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Euss the graph of 2 (xZ 9) x²4 . こ must Show ALL Seven the graph

Euss the graph of 2 (xZ 9) x²4 . こ must Show ALL Seven the graph

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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**Educational Exercise: Analyzing the Graph of a Rational Function** **Objective:** Discuss the graph of the function \( f(x) = \frac{2(x^2 - 9)}{x^2 - 4} \). **Instructions:** Follow and demonstrate all seven analysis steps, and provide a detailed graph. **Steps for Discussion:** 1. **Simplify the Expression:** - Factor the numerator and denominator where possible. 2. **Identify Domain:** - Determine where the function is undefined by setting the denominator equal to zero and solving for \(x\). 3. **Find Intercepts:** - Calculate the x-intercepts by setting \(f(x) = 0\). - Calculate the y-intercept by evaluating \(f(0)\). 4. **Analyze Asymptotes:** - Determine vertical asymptotes from the roots of the denominator that do not cancel with the numerator. - Explore horizontal or slant asymptotes by comparing the degrees of the numerator and denominator. 5. **Evaluate Behavior around Asymptotes:** - Examine how the function behaves near the asymptotes, approaching positive or negative infinity. 6. **Identify Symmetry:** - Check for even or odd symmetry, testing \( f(-x) = f(x) \) for even, or \( f(-x) = -f(x) \) for odd. 7. **Determine End Behavior:** - Analyze the behavior of \( f(x) \) as \( x \to \pm \infty \). **Conclusion:** Translate these results into a graph for visualization. Ensure each feature (intercepts, asymptotes, symmetry) is clearly labeled and accurately represented. **Graph Explanation:** This function is a rational function with potential asymptotic behavior and specific intercepts. The exercise challenges the student to identify and graphically represent these characteristics, grounding their understanding in rational function behavior.
Transcribed Image Text:**Educational Exercise: Analyzing the Graph of a Rational Function** **Objective:** Discuss the graph of the function \( f(x) = \frac{2(x^2 - 9)}{x^2 - 4} \). **Instructions:** Follow and demonstrate all seven analysis steps, and provide a detailed graph. **Steps for Discussion:** 1. **Simplify the Expression:** - Factor the numerator and denominator where possible. 2. **Identify Domain:** - Determine where the function is undefined by setting the denominator equal to zero and solving for \(x\). 3. **Find Intercepts:** - Calculate the x-intercepts by setting \(f(x) = 0\). - Calculate the y-intercept by evaluating \(f(0)\). 4. **Analyze Asymptotes:** - Determine vertical asymptotes from the roots of the denominator that do not cancel with the numerator. - Explore horizontal or slant asymptotes by comparing the degrees of the numerator and denominator. 5. **Evaluate Behavior around Asymptotes:** - Examine how the function behaves near the asymptotes, approaching positive or negative infinity. 6. **Identify Symmetry:** - Check for even or odd symmetry, testing \( f(-x) = f(x) \) for even, or \( f(-x) = -f(x) \) for odd. 7. **Determine End Behavior:** - Analyze the behavior of \( f(x) \) as \( x \to \pm \infty \). **Conclusion:** Translate these results into a graph for visualization. Ensure each feature (intercepts, asymptotes, symmetry) is clearly labeled and accurately represented. **Graph Explanation:** This function is a rational function with potential asymptotic behavior and specific intercepts. The exercise challenges the student to identify and graphically represent these characteristics, grounding their understanding in rational function behavior.
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