Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions
Rational functions are expressions formed by the ratio of two polynomials. They can exhibit unique behaviors such as asymptotes, intercepts, and discontinuities. Understanding the structure of rational functions is essential for analyzing their graphs, particularly how they behave near their vertical and horizontal asymptotes.
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Intro to Rational Functions
Transformations of Functions
Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, adding a constant to the input of a function shifts the graph horizontally, while adding a constant to the output shifts it vertically. Recognizing these transformations helps in graphing new functions based on known ones, such as f(x) = 1/x.
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Domain & Range of Transformed Functions
Asymptotes
Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where the function is undefined, while horizontal asymptotes describe the behavior of the function as x approaches infinity. Identifying asymptotes is crucial for accurately sketching the graph of rational functions, as they dictate the overall shape and limits of the graph.
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Introduction to Asymptotes