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Chapter 6: Problem 108

Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?

Short Answer

Expert verified
The process of sketching a curve from parametric equations involves understanding parametric equations, calculating points, plotting these points with correct orientation on a graph, and then connecting the points following the same direction of orientation. Curve orientation refers to the direction a point 'travels' along the curve as the value of the parameter increases.

Step by step solution

01

Understand Parametric Equations

A parametric equation defines a group of quantities as functions of one or more independent variables called parameters. For a two-dimensional plane, these parameters are usually denoted as \(x\) and \(y\), with a third variable, generally \(t\), being the parameter. \(x\) and \(y\) are related to \(t\) by the two equations i.e. \(x= f(t)\) and \(y= g(t)\).
02

Calculating Points

By incrementally choosing typical values for \(t\), for example -3 to 3 with increments of 1, find the corresponding \(x\) and \(y\) values from the provided parametric equations. These pairs of \(x\) and \(y\) values will provide points on the graph.
03

Plotting the Points

Plot the points on a graph. Be sure to indicate the orientation - the path a point takes as \(t\) increases. NOTE: The values of \(t\) are usually chosen to start from a lower value gradually increase to a higher value. This normally indicates the orientation of the curves.
04

Connect The Points

Draw a curve through the plotted points, making sure to follow the indicated orientation. The drawn curve is the graphical representation of the parametric equations.
05

Understanding Curve Orientation

Curve orientation refers to the direction that a point would take as it moves along the curve. This is established by the order of the plotted points as the parameter increases. This will provide a sense of 'travel direction' on the curve. Pay attention, the same set of points can create different curves with different orientations when sequenced differently.

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