Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![## Graphing Rational Functions: A 7-Step Strategy
### Consider the function:
\[ f(x) = -\frac{2}{x^2 - 9} \]
### Steps to Follow:
1. **Symmetry**:
- Determine if the function has any symmetry (e.g., even, odd functions).
2. **X-intercept**:
- Find the points where the graph intersects the x-axis.
3. **Y-intercept**:
- Identify where the graph intersects the y-axis.
4. **Vertical Asymptotes**:
- Determine the values of \( x \) that make the function undefined, and where the graph approaches infinity.
5. **Horizontal Asymptotes**:
- Find the horizontal lines that the graph approaches as \( x \) goes to positive or negative infinity.
### Analysis of the Given Function:
- **X-intercept**: This function has no x-intercepts because the numerator does not equal zero.
- **Y-intercept**: Substitute \( x = 0 \) into the function.
- **Vertical Asymptotes**: Set the denominator equal to zero, solve \( x^2 - 9 = 0 \).
- **Horizontal Asymptotes**: Analyze the degrees of the polynomial in the numerator and the denominator.
Following these steps will aid in accurately sketching the graph of the given rational function \( f(x) = -\frac{2}{x^2 - 9} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc154e8f7-5981-4a34-85b0-b10a7e25c57a%2Ff711f5a5-57e2-49ad-ae8d-3177f754d979%2Ffiqu05_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Graphing Rational Functions: A 7-Step Strategy
### Consider the function:
\[ f(x) = -\frac{2}{x^2 - 9} \]
### Steps to Follow:
1. **Symmetry**:
- Determine if the function has any symmetry (e.g., even, odd functions).
2. **X-intercept**:
- Find the points where the graph intersects the x-axis.
3. **Y-intercept**:
- Identify where the graph intersects the y-axis.
4. **Vertical Asymptotes**:
- Determine the values of \( x \) that make the function undefined, and where the graph approaches infinity.
5. **Horizontal Asymptotes**:
- Find the horizontal lines that the graph approaches as \( x \) goes to positive or negative infinity.
### Analysis of the Given Function:
- **X-intercept**: This function has no x-intercepts because the numerator does not equal zero.
- **Y-intercept**: Substitute \( x = 0 \) into the function.
- **Vertical Asymptotes**: Set the denominator equal to zero, solve \( x^2 - 9 = 0 \).
- **Horizontal Asymptotes**: Analyze the degrees of the polynomial in the numerator and the denominator.
Following these steps will aid in accurately sketching the graph of the given rational function \( f(x) = -\frac{2}{x^2 - 9} \).
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