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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![**Title:** Graphing Asymptotes of Rational Functions
**Objective:** Understand how to identify and graph vertical and horizontal asymptotes of a rational function.
**Rational Function Example:**
\[ f(x) = \frac{9x + 2}{x^2 - 1} \]
**Instructions:**
1. **Vertical Asymptotes:**
- Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points).
- For the function \( f(x) = \frac{9x + 2}{x^2 - 1} \), set the denominator equal to zero:
\[ x^2 - 1 = 0 \]
- Solve for \( x \):
\[ (x-1)(x+1) = 0 \]
\[ x = 1 \quad \text{and} \quad x = -1 \]
- These are the vertical asymptotes.
2. **Horizontal Asymptotes:**
- Horizontal asymptotes are determined by comparing the degrees of the polynomial in the numerator and the denominator.
- Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is:
\[ y = 0 \]
**Graph Explanation:**
- The grid shows a coordinate system with dashed blue lines representing the asymptotes.
- Vertical blue dashed lines are at \( x = 1 \) and \( x = -1 \).
- A horizontal blue dashed line is at \( y = 0 \).
**Conclusion:** By setting the denominator to zero and comparing polynomial degrees, you can identify the asymptotes to understand the behavior of a rational function as it approaches certain points or extends towards infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05a83d1e-0628-4f64-b9f9-b112850cc2ab%2F8e30bf09-37b3-4f36-bbd7-059cdf457b10%2Ftm20fvh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title:** Graphing Asymptotes of Rational Functions
**Objective:** Understand how to identify and graph vertical and horizontal asymptotes of a rational function.
**Rational Function Example:**
\[ f(x) = \frac{9x + 2}{x^2 - 1} \]
**Instructions:**
1. **Vertical Asymptotes:**
- Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points).
- For the function \( f(x) = \frac{9x + 2}{x^2 - 1} \), set the denominator equal to zero:
\[ x^2 - 1 = 0 \]
- Solve for \( x \):
\[ (x-1)(x+1) = 0 \]
\[ x = 1 \quad \text{and} \quad x = -1 \]
- These are the vertical asymptotes.
2. **Horizontal Asymptotes:**
- Horizontal asymptotes are determined by comparing the degrees of the polynomial in the numerator and the denominator.
- Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is:
\[ y = 0 \]
**Graph Explanation:**
- The grid shows a coordinate system with dashed blue lines representing the asymptotes.
- Vertical blue dashed lines are at \( x = 1 \) and \( x = -1 \).
- A horizontal blue dashed line is at \( y = 0 \).
**Conclusion:** By setting the denominator to zero and comparing polynomial degrees, you can identify the asymptotes to understand the behavior of a rational function as it approaches certain points or extends towards infinity.
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