically...

original problem:

Eliminate the parameter to find a Cartesian equation of the curve

$\displaystyle

x = \frac{1}{2}\cos\theta , y = 2\sin\theta , 0 <= \theta <= \pi

$

My steps:

$\displaystyle 2x = \cos\theta$ and $\displaystyle \frac{y}{2} = \sin\theta$

1) $\displaystyle 2x^2 + \frac{1}{2}y^2 = 1$

2) $\displaystyle \frac{1}{2}y^2 = 1 - 2x^2$

3) $\displaystyle y^2 = 2(1-2x^2)$

4) $\displaystyle y^2 = 2 - 4x^2$

Solution) $\displaystyle y = \sqrt{2-4x^2}$

is this correct? If so how do I graph this appropriately? Do I just graph the function I have for the solution or do I have to plug back in the trigonometric x value?

Thanks