Skip to main content
Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.46
Chapter 12, Problem 12.1.46

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.
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.
Express the \(x\)-coordinate as a linear function of \(t\) that moves from 8 to −2. This can be written as \(x(t) = 8 + t(-2 - 8) = 8 - 10t\).
Since the \(y\)-coordinate is constant at 2, write \(y(t) = 2\) for all \(t\) in the interval.
State the parametric equations and the interval for \(t\): \(x(t) = 8 - 10t\), \(y(t) = 2\), with \(t\) in the interval \([0, 1]\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
Guided course
08:02
Parameterizing Equations

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.
Recommended video:
05:13
Slopes of Tangent Lines

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.
Recommended video:
Guided course
05:59
Eliminating the Parameter
Related Practice
Textbook Question

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?


Textbook Question

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 segment of the parabola y=2x ²−4, where −1≤x≤5

1
views
Textbook Question

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.


An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

1
views
Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

1
views
Textbook Question

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 line segment starting at P(0, 0) and ending at Q(2, 8)

1
views