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Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 8 - Complex Numbers, Polar Equations, and Parametric EquationsProblema 27
Capitolo 9, Problema 27

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. See Examples 1 and 2.


x = t + 2 , y = t ―4 , for t in (― ∞ , ∞)

Guida verificata passo dopo passo
1
Step 1: Understand the parametric equations given: \(x = t + 2\) and \(y = t - 4\), where \(t\) is a parameter that can take any real value from \(-\infty\) to \(\infty\).
Step 2: To graph the curve, recognize that as \(t\) varies, the point \((x, y)\) moves along the curve defined by these equations. Plot several points by choosing values of \(t\), calculating corresponding \(x\) and \(y\), and then sketch the curve through these points.
Step 3: To find a rectangular equation (an equation involving only \(x\) and \(y\)), eliminate the parameter \(t\) from the system. From the first equation, express \(t\) in terms of \(x\): \(t = x - 2\).
Step 4: Substitute \(t = x - 2\) into the second equation: \(y = (x - 2) - 4\).
Step 5: Simplify the expression to get the rectangular equation: \(y = x - 6\). This equation represents the same curve as the parametric equations but in terms of \(x\) and \(y\) only.

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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. Understanding how x and y depend on t allows you to describe and analyze curves that may not be functions in the traditional sense.
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Percorso guidato
08:02
Parameterizing Equations

Eliminating the Parameter to Find Rectangular Equations

To convert parametric equations into a rectangular (Cartesian) equation, you solve one equation for the parameter and substitute into the other. This process removes the parameter t, yielding a direct relationship between x and y.
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Percorso guidato
05:59
Eliminating the Parameter

Graphing Parametric Curves

Graphing parametric curves involves plotting points (x(t), y(t)) for various values of t. Recognizing the shape and direction of the curve helps in visualizing the relationship between x and y and verifying the rectangular equation.
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Percorso guidato
04:47
Introduction to Parametric Equations
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For each pair of polar coordinates, (c) give the rectangular coordinates for the point. See Examples 1 and 2(a).


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Graph each plane curve defined by the parametric equations for t in [0, 2π] Then find a rectangular equation for the plane curve. See Example 3.


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