/*! 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 14 Find the equation of the parabol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 14

Find the equation of the parabola defined by the given information. Sketch the parabola. Vertex: (2,3)\(;\) directrix: \(x=4\)

Short Answer

Expert verified
The equation is \((y - 3)^2 = 8(x - 2)\). The parabola opens to the left.

Step by step solution

01

Identify Parabola Characteristics

The vertex of the parabola is given as the point \((2, 3)\). The directrix is a vertical line given by \(x = 4\). Since the directrix is vertical, the parabola opens horizontally.
02

Determine the Direction and Equation Form

The parabola opens to the left because the directrix \(x = 4\) is to the right of the vertex \((2, 3)\). The standard form of a horizontally opening parabola is:\[(y - k)^2 = -4p(x - h)\]Here, \((h, k)\) is the vertex of the parabola, and \(p\) is the distance from the vertex to the directrix.
03

Calculate the Distance 'p'

The distance from the vertex to the directrix \(x = 4\) is 2 units because the vertex is at \(x = 2\). If the parabola opens to the left, \(p\) is negative. Thus, \(p = -2\).
04

Substitute Values into the Standard Form

Substitute \((h, k) = (2, 3)\) and \(p = -2\) into the equation for a horizontal parabola:\[(y - 3)^2 = -4(-2)(x - 2)\] Simplify this to get the equation of the parabola:\[(y - 3)^2 = 8(x - 2)\]
05

Verify and Sketch the Parabola

The parabola has its vertex at (2,3), opens to the left, and its directrix is the line \(x = 4\). Use these characteristics to sketch the parabola. The axis of symmetry is horizontal and passes through \(y = 3\).

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.

Vertex of a Parabola
Understanding the vertex of a parabola is crucial as it is essentially the starting point. The vertex is the peak point or the lowest point of a parabola depending on the direction it opens. In this context, it acts as a reference point to determine the shape and position of the parabola on the graph.
  • For our exercise, the vertex is at the point \(2, 3\), which is pivotal in forming the parabola equation.
  • It serves as the \((h, k)\) in the parabola's standard form equation, specifically, \((h, k) = (2, 3)\).
  • Knowing the vertex helps in sketching the parabola accurately on a graph.
The vertex also allows you to understand other properties of the parabola, like its axis of symmetry, which is a line that splits the parabola into two mirror images. For a horizontally opening parabola, this axis is a horizontal line passing through the y-value of the vertex.
Directrix of a Parabola
The directrix is an important component in defining a parabola. It is a line that, together with the focus, helps form the parabola's shape. In simple terms, a parabola is a set of points equidistant from a fixed point, the focus, and a line, the directrix.
  • In this exercise, the directrix is given as the equation \(x = 4\).
  • It lies vertically, indicating that our parabola opens in the horizontal direction.
  • The directrix is always perpendicular to the axis of symmetry of the parabola.
For our specific example, the vertex is to the left of the directrix, leading the parabola to open leftwards. The distance from the vertex to the directrix is crucial for determining the value of \(p\), which influences the equation's form and graph orientation.
Horizontal Parabola Standard Form
The standard form of the parabola varies depending on its opening direction. For a horizontally oriented parabola, the standard form is \[(y - k)^2 = 4p(x - h)\]. Here, \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus or directrix.
  • For our specific parabola exercise, \((h, k) = (2, 3)\), and \(p = -2\).
  • The negative \(p\) value indicates the parabola opens to the left.
  • Plugging the values into the equation, we formed \((y - 3)^2 = 8(x - 2)\).
  • This equation helps in accurately sketching and verifying the parabola's position and direction.
Understanding how the standard form changes with the parabola orientation is essential for correctly plotting and analyzing parabolas on a graph, allowing deeper insight into their mathematical and practical applications.

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

A solid of revolution is described. Find or approximate its surface area as specified. Find the surface area of the sphere formed by rotating the circle \(x=2 \cos t, \quad y=2 \sin t\) about: (a) the \(x\) -axis and (b) the \(y\) -axis.

Find: (a) \(\frac{d y}{d x}\) (b) the equation of the tangent and normal lines to the curve at the indicated \(\theta\) -value. \(r=\cos (3 \theta) ; \quad \theta=\pi / 6\)

Graph the polar function on the given interval. \(r=2, \quad 0 \leq \theta \leq \pi / 2\)

Find: (a) \(\frac{d y}{d x}\) (b) the equation of the tangent and normal lines to the curve at the indicated \(\theta\) -value. \(r=1 ; \quad \theta=\pi / 4\)

Find the arc length of the graph of the parametric equations on the given interval(s). \(x=e^{t / 10} \cos t, \quad y=e^{t / 10} \sin t\) on \([0,2 \pi]\) and \([2 \pi, 4 \pi]\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.