91Ó°ÊÓ

Parametric equations allow us to define a curve in a plane by setting both the x and y coordinates in terms of a third variable, often labeled as \( t \). In this exercise, \( x = 2 + \sin t \) and \( y = 3 + \cos t \) are our parametric equations. Here, \( t \) can be thought of as a parameter that controls the position on the curve as it changes, like time ticking along a path.

To graph these parametric equations, the key steps involve picking a range for \( t \), calculating points by substituting values of \( t \), and then plotting those points on a Cartesian plane. In our example, choosing \( t \) between \( 0 \) and \( 2\pi \) encompasses one full cycle of the sine and cosine functions, giving complete information about the shape of the curve.

• Pick values of \( t \) within the range, like \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \).
• Substitute each \( t \) value into the parametric equations to find corresponding \( x \) and \( y \) values.
• Plot these \( (x, y) \) points on a graph.

These steps will form the basis for sketching the particular path described by your parametric equations.

A plane curve is a collection of points that can be connected smoothly in a 2D plane. Parametric equations are excellent tools to describe such curves, as they combine two equations to determine position through a dynamic parameter \( t \). In our scenario, the parametric equations \( x = 2 + \sin t \) and \( y = 3 + \cos t \) take advantage of the periodic nature of sine and cosine to describe motion along a closed path, in this case, an ellipse-like shape centered around a point in the Cartesian plane.

Every curve has unique characteristics:

• Curvature, which typically means how the direction of the curve changes along its length.
• Smoothness, indicating that the path can generally be drawn without lifting your pen and without any sharp corners.
• Closedness, showing whether the curve forms a loop, as our example does.

In the context of parametric equations, smooth continuous curves are typically guaranteed due to the gentle transitions of sine and cosine functions unless the range of \( t \) covers discontinuities. Here, the choice to use both functions results in a predictable, well-behaved plane curve.

Understanding the orientation of a curve refers to knowing the direction in which the curve is traced as the parameter \( t \) increases. In cases of closed curves formed by parametric equations, like the one given here, orientation becomes especially relevant to comprehend movement direction.

For \( x = 2 + \sin t \) and \( y = 3 + \cos t \), increasing values of \( t \) from \( 0 \) to \( 2\pi \) will adjust the (x, y) coordinates around the plane curve. We use arrows along the curve to demonstrate its directionality.

• Starting at \( t = 0 \) corresponds to \( (x, y) = (2, 4) \).
• Moving on to \( t = \frac{\pi}{2} \) takes us to \( (3, 3) \).
• Progressing further to \( t = \pi \), returning to \( (2, 2) \).
• The curve then shifts back up through \( (1, 3) \) at \( t = \frac{3\pi}{2} \).
• Finally, it closes its loop completing at \( t = 2\pi \) with \( (2, 4) \).

This path is traced in a counter-clockwise orientation. The choice of moving \( t \) from \( 0 \) to \( 2\pi \) without "backtracking" ensures the curve is smooth and continuously oriented.