Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Functions
Logarithmic functions, such as f(x) = log₂(x), are the inverses of exponential functions. They express the power to which a base must be raised to obtain a given number. Understanding the properties of logarithms, including their domain (x > 0) and range (all real numbers), is essential for analyzing their graphs.
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Graphs of Logarithmic Functions
Graphing Logarithmic Functions
The graph of a logarithmic function typically has a vertical asymptote at x = 0 and passes through the point (1, 0), since log₂(1) = 0. As x increases, the function rises slowly, reflecting the nature of logarithms growing without bound but at a decreasing rate. Familiarity with these characteristics aids in matching the function to its graph.
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Graphs of Logarithmic Functions
Transformations of Functions
Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For logarithmic functions, horizontal shifts can occur when the input is adjusted (e.g., log₂(x - h)), while vertical shifts occur when a constant is added or subtracted from the function (e.g., log₂(x) + k). Recognizing these transformations is crucial for accurately interpreting and matching graphs.
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Domain & Range of Transformed Functions