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describe now you can tell from the equation where to graph the branches of this hyperbola & write the equations of the asymprores Y= 4x-2 3x+1

describe now you can tell from the equation where to graph the branches of this hyperbola & write the equations of the asymprores Y= 4x-2 3x+1

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.2: Introduction To Conics: parabolas
Problem 72E
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# Understanding Hyperbola Asymptotes from the Equation ## Learning Objective Describe how you can tell from the equation where to graph the branches of the hyperbola and write the equations of the asymptotes. ### Key Concept When given a hyperbola equation, certain properties of the equation can help determine the asymptotes. ### Example Consider the following fraction representing the equation of a hyperbola: \[ y = \frac{4x - 2}{3x + 1} \] ### Steps to Identify the Asymptotes 1. **Identify the equation of the asymptotes**: Asymptotes of a hyperbola can be found by taking the dominant terms in the numerator and denominator when divided. 2. **Factors**: - Numerator: 4x - 2 - Denominator: 3x + 1 ### Calculating the Asymptotes To find the asymptote, divide the leading coefficients of \( x \): \[ y = \frac{4x - 2}{3x + 1} \] As \( x \rightarrow \pm \infty \), the fraction can be approximated by: \[ y \approx \frac{4x}{3x} = \frac{4}{3} \] Thus, the equation of the asymptote is \( y = \frac{4}{3}x \). ### Additional Notes A portion of the given handwriting has been crossed out and is unreadable. Understanding and identifying asymptotes provides significant insight into graphing the behavior and branch locations for hyperbolas. This is essential for more complex studies and applications in mathematics, particularly in calculus and analytic geometry.
Transcribed Image Text:# Understanding Hyperbola Asymptotes from the Equation ## Learning Objective Describe how you can tell from the equation where to graph the branches of the hyperbola and write the equations of the asymptotes. ### Key Concept When given a hyperbola equation, certain properties of the equation can help determine the asymptotes. ### Example Consider the following fraction representing the equation of a hyperbola: \[ y = \frac{4x - 2}{3x + 1} \] ### Steps to Identify the Asymptotes 1. **Identify the equation of the asymptotes**: Asymptotes of a hyperbola can be found by taking the dominant terms in the numerator and denominator when divided. 2. **Factors**: - Numerator: 4x - 2 - Denominator: 3x + 1 ### Calculating the Asymptotes To find the asymptote, divide the leading coefficients of \( x \): \[ y = \frac{4x - 2}{3x + 1} \] As \( x \rightarrow \pm \infty \), the fraction can be approximated by: \[ y \approx \frac{4x}{3x} = \frac{4}{3} \] Thus, the equation of the asymptote is \( y = \frac{4}{3}x \). ### Additional Notes A portion of the given handwriting has been crossed out and is unreadable. Understanding and identifying asymptotes provides significant insight into graphing the behavior and branch locations for hyperbolas. This is essential for more complex studies and applications in mathematics, particularly in calculus and analytic geometry.
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