In Exercises 25-34, 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+1) – 1

Key Concepts

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.

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.

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.