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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.41
Chapter 12, Problem 12.1.41

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The line segment starting at P(0, 0) and ending at Q(2, 8)

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Identify the points P(0, 0) and Q(2, 8) as the start and end points of the line segment. We want parametric equations that describe all points on the line segment between these two points.
Recall that a parametric equation for a line segment from point P(x_0, y_0) to Q(x_1, y_1) can be written as: \(x = x_0 + t(x_1 - x_0)\) \(y = y_0 + t(y_1 - y_0)\), where the parameter \(t\) varies over an interval.
Substitute the coordinates of P and Q into the parametric form: \(x = 0 + t(2 - 0) = 2t\) \(y = 0 + t(8 - 0) = 8t\).
Determine the interval for the parameter \(t\). Since the line segment starts at P when \(t=0\) and ends at Q when \(t=1\), the interval for \(t\) is \(0 \leq t \leq 1\).
Thus, the parametric equations for the line segment are \(x = 2t\), \(y = 8t\), with \(t\) in the interval \([0, 1]\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing representation of curves and line segments in a flexible way.
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Parameterizing Equations

Line Segment Parameterization

To parameterize a line segment between two points P and Q, use a parameter t that varies over an interval, typically [0,1]. The coordinates are given by linear interpolation: x(t) = x_P + t(x_Q - x_P), y(t) = y_P + t(y_Q - y_P), tracing the segment as t moves from 0 to 1.
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Parameter Interval

The parameter interval defines the range of t values for which the parametric equations describe the curve. For a line segment, choosing t in [0,1] ensures the curve starts at P when t=0 and ends at Q when t=1, covering all points in between.
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Eliminating the Parameter
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