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 12.1.51
Chapter 12, Problem 12.1.51

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 upper half of the parabola x=y ², originating at (0, 0)

Verified step by step guidance
1
Identify the given curve equation: \(x = y^2\). Since we want the upper half of the parabola, this corresponds to the part where \(y \geq 0\).
Choose a parameter to express both \(x\) and \(y\) in terms of it. A natural choice is to let \(t = y\), since \(y\) is already isolated in the equation.
Express \(x\) in terms of \(t\): since \(x = y^2\), substituting \(y = t\) gives \(x = t^2\).
Write the parametric equations as \(x = t^2\) and \(y = t\), where \(t\) represents the parameter.
Determine the interval for \(t\): because we want the upper half of the parabola starting at \((0,0)\) and \(y \geq 0\), the parameter \(t\) should satisfy \(t \geq 0\). So, the interval is \(t \in [0, \infty)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
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 are defined in terms of t, allowing more flexible representations of curves, including those that are not functions in the traditional sense.
Recommended video:
Guided course
08:02
Parameterizing Equations

Parabola and Its Equation

A parabola is a conic section defined by a quadratic relationship between x and y. In this case, the parabola is given by x = y², which opens along the x-axis. Understanding this relationship helps in choosing a parameter that can generate points on the curve.
Recommended video:
7:42
Properties of Parabolas

Parameter Interval and Curve Orientation

The parameter interval defines the range of values for the parameter t that traces the desired portion of the curve. For the upper half of the parabola x = y² starting at (0,0), the parameter must be chosen so that y ≥ 0, ensuring only the upper half is represented.
Recommended video:
02:59
Finding Area Between Curves that Cross on the Interval
Related Practice
Textbook Question

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.


(1, √3)

1
views
Textbook Question

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(2, 7π/4)

1
views
Textbook Question

45–60. Areas of regions Find the area of the following regions.


The region inside the lemniscate r² = 6 sin 2θ

Textbook Question

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

Textbook Question

23–24. Radar Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane, typically measured in degrees clockwise from north.

A plane is 50 miles from a radar station at an angle of 10 dgeree clockwise from north. Find polar coordinates for the location of the plane.

1
views
Textbook Question

Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.

1
views