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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 12.4.35
Chapter 12, Problem 12.4.35

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola symmetric about the y-axis that passes through the point (2, -6)

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Identify the general form of the parabola symmetric about the y-axis with vertex at the origin. This form is given by \(y = ax^2\), where \(a\) is a constant to be determined.
Substitute the coordinates of the given point \((2, -6)\) into the equation \(y = ax^2\) to find the value of \(a\). This means plugging in \(x = 2\) and \(y = -6\).
Write the equation after substitution: \(-6 = a \times (2)^2\).
Solve for \(a\) by dividing both sides of the equation by \(4\) (since \(2^2 = 4\)), which gives \(a = \frac{-6}{4}\).
Write the final equation of the parabola by substituting the value of \(a\) back into the general form: \(y = ax^2\).

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Key Concepts

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

Standard Form of a Parabola

A parabola symmetric about the y-axis with its vertex at the origin can be expressed as y = ax². This form shows that y depends on the square of x, and the coefficient a determines the parabola's width and direction (upward if a > 0, downward if a < 0).
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Circles in Standard Form Example 1

Using a Point to Find the Coefficient

To find the specific equation of a parabola, substitute the coordinates of a known point on the curve into the standard form. This allows solving for the coefficient a, tailoring the general equation to the given parabola.
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Critical Points

Symmetry About the y-Axis

A parabola symmetric about the y-axis means its graph is mirrored on either side of the y-axis. This symmetry implies the equation involves only even powers of x, ensuring that f(x) = f(-x) for all x.
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Disk Method Using y-Axis
Related Practice
Textbook Question

Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge? 

Textbook Question

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


(1, 2π/3)

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Textbook Question

Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.

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Textbook Question

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves


r = 3 sin θ and r = 3 cos θ

Textbook Question

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


The region inside the curve r = √(cos θ) and inside the circle r = 1/√2 in the first quadrant

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Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2