Sketch the graph of y=g(x) by starting with the graph of...

Sketch the graph of $y=g(x)$ by starting with the graph of $y=f(x)$ and using transformations. Track at least three points of your choice and the horizontal asymptote through the transformations. State the domain and range of $g$.
$f(x)=2^{x}, g(x)=2^{x}-1$

Key Concepts

Exponential Functions

Exponential functions are mathematical functions in which a constant base is raised to a variable exponent. They exhibit rapid growth or decay and are characterized by properties such as passing through the point (0, 1) in their standard form. Such functions are key to understanding many real-world phenomena and their behavior under various transformations.

Function Transformations

Function transformations involve shifting, stretching, compressing, or reflecting the graph of a base function to produce a new function. They provide a systematic way to understand how alterations in an equation translate into movements or changes in the associated graph, maintaining the overall structure while modifying its position or scale.

Vertical Translation

A vertical translation is a type of transformation where a constant is added to or subtracted from the function, effectively shifting the entire graph up or down. This operation does not affect the shape of the graph but changes key features such as the vertical position and the location of the horizontal asymptote.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches as the independent variable goes to infinity or negative infinity. In the context of exponential functions, transformations can move this asymptote, which is essential in understanding the function's long-term behavior.

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. Determining the domain and range is crucial, especially when the function undergoes transformations, as these operations can affect the output values even if the set of inputs remains unchanged.

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