The parametric equation $(x(t), y(t)) = (2 + \cos t, 1 + \sin t)$ represents a circle in the plane. Let's analyze this step by step.

Parametric Equations:

$x(t) = 2 + \cos t$ $y(t) = 1 + \sin t$

Rewriting into a Standard Circle Form:

1. We know that $\cos^2 t + \sin^2 t = 1$ for all $t$.
2. Subtract 2 from $x(t)$ and 1 from $y(t)$: $x - 2 = \cos t \quad \text{and} \quad y - 1 = \sin t$
3. Now, square both sides and add them: $(x - 2)^2 + (y - 1)^2 = \cos^2 t + \sin^2 t = 1$

Conclusion:

This is the equation of a circle with center at $(2, 1)$ and radius $1$.

Graphical Description:

The curve described by the parametric equation is a circle centered at $(2, 1)$ with radius $1$.

Would you like further details or have any questions about this?

Here are 5 related questions to expand your understanding:

1. How would the graph change if the parametric equation was $(x, y) = (a + \cos t, b + \sin t)$?
2. What would the graph look like if you changed $\cos t$ to $\cos(2t)$ in the parametric equation?
3. Can we use a different parametric representation for the same circle?
4. How can you generalize parametric equations to describe ellipses instead of circles?
5. How would you describe the curve if we replaced $\cos t$ and $\sin t$ with $\tan t$ and $\cot t$?

Tip: Parametric equations are often used in computer graphics to model smooth curves such as circles and ellipses because they easily define continuous motion.