Problem 51 - College Algebra Essentials

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 asymptote of \(f(x)=(1/2)^{x}\) is y=0 and of \(g(x)=(1/2)^{x-1}+1\) is y=1. The graphs of \(f(x)\) and \(g(x)\) are similar in shape, but \(g(x)\) is shifted 1 unit upwards and 1 unit to the right compared to \(f(x)\).

Step 1: Analyze Function Structure

Notice that both functions are exponential functions with base \(1/2\), with function \(g(x)\) being a vertical shift of function \(f(x)\). This means that the overall shape of the two will be similar, but the graph of \(g(x)\) will be shifted 1 unit upwards and 1 unit to the right.

Step 2: Identify Asymptotes

The graphs of exponential functions approach but never reach the x-axis, which is a horizontal asymptote. Therefore, the asymptote of the graph of \(f(x)=(1/2)^{x}\) is the x-axis, or y=0, and the asymptote of the graph of \(g(x)=(1/2)^{x-1}+1\) is y=1.

Step 3: Plot the Graphs

Begin by plotting some points for both functions. Apply the vertical transformations (shifting up by 1 and right by 1) to plot \(g(x)\). Compare the graphs to check they're vertically shifted versions of each other, and both approach their respective asymptotes as \(x\) tends to infinity.

Step 4: Confirm with Graphing Utility

Use a graphing tool to confirm that the hand-drawn graphs are accurate. Input the functions into the graphing tool and compare the results with the hand-drawn sketches.

Key Concepts

Asymptotes

In the context of exponential functions, asymptotes are lines that the graph of the function approaches but never actually meets. For the function \( f(x) = \left(\frac{1}{2}\right)^{x} \), the graph will gradually get closer and closer to the x-axis as \( x \) decreases, but it will never touch the axis.
This indicates that the x-axis, or the line \( y = 0 \), is a horizontal asymptote for this function. Likewise, for \( g(x) = \left(\frac{1}{2}\right)^{x-1}+1 \), due to the vertical shift upwards by 1, the asymptote shifts from \( y = 0 \) to \( y = 1 \).

Horizontal asymptotes are key in understanding the long-term behavior of a graph and identifying them is straightforward in exponential functions as they depend on the vertical transformation applied to the original function.

Graphing Techniques

Graphing exponential functions requires an understanding of how certain transformations affect the graph. Let's start by plotting points. For the function \( f(x) = \left(\frac{1}{2}\right)^{x} \), selecting values of \( x \) such as -2, -1, 0, 1, and 2 will give you points to plot:

• \( f(-2) = 4 \)
• \( f(-1) = 2 \)
• \( f(0) = 1 \)
• \( f(1) = 0.5 \)
• \( f(2) = 0.25 \)

For the function \( g(x) = \left(\frac{1}{2}\right)^{x-1}+1 \), apply the transformations of shifting right by 1 and upward by 1. After these shifts, you'll plot equivalent points:

• \( g(-1) = 3 \)
• \( g(0) = 2 \)
• \( g(1) = 1.5 \)
• \( g(2) = 1.25 \)
• \( g(3) = 1.125 \)

Using a graphing utility can solidify your understanding by visually confirming these transformations and seeing the curves more clearly.

Function Transformations

Function transformations alter the appearance of a graph without changing its basic shape, much like moving a picture up, down or sideways. Understanding these changes is crucial for mastering graphing techniques for exponential functions.

An exponential graph like \( f(x) = \left(\frac{1}{2}\right)^{x} \) can be transformed in several ways:

• A vertical shift involves adding or subtracting a constant. For \( g(x) = \left(\frac{1}{2}\right)^{x-1}+1 \), the '+1' shifts the graph up by one unit.
• A horizontal shift involves changing the exponent by adding/subtracting inside the function base. In \( g(x) = \left(\frac{1}{2}\right)^{x-1}+1 \), the 'x-1' means a shift to the right by one unit.

Understanding these simple transformations allows you to predict and sketch graphs accurately. They account for the relative positioning of asymptotes and help you visualize how each point on the graph shifts from its original state.