Problem 97 - 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 graph of the function \(f(x) = 2^{-x} - 1\) will show a decreasing pattern and will have the horizontal asymptote at \(y = -1\).

Step 1: Construct a Table of Values

Choose a range of x-values, from negative to positive, and calculate the corresponding f(x) values using the function \(f(x) = 2^{-x} - 1\). You can use a graphing calculator or utility to do this. The table might look something like this: \[ \begin{array}{|c|c|}\hlinex & f(x) \\hline-2 & 3 \-1 & 1 \0 & -1 \1 & -0.5 \2 & -0.75 \\hline \end{array} \]

Step 2: Sketch the Graph

Now plot these values on a graph, where the x-axis represents the x-values and the y-axis represents the \(f(x)\) values. The points from the table will form the shape of the graph.

Step 3: Identify Asymptotes

An asymptote is a line that the graph approaches but never reaches. The horizontal asymptote for this function is \(y = -1\), this can be inferred from the equation, where as \(x\) approaches infinitely positive or infinitely negative, the value of \(f(x)\) gets closer to -1 but never reaches it.

Key Concepts

Understanding Asymptotes in Graphs

Asymptotes are lines that a graph approaches but never crosses or touches. They serve as imaginary boundaries that guide the shape of a graph. In the context of the function \(f(x) = 2^{-x} - 1\), we are dealing with a horizontal asymptote. To identify it, observe what happens to \(f(x)\) as \(x\) becomes extremely large or extremely small. As \(x\) heads towards positive infinity or negative infinity, \(2^{-x}\) becomes exceedingly small, approaching zero. Consequently, \(f(x)\) nears \(-1\), since \(f(x) = 2^{-x} - 1\). Hence, the horizontal asymptote here is \(y = -1\). As a general rule, the horizontal asymptote of a function shows where the graph will settle or level off as \(x\) extends to extreme values. Recognizing these lines helps in sketching and understanding the function's long-term behavior.

Creating a Function Table

A function table is an essential tool for visualizing the relationship between \(x\) and \(f(x)\). It involves selecting various \(x\)-values, plugging them into the function to compute \(f(x)\). For our function \(f(x) = 2^{-x} - 1\), try picking values like \(-2, -1, 0, 1,\) and \(2\). Calculate \(f(x)\) for these \(x\)-values to acquire points that can be plotted on a graph. These calculated coordinates represent the curve's formation when connected.
The table effectively gives a glimpse into the behavior of the function. A clear trend often emerges, hinting at the general direction and curvature of the graph. When preparing a function table:

• Ensure a wide enough range of \(x\)-values to capture the graph's essential features.
• Focus on both negative and positive values to see the full spectrum of the function's impact.
• Use technology, like graphing utilities, for quick computations and broader range exploration.

Graph Sketching Tips

Graph sketching allows us to visually interpret mathematical relationships. With a set of points from a function table, you can plot these on a coordinate plane to sketch the graph of \(f(x) = 2^{-x} - 1\).
Begin by marking precise points like \((-2, 3), (-1, 1), (0, -1), (1, -0.5), (2, -0.75)\) on the graph. These arise from the function table calculations.

• Connect the points smoothly, respecting the natural curve suggested by the function.
• Look for symmetry or repeating patterns that simplify sketching.
• Pay attention to the horizontal asymptote \(y = -1\). This line helps guide the curvature as it nears the boundary but shouldn’t be crossed.

Notice how the curve tends to flatten as it approaches the horizontal asymptote, especially for extreme \(x\)-values. As you practice sketching, you'll gain intuition about how different functions behave and interact visually, enhancing your mathematical understanding.