Ch. 7 - Conic Sections

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

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 25

Chapter 8, Problem 25

Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: (2, - 3); Focus: (2, - 5)

Verified step by step guidance

Identify the vertex $ V = (2, -3) $ and the focus $ F = (2, -5) $. Since the x-coordinates are the same, the parabola opens vertically (either up or down).

Calculate the distance $ p $ between the vertex and the focus. This distance determines how far the parabola opens from the vertex. Use the formula $ p = y_{focus} - y_{vertex} $.

Write the standard form of the parabola that opens vertically: $ (x - h)^2 = 4p(y - k) $, where $ (h, k) $ is the vertex.

Substitute the vertex coordinates $ (h, k) = (2, -3) $ and the value of $ p $ found in step 2 into the equation.

Simplify the equation to get the standard form 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 when the vertex and focus are known.

Standard Form of a Parabola

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

Finding the Parameter p

The parameter p represents the distance from the vertex to the focus (or directrix) and determines the parabola's width and direction. Calculating p from the given vertex and focus coordinates is essential to write the equation in standard form.

Recommended video:

3:43

Finding the Domain of an Equation

Related Practice

Textbook Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-5, 0), (5, 0); vertices: (-8, 0), (8,0)

views

Textbook Question

Find the standard form of the equation of each hyperbola.

Textbook Question

Explain why it is not possible for a hyperbola to have foci at (0,-2) and (0,2) and vertices at (0,-3) and (0,3).

views

rank

Textbook Question

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (-8,0), (8,0); Vertices: (-3,0), (3,0)

views

Textbook Question

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $y = \pm \sqrt{x^{2} - 2}$

views

Textbook Question

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (0,-4), (0,4); Vertices: (0, -2), (0,2)

views