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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![**Exploration of Asymptotes and Their Intersection with Functions**
This text poses the question: "Does it cross HA/slant asymptotes?"
A mathematical expression is presented:
\[
\frac{x^3 - 4x}{x^2 - 3x + 12}
\]
### Analysis
When analyzing whether a rational function crosses its horizontal or slant asymptotes, one must typically consider the following steps:
1. **Determine the Horizontal Asymptote (HA):**
- For rational functions, if the degree of the numerator is equal to or less than the degree of the denominator, check the coefficients of the highest degree terms.
- If the degree of the numerator is greater than the degree of the denominator, there is no HA, but there may be a slant asymptote.
2. **Determine the Slant Asymptote:**
- If the degree of the numerator is exactly one more than the degree of the denominator, perform polynomial long division to find the slant asymptote.
3. **Check for Intersection with Asymptotes:**
- Set the function equal to the asymptote and solve for x to see if any real solutions exist. If solutions exist, the function crosses the asymptote at those points.
### Further Considerations
These concepts are critical within calculus and algebra for understanding the behavior of functions and their graphical representations. Further exploration might include graphing the function and numerically calculating important points to verify theoretical deductions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd560ec21-30e6-4e1a-8bd1-aa71f003b0e3%2F66d1b92c-c61e-4231-a157-6687f4a68f98%2Fk4nd491_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exploration of Asymptotes and Their Intersection with Functions**
This text poses the question: "Does it cross HA/slant asymptotes?"
A mathematical expression is presented:
\[
\frac{x^3 - 4x}{x^2 - 3x + 12}
\]
### Analysis
When analyzing whether a rational function crosses its horizontal or slant asymptotes, one must typically consider the following steps:
1. **Determine the Horizontal Asymptote (HA):**
- For rational functions, if the degree of the numerator is equal to or less than the degree of the denominator, check the coefficients of the highest degree terms.
- If the degree of the numerator is greater than the degree of the denominator, there is no HA, but there may be a slant asymptote.
2. **Determine the Slant Asymptote:**
- If the degree of the numerator is exactly one more than the degree of the denominator, perform polynomial long division to find the slant asymptote.
3. **Check for Intersection with Asymptotes:**
- Set the function equal to the asymptote and solve for x to see if any real solutions exist. If solutions exist, the function crosses the asymptote at those points.
### Further Considerations
These concepts are critical within calculus and algebra for understanding the behavior of functions and their graphical representations. Further exploration might include graphing the function and numerically calculating important points to verify theoretical deductions.
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