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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 12.2.13
Chapter 12, Problem 12.2.13

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-4, 3π/2)

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Recall that a point in polar coordinates is given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
Plot the point \((-4, \frac{3\pi}{2})\) by first considering the angle \(\frac{3\pi}{2}\), which corresponds to the downward direction along the negative y-axis.
Since \(r\) is negative, move in the opposite direction of the angle \(\frac{3\pi}{2}\). This means you move 4 units upward (opposite to downward) along the positive y-axis.
To find two alternative representations, use the fact that adding \(2\pi\) to the angle does not change the point, and changing the sign of \(r\) while adding \(\pi\) to the angle gives the same point. So, the alternatives are:
1) \((r, \theta + 2\pi) = (-4, \frac{3\pi}{2} + 2\pi)\) and 2) \((-r, \theta + \pi) = (4, \frac{3\pi}{2} + \pi)\). Write these explicitly as your alternative polar coordinates.

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Key Concepts

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

Polar Coordinates System

Polar coordinates represent points in a plane using a radius and an angle, denoted as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. This system is useful for describing locations in circular or rotational contexts.
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Intro to Polar Coordinates

Graphing Points in Polar Coordinates

To graph a point (r, θ), start at the origin, rotate counterclockwise by angle θ, then move outward (or inward if r is negative) by distance r. Negative radius values mean moving in the opposite direction of the angle, which affects the point's position on the plane.
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Alternative Representations of Polar Coordinates

A single point can have multiple polar coordinate representations by adding or subtracting full rotations (2π) to the angle or by changing the sign of the radius and adjusting the angle by π. For example, (r, θ) is equivalent to (-r, θ + π) and (r, θ + 2πk) for any integer k.
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Related Practice
Textbook Question

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

A parabola with focus at (3, 0)

Textbook Question

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

Textbook Question

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r cos θ = -4

Textbook Question

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

A parabola that opens to the right with directrix x = -4

Textbook Question

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 4 cos t, y = 4 sin t; slope = 1/2

Textbook Question

Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2

c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

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