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Chapter 6: Problem 42

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

Short Answer

Expert verified
The graph generated by the parametric equations \(x = 4 + 3\cos\theta\) and \(y = -2 + 2\sin\theta\) with \(\theta\) varying from \(0\) to \(2\pi\) is a circle centered at \( (4, -2) \) with radius \(2\).

Step by step solution

01

Understanding Parametric Equations

Parametric equations are a set of equations that express the coordinates of the points of a curve or surface as functions of one or more parameters. In this case, the parameter is \(\theta\), and it's varying values will give different points on the curve represented by \(x\) and \(y\).
02

Set up the Parametric Equations in the Graphing Utility

Enter the given equations into your graphing utility. Make sure to replace \(\theta\) with the variable that your specific tool uses. Typically, this is 't' for time. So the equations will look like: \(x = 4 + 3\cos(t)\) and \(y = -2 + 2\sin(t)\).
03

Set the Parameter Range

In order to cover all possible points, make sure to set the range of the parameter \(\theta\) or 't' to cover the full circle, that is \(0\) to \(2\pi\). This is because \(\theta\) is the angle in the unit circle, and a full circle's angle measurement is \(2\pi\) radians.
04

Generate the Graph

After setting up the equations and the range correctly, generate the graph. This will provide a visual representation of all the points \((x, y)\) formed by various values of \(\theta\) from \(0\) to \(2\pi\).

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