A vertical asymptote is a vertical line that a graph of a function approaches but never touches. It represents the value of the independent variable where the function is undefined or where the function's value becomes infinitely large or small.
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Vertical asymptotes occur when the denominator of a rational function is equal to zero, causing the function to be undefined at that point.
The location of a vertical asymptote is determined by the roots of the denominator of the rational function.
Vertical asymptotes can be used to sketch the graph of a rational function, as the function will approach but never touch the asymptote.
Identifying the vertical asymptotes of a function is an important step in evaluating and graphing logarithmic functions, as logarithmic functions can be expressed as rational functions.
Understanding vertical asymptotes is crucial for analyzing the behavior and properties of logarithmic functions, such as their range and end behavior.
Review Questions
Explain how the concept of a vertical asymptote relates to the evaluation and graphing of logarithmic functions.
Vertical asymptotes are closely tied to the evaluation and graphing of logarithmic functions because logarithmic functions can be expressed as rational functions. The vertical asymptotes of a logarithmic function correspond to the values of the independent variable where the denominator of the rational function representation is equal to zero, causing the function to be undefined at those points. Identifying the vertical asymptotes is an important step in sketching the graph of a logarithmic function, as the function will approach but never touch the asymptote.
Describe how the domain of a function is affected by the presence of a vertical asymptote.
The presence of a vertical asymptote in a function's graph indicates that the function is undefined at the value of the independent variable corresponding to the asymptote. This means that the domain of the function, which is the set of all possible input values, will exclude the value(s) where the vertical asymptote occurs. The function is not defined at those points, and the graph of the function will approach but never touch the vertical asymptote.
Analyze how the behavior of a logarithmic function near a vertical asymptote can be used to draw conclusions about the function's properties, such as its range and end behavior.
The behavior of a logarithmic function near a vertical asymptote provides important insights into the function's properties. As the function approaches the vertical asymptote, its value becomes infinitely large or small, indicating that the function's range is affected by the presence of the asymptote. Additionally, the way the function approaches the asymptote, either from above or below, can reveal information about the function's end behavior. Understanding the relationship between vertical asymptotes and the properties of logarithmic functions is crucial for accurately evaluating and graphing these functions.
A line that a curve approaches but never touches, either as the curve gets closer and closer to the line or as the values of the variables become larger or smaller.