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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)

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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 \(t \in [0, \infty)\).

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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.
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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.
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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.
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Finding Area Between Curves that Cross on the Interval
Related Practice
Textbook Question

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

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Textbook Question

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


(1, √3)

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Textbook Question

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


The region inside the lemniscate r² = 6 sin 2θ

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Textbook Question

What is the polar equation of a circle of radius √(a²+b²) centered at (a, b)?

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Textbook Question

75–76. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

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Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = 2 - 2 sin θ  b

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