Final answer:
To find a Cartesian equation from parametric equations x = t² - 2, y = t + 3, we solve for t in the second equation and substitute into the first. The resulting Cartesian equation is a parabola x = y² - 6y + 7, which opens to the right and has a vertex within the range of y values 0 to 6.
Step-by-step explanation:
To eliminate the parameter and find the Cartesian equation of the curve defined by the parametric equations x = t² - 2 and y = t + 3, we can solve the second equation for t and then substitute into the first equation. So, t = y - 3. Substituting this into the equation for x, we get x = (y - 3)² - 2. Expanding this, we have the Cartesian equation x = y² - 6y + 7.
The curve described by this equation for 0 ≤ y ≤ 6 is a parabola that opens to the right due to the positive y² term. Since the coefficient of y is negative, the vertex of the parabola is located at a value of y greater than 3, thus within the specified range. For y values between 0 and 6, we can also deduce that the corresponding x values will increase quadratically as y moves away from the vertex of the parabola.