91Ó°ÊÓ

Graph transformations involve changing the position or shape of a graph in various ways. When dealing with exponential functions like \( f(x) = 2^{x} \), transformations can include shifts, stretches, or reflections.

In the given exercise, we are transforming the graph of \( f(x) = 2^{x} \) to obtain the graph of \( h(x) = 2^{x+2} - 1 \). Here’s what these transformations mean:

• **Horizontal Shift:** The graph moves 2 units to the left. This is indicated by the \( x+2 \) in the exponent.
• **Vertical Shift:** The graph moves 1 unit down. This is shown by the \( -1 \) outside the exponent.

Each transformation affects specific points on the graph. For instance, the point \( (0,1) \) on \( f(x) \) shifts to \( (-2,0) \) on \( h(x) \). These defined movements are crucial in plotting the correct graph for \( h(x) \).

The domain and range of a function tell us the possible values for \( x \) and \( y \) that can exist within a graph. For exponential functions, understanding these is key for interpreting their nature.

**Domain:**
Exponential functions like \( f(x) = 2^{x} \) or \( h(x) = 2^{x+2} - 1 \) have domains consisting of all real numbers. This means that we can input any real number into these functions and obtain an output.

**Range:**
The range details the possible \( y \) values that the graph will output. For the function \( h(x) = 2^{x+2} - 1 \), the vertical shift downwards changes the range to \( y > -1 \). This shift indicates that although \( y \) values start just above \( -1 \) and can increase indefinitely, they will never actually touch \(-1\). Understanding domain and range helps in comprehensively visualizing the graph's output bounds.

Horizontal asymptotes are horizontal lines that the graph of a function approaches as \( x \) tends towards positive or negative infinity. In the context of exponential functions, they help to identify the long-term behavior of the graph.

For the original function \( f(x) = 2^{x} \), the horizontal asymptote is \( y = 0 \). It indicates that as \( x \) decreases, the function values get closer and closer to zero but never actually reach it.

With the function \( h(x) = 2^{x+2} - 1 \), the transformations shift the horizontal asymptote down to \( y = -1 \). This shift occurs due to the \( -1 \) value in the function, which essentially lowers every point on the graph, pulling the asymptote downwards.

Horizontal asymptotes are crucial for understanding how exponential functions behave over long ranges, indicating values that the function will approach but never meet.