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What specific (include direction/opening and vertex/center) conic section is 4x + y + 72x + 14y + 273 = 0? 2. Explain using sentences and prove with mathematical evidence.

What specific (include direction/opening and vertex/center) conic section is 4x + y + 72x + 14y + 273 = 0? 2. Explain using sentences and prove with mathematical evidence.

Calculus: Early Transcendentals
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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**Question for Exploration:** What specific characteristics (including direction/opening and vertex/center) does the conic section exhibit in the equation \(4x^2 + y^2 + 72x + 14y + 273 = 0\)? **Instruction:** Explain using sentences and prove with mathematical evidence. **Guidance Overview:** This task requires analyzing the given equation of a conic section. You should identify the type of conic section, whether it is a parabola, ellipse, circle, or hyperbola. Additionally, determine key features such as direction of opening (if applicable) and coordinates of the vertex or center. 1. **Rearrange and Simplify the Equation** - Start by attempting to rearrange the given equation into a standard form to identify the conic section type. 2. **Identify the General Form** - Recognize coefficients associated with squared terms (\(x^2\), \(y^2\)) and correlate these with standard forms for conic sections. 3. **Calculate the Vertex or Center** - Use methods like completing the square to identify the vertex or center. 4. **Direction and Opening** - Based on the standard form discovered, deduce the orientation or direction. 5. **Mathematical Proof** - Provide mathematical steps and calculations to substantiate the identification and characteristics. This analytical approach will yield a comprehensive understanding of the conic section represented by the given equation.
Transcribed Image Text:**Question for Exploration:** What specific characteristics (including direction/opening and vertex/center) does the conic section exhibit in the equation \(4x^2 + y^2 + 72x + 14y + 273 = 0\)? **Instruction:** Explain using sentences and prove with mathematical evidence. **Guidance Overview:** This task requires analyzing the given equation of a conic section. You should identify the type of conic section, whether it is a parabola, ellipse, circle, or hyperbola. Additionally, determine key features such as direction of opening (if applicable) and coordinates of the vertex or center. 1. **Rearrange and Simplify the Equation** - Start by attempting to rearrange the given equation into a standard form to identify the conic section type. 2. **Identify the General Form** - Recognize coefficients associated with squared terms (\(x^2\), \(y^2\)) and correlate these with standard forms for conic sections. 3. **Calculate the Vertex or Center** - Use methods like completing the square to identify the vertex or center. 4. **Direction and Opening** - Based on the standard form discovered, deduce the orientation or direction. 5. **Mathematical Proof** - Provide mathematical steps and calculations to substantiate the identification and characteristics. This analytical approach will yield a comprehensive understanding of the conic section represented by the given equation.
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