Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. These functions exhibit rapid growth or decay, depending on the base. Understanding the basic shape and properties of the graph of f(x) = 2^x is essential, as it serves as the foundation for applying transformations to graph related functions.
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Transformations of Functions
Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For the function h(x) = 2^(x+1) - 1, the '+1' indicates a horizontal shift to the left by 1 unit, while the '-1' indicates a vertical shift downward by 1 unit. Mastery of these transformations allows for accurate graphing of modified functions based on the original exponential function.
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Domain & Range of Transformed Functions
Asymptotes
Asymptotes are lines that a graph approaches but never touches. For exponential functions, the horizontal asymptote is typically found at y = k, where k is a constant that the function approaches as x approaches infinity or negative infinity. In the case of h(x), the horizontal asymptote is y = -1, which is crucial for determining the function's behavior and for accurately graphing the function.
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