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Chapter 6: Problem 41

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=4+3 \cos \theta\\\ &y=-2+\sin \theta \end{aligned}$$

Short Answer

Expert verified
The graph is in the shape of an ellipse, shifted 4 units to the right and 2 units down, with width 6 and height 2.

Step by step solution

01

Assign the Equations to Axes

First off, assign the equation \(x=4+3 \cos \theta\) to the x-axis and the equation \(y=-2+\sin \theta\) to the y-axis. These equations gives the value of the x and y coordinates for any given value of \(\theta\).
02

Plot the Curve using a Graphing Utility Tool

Next, use a graphing utility tool to plot these two equations. Set the parameter \(\theta\) and observe the shape of the plot changes with the range of \(\theta\). Typically, for a simple review of the shape of the plot, a range of \(0 \leq \theta \leq 2\pi \) is usually enough.
03

Interpreting the Generated Graph

Finally, interpret the generated graph. The shape of the graph should resemble a circle or an ellipse, shifted 4 units to the right and 2 units down, due to the constants in the equations. The constants \(3\) and \(1\) in the equations determine the width and height of the ellipse, respectively.

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