Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 29

Chapter 8, Problem 29

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (- 3, 4); Directrix: y = 2

Verified step by step guidance

Recall that the standard form of a parabola with a vertical axis of symmetry is given by the equation \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus (or directrix).

Identify the focus \( F(-3, 4) \) and the directrix \( y = 2 \). The vertex \( V \) lies exactly halfway between the focus and the directrix along the vertical line.

Calculate the vertex coordinates by finding the midpoint between the focus and the directrix: \( k = \frac{4 + 2}{2} \) for the \( y \)-coordinate, and \( h = -3 \) (same as the focus's \( x \)-coordinate).

Determine the value of \( p \), which is the distance from the vertex to the focus (or directrix). Since the parabola opens vertically, \( p = 4 - k \) (distance from vertex to focus).

Substitute \( h \), \( k \), and \( p \) into the standard form equation \( (x - h)^2 = 4p(y - k) \) to write the equation of the parabola.

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.

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this definition helps in deriving the equation of the parabola based on given focus and directrix.

Standard Form of a Parabola

The standard form of a parabola's equation depends on its orientation. For a vertical parabola, it is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix. This form is essential for writing the equation once the vertex and p are known.

Finding the Vertex and Parameter p

The vertex lies midway between the focus and directrix. The distance p is the distance from the vertex to the focus (positive if the parabola opens upward, negative if downward). Calculating these allows you to write the parabola's equation in standard form.

Recommended video:

08:07

Vertex Form

Related Practice

Textbook Question

Find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). (y - 1)² = 4(x - 1)

a. b. c. d.

Textbook Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3

Textbook Question

Identify the conic represented by the equation without completing the square. y^2 + 4x + 2y - 15 = 0

Textbook Question

Find the standard form of the equation of each hyperbola.

Textbook Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (3, 2); Directrix: x = - 1

views

Textbook Question

Identify the conic represented by the equation without completing the square. 4x^2 - 9y^2 - 8x + 12y - 144 = 0