Problem 51 - Algebra and Trigonometry

Question

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=\left(\frac{1}{2}\right)^{x} \text { and } g(x)=\left(\frac{1}{2}\right)^{x-1}+1 $$

Step-by-Step Solution

Answer

The graph of \(f(x) = (\frac{1}{2})^x\) is an exponential decay starting from y = \infty and approaching the horizontal asymptote at \(y = 0\). The graph of \(g(x) = (\frac{1}{2})^{x-1} + 1\) is the same shape, but shifted one unit to the right and one unit up, with a new horizontal asymptote at \(y = 1\).

Step 1: Graphing the function \(f(x)\)

The function \(f(x) = (\frac{1}{2})^x\) is an exponential function. Its graph starts from the upper left, at \(y = \infty\), when \(x = -\infty\) and goes downwards, getting closer and closer to the x-axis without ever touching it (as \(x\) approaches \(\infty\)). The x-axis serves as a horizontal asymptote, defined by the line \(y = 0\).

Step 2: Graphing the function \(g(x)\)

\(\(g(x) = (\frac{1}{2})^{x-1} + 1\) is a transformation of the function \(f(x)\). Specifically, it's the graph of \(f(x)\) shifted one unit to the right and one unit up. It has the same general shape as \(f(x)\), starting from the upper left (y = \infty when \(x = -\infty + 1\)) and going downwards towards a horizontal asymptote. Due to the upward shift, the horizontal asymptote for \(g(x)\) is at \(y = 1\), represented by the line \(y = 1\).

Step 3: Use graphing utility

To confirm these results, use a graphing utility to plot both functions. The graph should align with these descriptions and clearly show the differences between the graphs and the asymptotes.

Key Concepts

Exponential Functions

We often encounter exponential functions in mathematics and various real-world applications such as population growth, radioactive decay, and even financial models. An exponential function is defined as a function of the form
\( f(x) = a^{x} \),
where a is a positive constant called the 'base' of the exponential function. The base determines the rate and direction of growth or decay.

In our exercise, we look at exponential functions with a base of \(\frac{1}{2}\), which implies a decay since the base is less than one. The function \(f(x) = (\frac{1}{2})^{x} \) decreases as x increases, which means the value of f(x) gets closer to zero - but importantly, it never reaches zero, therefore establishing the concept of a horizontal asymptote, which we will discuss further in a separate section.
These functions are vital in understanding exponential behavior, and graphing them requires recognition of their special properties and how shifts in their equations result in transformations on their graphs.

Horizontal Asymptote

A horizontal asymptote can be thought of as a line that a graph approaches but never actually touches or crosses. It represents a value that the function will get closer and closer to as the input either increases or decreases without bound, but the function will never actually reach this value.

For exponential functions like f(x) and g(x) in our exercise, the horizontal asymptotes help us understand the behavior of the graph at extreme values of x. For the function
\( f(x) = (\frac{1}{2})^{x} \),
the x-axis (or the line y = 0) is the horizontal asymptote, which tells us that as x goes to infinity, f(x) gets closer to zero. On the other hand, for
\( g(x) = (\frac{1}{2})^{x-1} + 1 \),
the horizontal asymptote is y = 1, indicating that as x goes to infinity, g(x) approaches 1. Recognizing horizontal asymptotes is key for sketching accurate graphs and understanding the long-term behavior of functions.

Graph Transformations

The concept of graph transformations is pivotal in understanding how modifications to a function's formula result in changes to its graph. Basically, transformations involve shifting, stretching, compressing, or reflecting the graph of a function.

For our functions f(x) and g(x), transformation is a core part of understanding how their graphs relate. The function
\( g(x) = (\frac{1}{2})^{x-1} + 1 \)
is actually the result of transforming the function f(x) by shifting it one unit to the right and one unit upwards. This shift to the right is due to the (x-1) component, which effectively delays the function's decay by one unit of x. Similarly, adding 1 to the entire function raises it up on the y-axis by one unit, changing its horizontal asymptote from y = 0 to y = 1. Recognizing these transformations allows us to quickly sketch the altered graph without plotting numerous points.

Rectangular Coordinate System

As the stage where we plot our functions, the rectangular coordinate system is a two-dimensional plane formed by two perpendicular axes: the horizontal x-axis and the vertical y-axis. The intersection point of these axes is called the origin, denoted as (0,0). Each point on the plane is identified by an ordered pair of numbers known as coordinates, which signify its position relative to the two axes.

When graphing the exponential functions f(x) and g(x), we use this coordinate system to visualize their behaviors. For instance, as we have seen, the function
\( f(x) = (\frac{1}{2})^{x} \) approaches the x-axis (or y = 0) but never touches it – this x-axis is an integral part of the coordinate system which, in this case, doubles as the horizontal asymptote. The rectangular coordinate system is essential for graphically solving equations, exploring functions, and analyzing their properties in a visual and intuitive way.