(a) To obtain the graph of g(x)=2^x-1, we start with the...

(a) To obtain the graph of $g(x)=2^{x}-1,$ we start with the graph of $f(x)=2^{x}$ and shift it _____ (upward/downward) 1 unit.
(b) To obtain the graph of $h(x)=2^{x-1}$, we start with the graph of $f(x)=2^{x}$ and shift it to the _____ (left/right) 1 unit.

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Key Concepts

Exponential Functions

Exponential functions are functions of the form f(x) = a^x, where the variable appears as an exponent. These functions are known for their rapid growth or decay and unique characteristics such as a horizontal asymptote and a defined starting value when x equals zero.

Horizontal Shifts

Horizontal shifts happen when a constant is added to or subtracted from the input variable of a function. This effectively moves the graph left or right; for example, replacing x with (x - 1) shifts the graph to the right by one unit.

Vertical Shifts

Vertical shifts occur when a constant is added or subtracted from the entire function, moving the graph up or down. For instance, subtracting a constant from an exponential function results in the entire graph moving downward by that constant amount.

Graph Transformations

Graph transformations refer to the operations that move or change the shape of a graph in a predictable way. They include shifts, reflections, stretches, and compressions, and help in understanding how changes to the algebraic form of a function affect its graph.

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