/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 Write an equation for each parab... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 42

Write an equation for each parabola. vertex \((-2,1),\) focus \((-2,-3)\)

Short Answer

Expert verified
(x+2)^2 = -16(y-1)

Step by step solution

01

Determine the direction of the parabola

The vertex is (-2,1), and the focus is (-2,-3). Since the focus is below the vertex, the parabola opens downwards.
02

Calculate the distance from the vertex to the focus

The distance from the vertex to the focus (p) is the vertical distance between -2 and -3, which is |1 - (-3)| = 4.
03

Write the standard form of the parabola equation

For a downward opening parabola with vertex (h, k) and distance 4 : (y - k)^2 = 4p(x - h), so we use (h, k) = (-2, 1) and p = -4.Equation: (x+2)^2 = -16(y-1)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

parabola properties
A parabola is a U-shaped curve that can open upwards, downwards, left, or right. The direction in which it opens depends on its orientation. For parabolas that open upwards or downwards, their equations are in the form \(\( (x - h)^2 = 4p(y - k) \)\).
If it opens upwards, \(\( p > 0 \)\). If it opens downwards, \(\( p < 0 \)\).
For parabolas that open left or right, the standard form is \(\( (y - k)^2 = 4p(x - h) \)\), where the direction is determined by the sign of \(\( p \)\).
Besides direction, some core properties include:
  • Vertex: The highest or lowest point of the parabola.
  • Focus: A fixed point, which shows how
vertex and focus relationship
The relationship between the vertex and the focus is crucial for understanding the geometry of a parabola. Each parabola is uniquely determined by its vertex and focus.
The vertex \(\( (h, k) \)\) is the point where the parabola changes direction. It is the midpoint between the directrix (a line) and the focus (a point).
The focus \(\( (h, k \pm p) \)\) lies inside the parabola, and is a fixed point that influences the shape.
The parameter \(\( p \)\) is the distance between the vertex and the focus. If \(\( p > 0 \)\), the parabola opens upwards or to the right, while if \(\( p < 0 \)\), it opens downwards or to the left depending on the equation's form.
standard form of a parabola
The standard form of a parabola helps simplify its graphing and interpretation. The equation differs based on whether the parabola opens vertically or horizontally. For a vertical parabola, the equation is:
\( (x - h)^2 = 4p(y - k) \)
  • Where \(\( (h, k) \)\) is the vertex.
  • \(\( p \)\) is the distance from the vertex to the focus. Ensure you consider its sign accurately.

For horizontal parabolas:
\( (y - k)^2 = 4p(x - h) \)
  • Where \(\( (h, k) \)\) is still the vertex.
  • The distance \(\( p \)\) now determines the horizontal direction.

By identifying these components, you can quickly write the equation of any parabola given its vertex and focus.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

$$\text {Give the focus, directrix, and axis of symmetry for each parabola.}$$ $$y^{2}=-4 x$$

$$\text {Graph each horizontal parabola, and give the domain and range.}$$ $$-x=3 y^{2}+6 y+2$$

Write an equation for each hyperbola. vertices at \((5,-2),(1,-2) ;\) asymptotes \(y=\pm \frac{3}{2}(x-3)-2\)

Write an equation for each parabola with vertex at the origin. through \((3,2),\) symmetric with respect to the \(x\) -axis

Write an equation for each hyperbola. vertices at \((4,5),(4,1) ;\) asymptotes \(y=\pm 7(x-4)+3\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.