1. **State the problem:** We are given parametric equations $x(t) = t^2 - 2t$ and $y(t) = t + 1$ for $-2 \leq t \leq 4$. We need to sketch the curve, indicate the direction, and find the Cartesian equation. 2. **Find the Cartesian equation:** From $y(t) = t + 1$, solve for $t$: $$t = y - 1$$ 3. Substitute $t = y - 1$ into $x(t)$: $$x = (y - 1)^2 - 2(y - 1)$$ 4. Expand and simplify: $$x = (y^2 - 2y + 1) - 2y + 2 = y^2 - 4y + 3$$ 5. **Cartesian equation:** $$x = y^2 - 4y + 3$$ 6. **Direction:** As $t$ increases from $-2$ to $4$, $y = t + 1$ increases from $-1$ to $5$, so the curve is traced from $y = -1$ to $y = 5$. 7. **Summary:** The curve is a parabola opening rightwards described by $$x = y^2 - 4y + 3$$ with $y$ ranging from $-1$ to $5$, and the direction is from lower to higher $y$ values.