91Ó°ÊÓ

In this exercise, we deal with the transformation of functions to understand how they deviate from their base forms. Our base function is the well-known exponential function, which is expressed as \(y = 2^x\). This function indicates rapid growth as x increases. However, when the function is transformed to \(f(x) = 2^{x-1} + 5\), some significant changes occur.

The term \(x - 1\) inside the exponent denotes a horizontal shift. Specifically, this results in shifting the graph of \(y = 2^x\) one unit to the right. The transformation affects each point on the original graph in this way. Furthermore, the addition of \(+5\) to the entire function shifts the graph vertically. This means that every point on the graph is moved five units upwards on the y-axis.

These transformations can be described with two simple rules:

• Horizontal shifts occur when changes are made inside the function, i.e., with the x-variable.
• Vertical shifts occur when values are added or subtracted outside of the function.

Understanding these transformations is essential in transforming the normal exponential curves to match reality as expressed by our modified function.

An important feature of exponential graphs is the horizontal asymptote. A horizontal asymptote is a line that the graph approaches but never actually touches. In our function \(f(x) = 2^{x-1} + 5\), we find that the horizontal asymptote is at \(y = 5\).

But why does the graph have a horizontal asymptote at this specific value? This line is derived from the vertical shift of the function. When we add 5 to the entire function, it means that the base graph of \(2^x\) has been elevated 5 units up. This shift effectively repositions the horizontal asymptote from \(y=0\) (in the parent function \(y=2^x\)) to \(y=5\) in our transformed function.

Remember, the horizontal asymptote is significant because it shows the bounding line that the exponential graph approaches as \(x\) becomes very large or very negative. It forms one of the fundamental considerations when sketching any graph of an exponential function.

Once our understanding of the function's behavior is solid, plotting points becomes a hands-on way to bring our graph to life. In the function \(f(x) = 2^{x-1} + 5\), plotting points helps to illustrate the transformations and confirm the presence of the horizontal asymptote.

What you do is take selected values of \(x\) and calculate the corresponding \(f(x)\). For instance, substituting \(x = 0\) gives \(f(x) = 5.5\), \(x = 1\) results in \(f(x) = 6\), and \(x = 2\) produces \(f(x) = 7\). Plotting these points onto a graph will begin to show the curve of this exponential function.

Here's a quick guide:

• Choose a range of x-values, including negative numbers for completeness.
• Calculate the corresponding y-values using the function.
• Mark these points on the graph and sketch the curve.

Plotting confirms the rightward shift and vertical translation visually, plus clearly depicts the horizontal asymptote at \(y = 5\) as the graph approaches this line. This graphing process solidifies our understanding of the exponential function and its transformations.