Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.4.38

Chapter 12, Problem 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.

Verified step by step guidance

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 \[x^2 = -8(y - 2)\].

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

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.

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.

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.

Recommended video:

5:47

Solving Exponential Equations Using Logs

Related Practice

Textbook Question

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)

A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?

Textbook Question

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

Textbook Question

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.

An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

views

Textbook Question

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)

views

Textbook Question

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 horizontal line segment starting at P(8, 2) and ending at Q(−2, 2)

views

Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

x² + y²/9 = 1