Solution for Part (b): Writing the Parametric Equations in Cartesian Form

The given parametric equations are: $x(t) = t^2 - 2t$ $y(t) = t + 1$

Step 1: Solve for $t$ in terms of $y$ From the second equation: $y = t + 1 \implies t = y - 1$

Step 2: Substitute $t = y - 1$ into $x(t)$ Substitute $t = y - 1$ into $x(t) = t^2 - 2t$: $x = (y - 1)^2 - 2(y - 1)$

Expand $(y - 1)^2$ and simplify: $x = (y^2 - 2y + 1) - 2(y - 1)$ $x = y^2 - 2y + 1 - 2y + 2$ $x = y^2 - 4y + 3$

Final Cartesian Equation:

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

Let me know if you need a graph or further clarification!

Questions for Deeper Understanding:

1. What is the geometric shape described by the Cartesian equation $x = y^2 - 4y + 3$?
2. How does the parameter $t$ define the direction of motion along the curve?
3. Can you determine the domain of $y$ based on the range of $t$?
4. What are the implications of reversing the parametric equations for the graph?
5. How do you calculate the slope of the tangent to the curve at a given point?

Tip: Always verify your substitution step when transforming parametric to Cartesian forms to ensure accuracy!