Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problema 12.4.38

Capitolo 12, Problema 12.4.38

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

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Identify the vertex of the parabola, which is given as (0, 2). This is the highest point on the parabola since it opens downward.

Note the directrix line, which is horizontal and given by the equation \(y = 4\). The parabola opens downward because the vertex is below the directrix.

Find the focus of the parabola. The focus lies the same distance from the vertex as the directrix but on the opposite side. Calculate the distance \(p\) between the vertex and the directrix: \(p = 4 - 2 = 2\). Since the parabola opens downward, the focus is at \((0, 2 - 2) = (0, 0)\).

Use the standard form of a vertical parabola with vertex \((h, k)\) and focus distance \(p\(: \[(x - h)^2 = 4p(y - k)\]. Here, \)h = 0\), \(k = 2\), and \(p = -2\) (negative because it opens downward). Substitute these values to get the equation.

Write the final equation of the parabola as \[x^2 = 4(-2)(y - 2)\] or simplified to \[x^2 = -8(y - 2)\].

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Parabola Definition and Properties

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. The vertex is the midpoint between the focus and directrix. The parabola opens away from the directrix, and its shape depends on the distance between the vertex and the focus.

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Properties of Parabolas

Equation of a Parabola with Vertical Axis

For a parabola with a vertical axis of symmetry, the standard form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus. If p is positive, the parabola opens upward; if negative, it opens downward.

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Horizontal Parabolas

Using the Directrix to Find the Parabola Equation

The directrix is a line equidistant from the vertex as the focus but in the opposite direction. Knowing the directrix's equation helps determine p and the vertex's position, which are essential to write the parabola's equation accurately.

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Solving Exponential Equations Using Logs

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

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

x = 2 + √t, y = 2 - 4t; slope = -8

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

r = sin 3θ

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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)

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What is the slope of the line θ=π/3?

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53–56. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.

An ellipse with vertices (0, ±9) and eccentricity ¼

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