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Graph the rational function. 2x2 7) f(x) =- x2 - 4 10 -10 -5 10 x

Graph the rational function. 2x2 7) f(x) =- x2 - 4 10 -10 -5 10 x

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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**Graph the Rational Function** Consider the rational function: \[ f(x) = \frac{2x^2}{x^2 - 4} \] **Graph Explanation:** - **Axes:** The graph features a standard Cartesian coordinate system with the x-axis and y-axis intersecting at the origin (0,0). - **Scale:** Both axes are labeled with intervals ranging from -10 to 10. **Function Characteristics:** - **Vertical Asymptotes:** These will occur at the values of \( x \) that make the denominator zero, which are \( x = \pm 2 \). - **Horizontal Asymptote:** Since the degrees of the numerator and denominator are equal, the horizontal asymptote can be found by dividing the leading coefficients. Thus, \( y = 2 \). **Graphing Approach:** 1. **Identify Asymptotes:** Sketch the vertical asymptotes at \( x = -2 \) and \( x = 2 \), and a horizontal asymptote at \( y = 2 \). 2. **Plot Points:** Choose points on either side of the asymptotes to understand the behavior of the function. 3. **Behavior at Infinity:** As \( x \) approaches infinity or negative infinity, the function approaches the horizontal asymptote, \( y = 2 \). This graph provides insights into the behavior of the rational function near its asymptotes and intercepts.
Transcribed Image Text:**Graph the Rational Function** Consider the rational function: \[ f(x) = \frac{2x^2}{x^2 - 4} \] **Graph Explanation:** - **Axes:** The graph features a standard Cartesian coordinate system with the x-axis and y-axis intersecting at the origin (0,0). - **Scale:** Both axes are labeled with intervals ranging from -10 to 10. **Function Characteristics:** - **Vertical Asymptotes:** These will occur at the values of \( x \) that make the denominator zero, which are \( x = \pm 2 \). - **Horizontal Asymptote:** Since the degrees of the numerator and denominator are equal, the horizontal asymptote can be found by dividing the leading coefficients. Thus, \( y = 2 \). **Graphing Approach:** 1. **Identify Asymptotes:** Sketch the vertical asymptotes at \( x = -2 \) and \( x = 2 \), and a horizontal asymptote at \( y = 2 \). 2. **Plot Points:** Choose points on either side of the asymptotes to understand the behavior of the function. 3. **Behavior at Infinity:** As \( x \) approaches infinity or negative infinity, the function approaches the horizontal asymptote, \( y = 2 \). This graph provides insights into the behavior of the rational function near its asymptotes and intercepts.
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