Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 25

Capitolo 5, Problema 25

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 = √t, y = t − 1

Guida verificata passo dopo passo

Start with the given parametric equations: \(x = \sqrt{t}\) and \(y = t - 1\).

Express \(t\) in terms of \(x\) from the first equation: since \(x = \sqrt{t}\), then \(t = x^2\) (note that \(x \geq 0\) because square roots are non-negative).

Substitute \(t = x^2\) into the second equation to eliminate the parameter: \(y = x^2 - 1\).

Recognize that the rectangular equation is \(y = x^2 - 1\), which is a parabola opening upwards, shifted down by 1 unit.

To sketch the curve, plot the parabola \(y = x^2 - 1\) for \(x \geq 0\) (since \(x = \sqrt{t}\) implies \(x \geq 0\)), and use arrows pointing in the direction of increasing \(t\) (which corresponds to increasing \(x\) and \(y\) values).

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Parametric Equations and Parameters

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Understanding how the parameter t controls the position on the curve is essential for analyzing the curve's shape and orientation.

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Percorso guidato

05:59

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to form a single rectangular equation in terms of x and y. This process helps to identify the curve's equation in the Cartesian plane, making it easier to analyze and sketch.

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Percorso guidato

05:59

Eliminating the Parameter

Sketching and Orientation of Curves

Sketching the curve requires plotting points that satisfy the rectangular equation and indicating the direction of increasing parameter t with arrows. Orientation shows how the curve is traced as t increases, which is important for understanding the curve's behavior.

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Percorso guidato

06:05

Eliminate the Parameter Example 2

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