Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
5. Rational Functions
Graphing Rational Functions
4:50 minutes
Problem 50c
Textbook Question
Textbook QuestionWork each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). ƒ(x)=-1/(x-2)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. The general form is ƒ(x) = P(x)/Q(x), where P and Q are polynomials. Understanding rational functions is crucial for analyzing their behavior, particularly in relation to asymptotes, which occur where the function is undefined.
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Vertical Asymptotes
Vertical asymptotes are vertical lines that represent values of x where a rational function approaches infinity or negative infinity. They occur at values of x that make the denominator zero, provided the numerator is not also zero at those points. In the given function ƒ(x) = -1/(x-2), the vertical asymptote is at x = 2.
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Graph Behavior Near Asymptotes
The behavior of a graph near an asymptote is essential for understanding how the function behaves as it approaches the asymptote. For the function ƒ(x) = -1/(x-2), as x approaches 2 from the left, the function value decreases without bound, while from the right, it increases without bound. This behavior helps in sketching the graph and identifying the correct representation among multiple choices.
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