Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Statement
**Question 5:**
Identify the graph of the equation
\[ 4x^2 + 16y - 16x + 32 = 0 \]
as a parabola, ellipse, or hyperbola.
### Instructions for Solving the Problem
To determine the nature of the conic section represented by the given equation, follow these steps:
1. **Rewrite the Equation in Standard Form**: Transform the given equation into a more recognizable form of a conic section by completing the square if necessary.
2. **Identify the Conic Section**:
- **Parabola**: An equation of the form \(Ax^2 + By = C\) or \(Ax + By^2 = C\).
- **Ellipse**: An equation where both \(x^2\) and \(y^2\) terms are present, and their coefficients have the same sign.
- **Hyperbola**: An equation where both \(x^2\) and \(y^2\) terms are present, and their coefficients have opposite signs.
### Detailed Steps and Solution
1. **Rewrite in Standard Form**:
Given equation:
\[ 4x^2 + 16y - 16x + 32 = 0 \]
Simplify and Complete the Square:
\[ 4x^2 - 16x + 16y + 32 = 0 \]
Group \(x\) terms and complete the square:
\[ 4(x^2 - 4x) + 16y + 32 = 0 \]
\[ 4(x^2 - 4x + 4) - 16 + 16y + 32 = 0 \]
\[ 4(x - 2)^2 + 16y + 16 = 0 \]
Simplify further:
\[ 4(x - 2)^2 + 16y = -16 \]
\[ (x - 2)^2 + 4y = -4 \]
2. **Identify the Conic Section**:
- The equation \((x - 2)^2 + 4y = -4\) (after simplification shows a squared term for \(x\) and a linear term for \(y\)) does NOT represent a standard form for ellipses or hyper](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c941f29-ddef-4885-9d8b-99e8246804cb%2F87bf5e30-703f-4077-8e18-ae0c3d596e0e%2F8qcp04g.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Question 5:**
Identify the graph of the equation
\[ 4x^2 + 16y - 16x + 32 = 0 \]
as a parabola, ellipse, or hyperbola.
### Instructions for Solving the Problem
To determine the nature of the conic section represented by the given equation, follow these steps:
1. **Rewrite the Equation in Standard Form**: Transform the given equation into a more recognizable form of a conic section by completing the square if necessary.
2. **Identify the Conic Section**:
- **Parabola**: An equation of the form \(Ax^2 + By = C\) or \(Ax + By^2 = C\).
- **Ellipse**: An equation where both \(x^2\) and \(y^2\) terms are present, and their coefficients have the same sign.
- **Hyperbola**: An equation where both \(x^2\) and \(y^2\) terms are present, and their coefficients have opposite signs.
### Detailed Steps and Solution
1. **Rewrite in Standard Form**:
Given equation:
\[ 4x^2 + 16y - 16x + 32 = 0 \]
Simplify and Complete the Square:
\[ 4x^2 - 16x + 16y + 32 = 0 \]
Group \(x\) terms and complete the square:
\[ 4(x^2 - 4x) + 16y + 32 = 0 \]
\[ 4(x^2 - 4x + 4) - 16 + 16y + 32 = 0 \]
\[ 4(x - 2)^2 + 16y + 16 = 0 \]
Simplify further:
\[ 4(x - 2)^2 + 16y = -16 \]
\[ (x - 2)^2 + 4y = -4 \]
2. **Identify the Conic Section**:
- The equation \((x - 2)^2 + 4y = -4\) (after simplification shows a squared term for \(x\) and a linear term for \(y\)) does NOT represent a standard form for ellipses or hyper
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