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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 39
Capitolo 5, Problema 39

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = 2ᵗ, y = 2⁻ᵗ; t ≥ 0

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Start with the given parametric equations: \(x = 2^{t}\) and \(y = 2^{-t}\), where \(t \geq 0\).
Express \(y\) in terms of \(x\) by eliminating the parameter \(t\). Since \(x = 2^{t}\), take the logarithm base 2 of both sides to get \(t = \log_{2} x\).
Substitute \(t = \log_{2} x\) into the equation for \(y\): \(y = 2^{-t} = 2^{-\log_{2} x}\).
Use the property of exponents and logarithms: \(2^{-\log_{2} x} = x^{-1} = \frac{1}{x}\). So the rectangular equation is \(y = \frac{1}{x}\).
Determine the domain for \(x\) based on \(t \geq 0\). Since \(x = 2^{t}\) and \(t \geq 0\), \(x \geq 1\). Sketch the curve \(y = \frac{1}{x}\) for \(x \geq 1\), and add arrows indicating the orientation as \(t\) increases (which corresponds to \(x\) increasing).

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y being directly related to 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 rewriting the parametric equations to form a single equation relating x and y directly. This is done by solving one equation for t and substituting into the other, enabling analysis and graphing in the rectangular coordinate system.
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Eliminating the Parameter

Graphing and Orientation of Parametric Curves

Graphing parametric curves requires plotting points (x(t), y(t)) for values of t in the given interval. Orientation is shown by arrows indicating the direction of increasing t, which helps understand the curve's traversal and behavior over the parameter range.
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04:47
Introduction to Parametric Equations
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