Begin by graphing f(x)=2^x. Then use transformations of this...

Begin by graphing $f(x)=2^{x}$. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ h(x)=2^{x+2}-1 $$

Key Concepts

Domain and Range

The domain of a function refers to all possible x-values (inputs) for which the function is defined, while the range refers to the set of possible y-values (outputs). Exponential functions typically have a domain of all real numbers and a range of positive values, though transformations can adjust the range. Analyzing these aspects is key to fully understanding the behavior of the function.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent. They exhibit rapid growth or decay depending on whether the base is greater than or less than one. Key properties include their continuous and smooth nature, and the fact that they are always positive, which is foundational for understanding their graphs and behavior.

Function Transformations

Function transformations involve shifting, stretching, compressing, or reflecting a given function's graph. In the context of exponential functions, transformations such as horizontal shifts (adjusting the input variable) and vertical shifts (adjusting the output) are used to move the graph without changing its overall shape. Understanding these transformations is crucial for graphing complex functions based on a basic exponential function.

Horizontal Asymptote

A horizontal asymptote is a line that a graph approaches as the input values become very large or very small. For basic exponential functions like f(x) = 2^x, the horizontal asymptote is typically y = 0. When a vertical shift is applied, the asymptote moves accordingly, which is important for identifying the long-term behavior of the function.

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