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
![**Educational Website Content**
**Graphing Rational Functions**
**Function:**
\[ f(x) = -\frac{1}{x-2} + 3 \]
**Description:**
We are given the function \( f(x) = -\frac{1}{x-2} + 3 \) and asked to identify its graph among two options displayed above. This function represents a rational function with a vertical asymptote and a horizontal transformation.
**Graphs Analysis:**
1. **First Graph (Left):**
- **Axes Range:** The x-axis ranges from -7 to 5, and the y-axis ranges from -5 to 5.
- **Curve Behavior:** The curve approaches a vertical line (asymptote) around \( x = -1 \) and runs off to positive and negative infinities. It curves over the x-axis, moving from top-left to bottom-right.
- **Horizontal Shift and Vertical Shift:** This graph displays a vertical asymptote at \( x = -1 \) and seems to have a horizontal asymptote as it approaches \( y = 0 \).
2. **Second Graph (Right):**
- **Axes Range:** The x-axis ranges from -5 to 7, and the y-axis ranges from -5 to 7.
- **Curve Behavior:** The curve has a vertical asymptote at \( x = 2 \) and another horizontal asymptote as it approaches \( y = 3 \). The curve moves from the top-right and bottom-left quadrants, which is consistent with the transformation of rational functions.
- **Horizontal Shift and Vertical Shift:** This graph accurately reflects the function's transformation with vertical shift upwards by 3 units.
**Correct Graph:** The second graph on the right accurately represents the function \( f(x) = -\frac{1}{x-2} + 3 \). It captures the vertical asymptote at \( x = 2 \) and the horizontal asymptote at \( y = 3 \). This is consistent with the properties of the given transformed rational function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9d23aa5-ac37-4015-8b63-d963ab01f4b6%2Fdbe13f15-fd24-492b-9226-b6e2637303fb%2Fo97rupp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content**
**Graphing Rational Functions**
**Function:**
\[ f(x) = -\frac{1}{x-2} + 3 \]
**Description:**
We are given the function \( f(x) = -\frac{1}{x-2} + 3 \) and asked to identify its graph among two options displayed above. This function represents a rational function with a vertical asymptote and a horizontal transformation.
**Graphs Analysis:**
1. **First Graph (Left):**
- **Axes Range:** The x-axis ranges from -7 to 5, and the y-axis ranges from -5 to 5.
- **Curve Behavior:** The curve approaches a vertical line (asymptote) around \( x = -1 \) and runs off to positive and negative infinities. It curves over the x-axis, moving from top-left to bottom-right.
- **Horizontal Shift and Vertical Shift:** This graph displays a vertical asymptote at \( x = -1 \) and seems to have a horizontal asymptote as it approaches \( y = 0 \).
2. **Second Graph (Right):**
- **Axes Range:** The x-axis ranges from -5 to 7, and the y-axis ranges from -5 to 7.
- **Curve Behavior:** The curve has a vertical asymptote at \( x = 2 \) and another horizontal asymptote as it approaches \( y = 3 \). The curve moves from the top-right and bottom-left quadrants, which is consistent with the transformation of rational functions.
- **Horizontal Shift and Vertical Shift:** This graph accurately reflects the function's transformation with vertical shift upwards by 3 units.
**Correct Graph:** The second graph on the right accurately represents the function \( f(x) = -\frac{1}{x-2} + 3 \). It captures the vertical asymptote at \( x = 2 \) and the horizontal asymptote at \( y = 3 \). This is consistent with the properties of the given transformed rational function.
Transcribed Image Text:### Analyzing Graphs of Rational Functions
The image contains two graphs displaying rational functions, each characterized by specific asymptotic behaviors.
#### Left Graph
- **Axes:** The graph is plotted with `x` on the horizontal axis ranging from -5 to 7, and `y` on the vertical axis ranging from -5 to 7.
- **Curve Description:**
- The graph consists of two branches.
- The branch in the first quadrant curves upward and to the right, indicating positive values of `y` as `x` increases.
- The second branch, located in the third quadrant, decreases as it moves leftwards, indicating negative values of `y`.
- **Asymptotes:**
- The vertical asymptote is likely around `x = 3`, suggesting the function approaches infinity as it gets closer to this line.
- A horizontal asymptote may exist around `y = 3`, revealing that as `x` approaches infinity, `y` approaches this value.
#### Right Graph
- **Axes:** The graph is plotted similarly, with `x` on the horizontal axis ranging from -7 to 5, and `y` on the vertical axis from -5 to 7.
- **Curve Description:**
- This graph also consists of two branches.
- The first branch, in the second quadrant, curves upward, suggesting a steep increase as `x` approaches the vertical asymptote.
- The second branch in the fourth quadrant decreases sharply, indicating negative `y` values as `x` approaches the vertical asymptote from the right.
- **Asymptotes:**
- The vertical asymptote is likely around `x = -2`, suggesting the function approaches infinity or negative infinity near this line.
- A horizontal asymptote appears around `y = 2`, indicating that as `x` increases positively or negatively, `y` nears this value.
### Conclusion
These graphs demonstrate classic behaviors of rational functions where vertical asymptotes occur at values excluded from their domain, and horizontal asymptotes represent long-term behavior as `x` approaches positive or negative infinity. Understanding these characteristics is essential in analyzing and interpreting rational functions.
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