To solve this problem, we will calculate $x$ and $y$ for each $t$ using the given parametric equations:

1. $x = 1 - t^2$
2. $y = 1 + t$

Here is the solution for each value of $t$:

For $t = 3$:

• $x = 1 - 3^2 = 1 - 9 = -8$
• $y = 1 + 3 = 4$

For $t = 2$:

• $x = 1 - 2^2 = 1 - 4 = -3$
• $y = 1 + 2 = 3$

For $t = 1$:

• $x = 1 - 1^2 = 1 - 1 = 0$
• $y = 1 + 1 = 2$

For $t = 0$:

• $x = 1 - 0^2 = 1 - 0 = 1$
• $y = 1 + 0 = 1$

For $t = -1$:

• $x = 1 - (-1)^2 = 1 - 1 = 0$
• $y = 1 + (-1) = 0$

For $t = -2$:

• $x = 1 - (-2)^2 = 1 - 4 = -3$
• $y = 1 + (-2) = -1$

For $t = -3$:

• $x = 1 - (-3)^2 = 1 - 9 = -8$
• $y = 1 + (-3) = -2$

Completed Table:

Let me know if you'd like me to plot the curve based on these points or if you have any questions!

Questions to Explore:

1. How do parametric equations differ from Cartesian equations?
2. What is the geometric interpretation of the curve in this example?
3. How can you eliminate the parameter $t$ to find the Cartesian equation?
4. Can the orientation of the curve be determined from the parametric equations?
5. How would the graph change if we altered the equations for $x$ or $y$?

Tip:

When sketching parametric curves, plotting points systematically as shown helps