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 = r − 1, y = r³; −4 ≤ r ≤ 4

검증된 단계별 안내

Identify the given parametric equations: \(x = r - 1\) and \(y = r^{3}\), with the parameter \(r\) ranging from \(-4\) to \(4\).

To eliminate the parameter \(r\), solve the first equation for \(r\): \(r = x + 1\).

Substitute \(r = x + 1\) into the second equation to express \(y\) solely in terms of \(x\): \(y = (x + 1)^{3}\).

Recognize that the resulting equation \(y = (x + 1)^{3}\) represents a cubic curve, which is a shifted cubic function along the x-axis.

For the positive orientation, note that as \(r\) increases from \(-4\) to \(4\), \(x\) increases from \(-5\) to \(3\), and \(y\) follows the cubic relationship accordingly, indicating the direction of the curve from left to right.

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주요 개념

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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 or r. Instead of y as a function of x, both x and y depend on the parameter, allowing representation of more complex curves and motions.

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 the parameter and substituting into the other, which helps in identifying the curve's shape in the xy-plane.

Curve Orientation and Description

Curve orientation refers to the direction in which the curve is traced as the parameter increases. Describing the curve involves identifying its shape and key features, while orientation indicates the path's direction, important for understanding motion or flow along the curve.

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 = r − 1, y = r³; −4 ≤ r ≤ 4