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To eliminate the parameter \( t \) and express the relationship between \( x \) and \( y \) in Cartesian form, we’ll start with the given parametric equations:

\[
x = t^2 - 2
\]
\[
y = t + 3
\]

1. **Solve for \( t \) in terms of \( y \)** from the second equation:
   \[
   y = t + 3 \Rightarrow t = y - 3
   \]

2. **Substitute \( t = y - 3 \)** into the first equation to express \( x \) in terms of \( y \):
   \[
   x = (y - 3)^2 - 2
   \]

3. Expand and simplify:
   \[
   x = (y^2 - 6y + 9) - 2
   \]
   \[
   x = y^2 - 6y + 7
   \]

So, the **Cartesian equation** of the curve is:
\[
x = y^2 - 6y + 7
\]

### Range of \( y \):
Given that \( -3 \leq t \leq 3 \), substituting into \( y = t + 3 \) gives:
\[
0 \leq y \leq 6
\]

Thus, the Cartesian equation is:
\[
x = y^2 - 6y + 7 \quad \text{for} \quad 0 \leq y \leq 6
\]

Would you like more details on this or have any further questions?

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### Related Questions

1. How can we find the range of \( x \) for this curve?
2. What is the vertex of the resulting parabola in terms of \( x \) and \( y \)?
3. How would the equation change if the parametric equations included a different transformation?
4. Can we find the derivative \( \frac{dy}{dx} \) of the curve given in Cartesian form?
5. How do we plot the Cartesian form of this parametric equation?

### Tip
When eliminating parameters, isolate the parameter in one equation and substitute it into the other to find the direct relationship.

Eliminate the parameter to find a Cartesian equation of the curve for 0 ≤ y ≤ 6.

Learn how to eliminate the parameter in a set of parametric equations to find a Cartesian equation of the curve. This solution covers techniques for transforming x = t^2 - 2 and y = t + 3, with constraints on y from 0 to 6.

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