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.

Learn how to eliminate the parameter in the parametric equations y = 3t - t^2 and x = 3t^2. This step-by-step guide simplifies the equations to express y in terms of x. Key concepts include substitution and algebraic manipulation, suitable for high school students (Grades 10-12).

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