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. $$ g(x)=\frac{1}{2} \cdot 2^{x} $$

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where the base 'a' is a positive constant not equal to 1. They exhibit rapid growth or decay depending on whether the base is greater than or less than one, and they are fundamental in applications ranging from finance to natural processes.

Graph Transformations

Graph transformations involve modifying the graph of a basic function via shifts, stretches, compressions, or reflections. This allows one to obtain the graph of a new function by applying a series of operations to the original function's graph, preserving the overall shape while altering its position or scale.

Vertical Stretch Transformation

A vertical stretch transformation multiplies the output of the function by a constant factor. In the context of exponential functions, multiplying by a constant such as 1/2 scales the graph vertically, making all y-values half as large while leaving the exponential growth or decay pattern unchanged.

Asymptotes

An asymptote is a line that a graph approaches but never touches. For exponential functions, a horizontal asymptote is typically present—commonly y = 0—which reflects the fact that the function values approach zero as x decreases, even though they never actually reach zero.

Domain and Range

The domain of a function is the set of all possible input values (usually, all real numbers for exponential functions), while the range is the set of all possible output values. For exponential functions, regardless of transformations, the domain remains all real numbers, and the range is limited to positive values if the function is never reflected vertically.

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