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(4) Sketch a graph of the function y || 2

(4) Sketch a graph of the function y || 2

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

Using derivatives/concavity/sign analysis

**Transcription:** (4) Sketch a graph of the function \( y = \frac{x^2}{x-2} \). **Explanation:** This task involves sketching the graph of the rational function \( y = \frac{x^2}{x-2} \). Key points to consider when sketching this graph include: 1. **Domain**: The function is undefined where the denominator is zero. Therefore, \( x = 2 \) is a vertical asymptote. 2. **Intercepts**: - **x-intercepts**: Set \( y = 0 \), which occurs when the numerator is zero. Hence, \( x^2 = 0 \), so \( x = 0 \). Thus, the x-intercept is at \( (0, 0) \). - **y-intercepts**: Set \( x = 0 \), which gives \( y = 0 \). So, the y-intercept is also at \( (0, 0) \). 3. **Horizontal Asymptote**: Generally determined by the degrees of the polynomial in the numerator and the denominator. Here, since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, there is an oblique asymptote. 4. **Behavior Near the Vertical Asymptote**: As \( x \) approaches 2 from the left and right, the function tends toward positive or negative infinity. 5. **End Behavior**: For \( x \) values approaching positive or negative infinity, analyze the leading terms. 6. **Oblique Asymptote**: Perform polynomial long division on \( x^2 \div (x-2) \) to find the slant asymptote. The result guides the general trend of the curve as \( x \) becomes large. Using these considerations, sketch the graph paying attention to the asymptotes and intercepts, and ensuring the correct behavior as \( x \) approaches the asymptotes and infinity.
Transcribed Image Text:**Transcription:** (4) Sketch a graph of the function \( y = \frac{x^2}{x-2} \). **Explanation:** This task involves sketching the graph of the rational function \( y = \frac{x^2}{x-2} \). Key points to consider when sketching this graph include: 1. **Domain**: The function is undefined where the denominator is zero. Therefore, \( x = 2 \) is a vertical asymptote. 2. **Intercepts**: - **x-intercepts**: Set \( y = 0 \), which occurs when the numerator is zero. Hence, \( x^2 = 0 \), so \( x = 0 \). Thus, the x-intercept is at \( (0, 0) \). - **y-intercepts**: Set \( x = 0 \), which gives \( y = 0 \). So, the y-intercept is also at \( (0, 0) \). 3. **Horizontal Asymptote**: Generally determined by the degrees of the polynomial in the numerator and the denominator. Here, since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, there is an oblique asymptote. 4. **Behavior Near the Vertical Asymptote**: As \( x \) approaches 2 from the left and right, the function tends toward positive or negative infinity. 5. **End Behavior**: For \( x \) values approaching positive or negative infinity, analyze the leading terms. 6. **Oblique Asymptote**: Perform polynomial long division on \( x^2 \div (x-2) \) to find the slant asymptote. The result guides the general trend of the curve as \( x \) becomes large. Using these considerations, sketch the graph paying attention to the asymptotes and intercepts, and ensuring the correct behavior as \( x \) approaches the asymptotes and infinity.
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