37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The horizontal line segment starting at P(8, 2) and ending at Q(−2, 2)

Verified step by step guidance

1. Identify the key characteristics of the curve: it is a horizontal line segment starting at point P(8, 2) and ending at point Q(−2, 2). Since the y-coordinate is constant (y = 2) for all points on this segment, the parametric equations will reflect this.
2. Choose a parameter, say \(t\), to represent the position along the line segment. A common choice is to let \(t\) vary from 0 to 1, where \(t=0\) corresponds to the starting point P and \(t=1\) corresponds to the ending point Q.
3. Express the \(x\)-coordinate as a linear function of \(t\) that moves from 8 to −2. This can be written as \(x(t) = 8 + t(-2 - 8) = 8 - 10t\).
4. Since the \(y\)-coordinate is constant at 2, write \(y(t) = 2\) for all \(t\) in the interval.
5. State the parametric equations and the interval for \(t\): \(x(t) = 8 - 10t\), \(y(t) = 2\), with \(t\) in the interval \([0, 1]\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing more flexible descriptions of curves, including lines and segments.

Representation of Line Segments

A line segment between two points can be represented parametrically by defining x and y as linear functions of a parameter t, typically ranging from 0 to 1. This parameterization moves from the start point to the end point as t increases.

Parameter Interval

The parameter interval specifies the range of t values for which the parametric equations describe the desired portion of the curve. For line segments, t usually varies over a closed interval like [0,1] to cover the entire segment between two points.