Question

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ \begin{array}{l} x=4+2 \cos \theta \\ y=-1+\sin \theta \end{array} $$

Short answer

The graph of the parametric equations follows a counterclockwise rotation. The corresponding rectangular equation is \((x-4)^2/4\) + \((y+1)^2 = 1\).

Step-by-step solution

Graph the Parametric Equations

Feed the equations into the graphing utility. For this, the inputs for x and y will be as follows: \(x=4+2 \cos \theta \) and \(y=-1+\sin \theta\). The plot of these equations will give us the look and the orientation of the curve.

Note the Orientation of the Curve

The curve will start from the point where \(\theta = 0\) and will continue in the direction as \(\theta\) increases. Notice that as \(\theta\) ranges from \(0\) to \(2\pi\), the curve completes one full rotation counterclockwise.

Eliminate the Parameter

To write a rectangular equation, we need to eliminate the parameter \(\theta\). As we know, \(\cos^2 \theta\) + \(\sin^2 \theta\) = 1. Therefore, the equation becomes \((x-4)^2/4\) + \((y+1)^2 = 1\).