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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 12.1.29
Chapter 12, Problem 12.1.29

15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.


x = 8 + 2t, y = 1; −∞ < t < ∞

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1
Identify the given parametric equations: \(x = 8 + 2t\) and \(y = 1\), where \(t\) is the parameter.
To eliminate the parameter \(t\), solve one of the equations for \(t\). From \(x = 8 + 2t\), isolate \(t\) by subtracting 8 and then dividing by 2: \(t = \frac{x - 8}{2}\).
Substitute the expression for \(t\) into the other equation. Since \(y = 1\) is constant and does not depend on \(t\), the equation in terms of \(x\) and \(y\) is simply \(y = 1\).
Interpret the resulting equation \(y = 1\): this represents a horizontal line in the \(xy\)-plane at the height \(y = 1\).
Determine the positive orientation by considering how \(x\) changes as \(t\) increases. Since \(x = 8 + 2t\), as \(t\) increases, \(x\) increases, so the curve is oriented from left to right along the line \(y = 1\).

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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 the description of more complex curves and motions.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This helps identify the geometric shape of the curve without referencing the parameter.
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Eliminating the Parameter

Curve Orientation

Curve orientation refers to the direction in which the curve is traced as the parameter increases. Understanding orientation is important for interpreting motion or direction along the curve, often indicated by arrows or parameter intervals.
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Summary of Curve Sketching
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