Graph the rational function. 8 x+2 f(x)= Start by drawing the vertical and horizontal asymptotes.
Graph the rational function. 8 x+2 f(x)= Start by drawing the vertical and horizontal asymptotes.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Graphing a Rational Function:
### Objective:
Learn how to graph the rational function and identify its vertical and horizontal asymptotes.
### Function to Graph:
\[ f(x) = \frac{8}{x+2} \]
### Step-by-Step Instructions:
1. **Identify Vertical Asymptotes:**
Determine where the function becomes undefined by setting the denominator equal to 0.
\[ x + 2 = 0 \]
Solve for x:
\[ x = -2 \]
Thus, the vertical asymptote is at \( x = -2 \).
2. **Identify Horizontal Asymptotes:**
Compare the degrees of the numerator and denominator. Since the denominator is of a higher degree than the constant numerator, the horizontal asymptote is at \( y = 0 \).
3. **Graphing the Asymptotes and Function:**
- Draw the vertical asymptote at \( x = -2 \), a dashed vertical line on the graph.
- Draw the horizontal asymptote at \( y = 0 \), a dashed horizontal line on the graph.
### Graph Explanation:
The graph provided is a standard coordinate plane with x-values ranging from -8 to 8 and y-values also ranging from -8 to 8. Key features include:
- The x-axis (horizontal line) and the y-axis (vertical line) crossing at the origin (0,0).
- The vertical asymptote, not explicitly shown but implied, should be drawn as a vertical dashed line intersecting \( x = -2 \).
- Similarly, the horizontal asymptote at \( y = 0 \) is indicated but should be represented as a horizontal dashed line coinciding with the x-axis.
By following these steps and using the provided graph, one can accurately graph the rational function \( f(x) = \frac{8}{x+2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbae73a37-719e-4892-8745-1f816a181ea7%2Fba96889a-63a3-4c53-913e-6cbbd176d16a%2Feer9dk8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Graphing a Rational Function:
### Objective:
Learn how to graph the rational function and identify its vertical and horizontal asymptotes.
### Function to Graph:
\[ f(x) = \frac{8}{x+2} \]
### Step-by-Step Instructions:
1. **Identify Vertical Asymptotes:**
Determine where the function becomes undefined by setting the denominator equal to 0.
\[ x + 2 = 0 \]
Solve for x:
\[ x = -2 \]
Thus, the vertical asymptote is at \( x = -2 \).
2. **Identify Horizontal Asymptotes:**
Compare the degrees of the numerator and denominator. Since the denominator is of a higher degree than the constant numerator, the horizontal asymptote is at \( y = 0 \).
3. **Graphing the Asymptotes and Function:**
- Draw the vertical asymptote at \( x = -2 \), a dashed vertical line on the graph.
- Draw the horizontal asymptote at \( y = 0 \), a dashed horizontal line on the graph.
### Graph Explanation:
The graph provided is a standard coordinate plane with x-values ranging from -8 to 8 and y-values also ranging from -8 to 8. Key features include:
- The x-axis (horizontal line) and the y-axis (vertical line) crossing at the origin (0,0).
- The vertical asymptote, not explicitly shown but implied, should be drawn as a vertical dashed line intersecting \( x = -2 \).
- Similarly, the horizontal asymptote at \( y = 0 \) is indicated but should be represented as a horizontal dashed line coinciding with the x-axis.
By following these steps and using the provided graph, one can accurately graph the rational function \( f(x) = \frac{8}{x+2} \).
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