You are working with the parametric equations:

$y = 3t - t^2 \quad \text{and} \quad x = 3t^2$

To eliminate the parameter $t$, let's solve for $t$ from the second equation:

1. From $x = 3t^2$, solve for $t$: $t^2 = \frac{x}{3}$ $t = \pm \sqrt{\frac{x}{3}}$

Now, substitute this expression for $t$ into the equation for $y$:

2. Substitute $t = \pm \sqrt{\frac{x}{3}}$ into $y = 3t - t^2$:

   $y = 3\left(\pm \sqrt{\frac{x}{3}}\right) - \left(\pm \sqrt{\frac{x}{3}}\right)^2$ Simplifying: $y = \pm 3\sqrt{\frac{x}{3}} - \frac{x}{3}$ $y = \pm \sqrt{3x} - \frac{x}{3}$

So, the relationship between $x$ and $y$ is:

$y = \pm \sqrt{3x} - \frac{x}{3}$

This represents the equation of a curve after eliminating the parameter $t$.

Would you like more details, or do you have any questions?

Here are five related questions to explore further:

1. What type of curve does this equation represent?
2. How do we find the slope of the curve at a given point?
3. How would we graph the parametric equations $y = 3t - t^2$ and $x = 3t^2$?
4. How can we calculate the velocity and acceleration if these parametric equations represent motion?
5. How do we determine the maximum or minimum points on this curve?

Tip: To analyze parametric equations, converting them to a Cartesian form often provides better insights into the nature of the curve.