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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 15
Capitolo 5, Problema 15

Eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve. x = √t , y = t + 1; −∞ < t < ∞

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Identify the given parametric equations: \(x = \sqrt{t}\) and \(y = t + 1\), with the parameter \(t\) ranging over all real numbers.
Since \(x = \sqrt{t}\), note that \(x\) is defined only for \(t \geq 0\) because the square root of a negative number is not a real number. This restricts the domain of \(t\) to \(t \geq 0\).
Express \(t\) in terms of \(x\) by squaring both sides of the equation \(x = \sqrt{t}\), which gives \(t = x^2\).
Substitute \(t = x^2\) into the equation for \(y\): \(y = t + 1\) becomes \(y = x^2 + 1\).
The Cartesian equation of the curve is \(y = x^2 + 1\) with \(x \geq 0\). To graph, plot this parabola starting from \(x=0\) (where \(y=1\)) and moving rightward, using arrows to indicate the direction as \(t\) increases.

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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 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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08:02
Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to express y directly in terms of x, removing t. This is done by solving one equation for t and substituting into the other, which helps in identifying the Cartesian form of the curve.
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05:59
Eliminating the Parameter

Orientation of Parametric Curves

Orientation indicates the direction in which the curve is traced as the parameter increases. Using arrows on the graph shows this direction, which is important for understanding the behavior and properties of the curve over the parameter's domain.
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04:47
Introduction to Parametric Equations
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