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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Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
![# Analyzing Rational Functions
### Task 9: Analyze the graph of each function.
#### Function \( b) \)
Given the function:
\[ F(x) = \frac{(x-1)(x+2)(x-3)}{x(x-4)^2} \]
### Analysis Steps
1. **Identify the Roots of the Numerator**:
- The function \( F(x) \) has zeros where the numerator is zero: \( x-1 = 0 \), \( x+2 = 0 \), \( x-3 = 0 \).
- Roots: \( x = 1, -2, 3 \).
2. **Identify the Roots and Behavior of the Denominator**:
- The function is undefined where the denominator is zero: \( x = 0 \) or \( x = 4 \).
- Vertical asymptotes occur at \( x = 0 \) and \( x = 4 \). The behavior at \( x = 4 \) is a double pole.
3. **Evaluate Horizontal Asymptotes**:
- The degree of the polynomial in the numerator is 3, and in the denominator is 3.
- Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. In this case, it is \( y = 1 \).
4. **Graph Layout**:
- The graph shows empty axes without any specific data plotted on it. A student would normally plot points and asymptotes based on the function’s behavior as noted above.
### Graphical Interpretation
When graphing, note key behaviors:
- **Zeros at \( x = 1, -2, 3 \) will cause the graph to cross the x-axis**.
- **Vertical asymptotes at \( x = 0 \) and a more significant effect at \( x = 4 \)**.
- **Horizontal asymptote at \( y = 1 \) describes end behavior**.
### Conclusion
By analyzing these steps and understanding the function's composition, one can accurately graph \( F(x) \) and understand its behavior across different intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F84ddc8be-a643-45f1-929d-52679633221d%2Fe9bf8a07-976e-4ea5-91b5-af351b5c6a19%2Fsw42iqb_processed.png&w=3840&q=75)
Transcribed Image Text:# Analyzing Rational Functions
### Task 9: Analyze the graph of each function.
#### Function \( b) \)
Given the function:
\[ F(x) = \frac{(x-1)(x+2)(x-3)}{x(x-4)^2} \]
### Analysis Steps
1. **Identify the Roots of the Numerator**:
- The function \( F(x) \) has zeros where the numerator is zero: \( x-1 = 0 \), \( x+2 = 0 \), \( x-3 = 0 \).
- Roots: \( x = 1, -2, 3 \).
2. **Identify the Roots and Behavior of the Denominator**:
- The function is undefined where the denominator is zero: \( x = 0 \) or \( x = 4 \).
- Vertical asymptotes occur at \( x = 0 \) and \( x = 4 \). The behavior at \( x = 4 \) is a double pole.
3. **Evaluate Horizontal Asymptotes**:
- The degree of the polynomial in the numerator is 3, and in the denominator is 3.
- Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. In this case, it is \( y = 1 \).
4. **Graph Layout**:
- The graph shows empty axes without any specific data plotted on it. A student would normally plot points and asymptotes based on the function’s behavior as noted above.
### Graphical Interpretation
When graphing, note key behaviors:
- **Zeros at \( x = 1, -2, 3 \) will cause the graph to cross the x-axis**.
- **Vertical asymptotes at \( x = 0 \) and a more significant effect at \( x = 4 \)**.
- **Horizontal asymptote at \( y = 1 \) describes end behavior**.
### Conclusion
By analyzing these steps and understanding the function's composition, one can accurately graph \( F(x) \) and understand its behavior across different intervals.
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