Problem 52 - 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. $$ f(x)=\left(\frac{1}{2}\right)^{x} \text { and } g(x)=\left(\frac{1}{2}\right)^{x-1}+2 $$

Step-by-Step Solution

Answer

Function \(f(x)\) has a horizontal asymptote at \(y=0\) and function \(g(x)\) has a horizontal asymptote at \(y=2\). There are no vertical asymptotes for either function. The function \(g(x)\) is a transformation of the function \(f(x)\), shifted one unit to the right and two units upwards.

Step 1: Graph function f

First, the function \(f(x)=(0.5)^x\) is graphed. This is a decreasing exponential function with an horizontal asymptote \(y=0\). As \(x\) approaches positive and negative infinity, \(f(x)\) approaches 0.

Step 2: Graph function g

Next, graph the function \(g(x)=(0.5)^{x-1}+2\). Notice that it is a transformation of the function \(f(x)\). The expression \(x-1\) means the function is shifted one unit to the right, and adding 2 means the function is shifted two units upwards.

Step 3: Asymptotes for function f and g

The horizontal asymptote for the function \(f(x)\) is \(y=0\). For the function \(g(x)\), due to the upward shift by two units, the horizontal asymptote is at \(y=2\). Both functions do not have any vertical asymptotes.

Step 4: Verification using a graphing utility

A graphing calculator or software can be used to graph these functions, and this will confirm the hand-drawn graphs and the locations of the asymptotes.

Key Concepts

Horizontal Asymptotes

Understanding horizontal asymptotes is crucial when graphing exponential functions. They are the lines that the function approaches but does not cross as the value of x grows larger, either positively or negatively. In our example, for the function f(x) = (1/2)^x, the line y = 0 is the horizontal asymptote. This means that as x increases or decreases indefinitely, f(x) gets closer and closer to zero but never actually reaches it.

When it comes to the transformed function g(x) = (1/2)^(x-1) + 2, the horizontal asymptote is affected by the vertical shift of the graph. Here, adding 2 to the function shifts the entire graph upwards by 2 units, changing the horizontal asymptote to y = 2. No matter how much x changes, g(x) will get infinitely close to 2 but not touch it. Asymptotes are essential in analyzing the long-term behavior of a function, allowing us to predict its direction and eventual leveling off point.

Transformations of Functions

When working with transformations of functions, the goal is to understand how modifications to the function's equation affect its graph. Simple transformations include shifts, stretches, and reflections. In our exercise, the transformation from f(x) = (1/2)^x to g(x) = (1/2)^(x-1) + 2 involves two key changes:

• The subtraction of 1 from x in the exponent results in a horizontal shift one unit to the right.
• The addition of 2 outside the exponent indicates a vertical shift upwards by two units.

Transformations enable us to derive new functions from the parent function while preserving the general shape. Knowing how each transformation affects the graph helps in quickly sketching the new function without plotting numerous points, saving time and ensuring accuracy.

Understanding the graphical relationship between the original function and its transformation aids in mastering the broader concept of function behavior, an essential skill for advanced mathematics.

Exponential Decay

Exponential decay refers to a situation where a quantity decreases at a rate proportional to its current value. This type of behavior is commonly found in real-life phenomena such as radioactive decay and depreciation of assets. In the context of our functions, f(x) = (1/2)^x and g(x) = (1/2)^(x-1) + 2, the base of the exponent, 1/2, is between 0 and 1, indicating exponential decay. As x increases, the value of f(x) gets smaller since we are repeatedly multiplying by a fraction.

Exponential decay models are incredibly powerful as they provide insight into how quantities change over time in a consistent pattern. Recognizing this pattern allows us to predict future values and understand the underlying principles governing the system's behavior. Through graphing these exponential decay functions, it becomes apparent how the function rapidly decreases initially but slows its descent as it approaches the horizontal asymptote, graphically demonstrating the decay process.