Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=2^{x-1}$$

Step-by-Step Solution

Answer

The table of values can include the points (-1,2), (0,1), (1,2), (2,4), and (3,8). The graph of \(f(x)=2^{x-1}\) swoops down close to but never touches the x-axis as it moves from left to right, hence the x-axis or y=0 is the horizontal asymptote.

Step 1: Construct the table of values

Take various values of \(x\), plug them into the function and calculate corresponding \(f(x)\). Choose a mix of negative, zero, and positive numbers for \(x\), to get a wide variety of points to graph. For instance, if you choose -1, 0, 1, 2, and 3 as \(x\) values, you will get the corresponding \(f(x)\) as 2, 1, 2, 4, and 8 respectively.

Step 2: Sketch the graph

Using the table of values you computed in the previous step, plot all the points and draw the graph. The graph of \(f(x)=2^{x-1}\) generally will start from a high point when \(x\) is a large negative number, swoop down close (but not touch) the x-axis and then start growing exponentially as \(x\) becomes positive.

Step 3: Identify the asymptote

The graph of \(f(x)=2^{x-1}\) will never touch or cross the x-axis, no matter how large or small \(x\) gets, making the x-axis (y=0) the horizontal asymptote of the function.

Key Concepts

Constructing a Table of Values

Understanding how to create a table of values is essential when graphing exponential functions. It begins by selecting strategic values for the independent variable, often denoted as \(x\), and calculating the corresponding dependent variable, represented as \(f(x)\). For an exponential function such as \(f(x)=2^{x-1}\), it's vital to choose a range of \(x\) values, including negative, zero, and positive numbers.

For example, when you take \(x\) values of -2, -1, 0, 1, and 2, you'll calculate the respective \(f(x)\) outputs as 0.25, 0.5, 1, 2, and 4. This methodical approach provides a clear picture of how the function behaves and starts to suggest the shape of the graph. When constructing your table, look for patterns in the outputs because this will help you predict and understand the general trend of the exponential function.

Sketching Graphs

Once you have a table of values, the next step is sketching the graph of the exponential function. This visual representation can help you grasp the function's behavior better. Using the points you've calculated, plot them on a coordinate plane. For \(f(x)=2^{x-1}\), as you plot each point, you will notice the characteristic 'J-shape' of an exponential growth function.

Make sure to draw smooth curves that pass through your plotted points, and remember that exponential functions increase very rapidly after a certain point. The left end of the graph will approach the x-axis without touching it, illustrating the concept of an asymptote. It's important to sketch beyond the calculated points, extending your graph to show the overall trend of the exponential behavior towards infinity and towards a decline into the asymptote.

Identifying Asymptotes

Asymptotes are lines that a graph approaches but never actually touches or crosses. Identifying asymptotes in exponential functions like \(f(x)=2^{x-1}\) is integral to understanding the function's limitations and behavior at extreme values of \(x\). The horizontal asymptote is particularly important.

In this case, the horizontal asymptote occurs where the function levels off as \(x\) becomes very negative. The graph gets infinitely close to the x-axis (which is the line \(y=0\)) yet never touches it, highlighting the horizontal asymptote of the function. Recognizing the asymptote allows you to see the boundary of the exponential growth or decay, providing vital insight into the function's properties and aiding in the accurate sketching of its graph.