Problem 42 - 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. $$y=2^{-x^{2}}$$

Step-by-Step Solution

Answer

The graph of the function \(y=2^{-x^{2}}\) is a curve that decreases as x values increase or decrease, approaching but never reaching y=0. The x-axis (y=0) is a horizontal asymptote of the graph. The function does not have any vertical asymptotes.

Step 1: Understanding the function

The function provided is an exponential function. The main property of this type of function is that the base, 2 in this case, is raised to the power of the expression inside the parentheses. The graph of this function represents a curve, which decreases as the value of x increases or decreases.

Step 2: Constructing a table of values

A table of values can be constructed by substituting a range of x values into the function. Choose a range of x values, for example -3 to 3. For each value of x, calculate the corresponding value of y by substituting x into the function. This will give a set of (x, y) pairs that can be used to plot the graph.

Step 3: Sketching the graph of the function

Now take those (x, y) pairs from the table of values and plot them on a graph. Drawing a curve through the points will give the graph of the function. The curve should decrease as x increases or decreases, approaching but never reaching 0 (this is due to the nature of exponential functions).

Step 4: Identifying the asymptotes of the graph

An asymptote is a line that the graph approaches but never meets. In this case, since the function approaches 0 but never reaches it, the x-axis (y=0) is a horizontal asymptote of the graph. If there were vertical asymptotes, they would be values of x that make the function undefined. In this case, there are no undefined x values in the function \(y=2^{-x^{2}}\), so it does not have any vertical asymptotes.

Key Concepts

Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions. Imagine it as an electronic graph paper that does the heavy lifting for you. With a graphing utility, you can input a function like the given exponential function, \(y=2^{-x^{2}}\), and see its graph displayed instantly. This visualization makes understanding the function's behavior much easier. When you use a graphing utility, you can instantly adjust parameters and examine different outputs.

• Enter the function into the graphing utility.
• Adjust the settings to view a range from \(-3\) to \(3\) for the x-values.
• Observe the plotted curve.

This process allows you to see how the graph behaves, providing insights that are not as easily seen on paper. It highlights important features like intercepts, symmetry, and the overall shape.

Asymptotes

Asymptotes are lines that a graph approaches but never quite touches. They act like invisible boundaries that the curve can't cross. In the function \(y=2^{-x^{2}}\), the graph approaches the x-axis as it moves away from the center in either direction, making the x-axis \((y=0)\) a horizontal asymptote. This tells us that the values of \(y\) get very close to zero but never actually reach it, no matter how large or small \(x\) becomes. Asymptotes help define the behavior of a graph at its extremes.

• Horizontal asymptotes occur when \(y\) approaches a constant value as \(x\) becomes extremely large or small.
• In this example, the horizontal asymptote is \(y=0\).
• No vertical asymptotes exist for \(y=2^{-x^{2}}\) because the expression is always defined for any real \(x\).

Understanding asymptotes gives you a clearer picture of the behavior at the ends of your graph and the limits to which the function can approach parabolic tendencies.

Table of Values

Creating a table of values is a foundational way to understand a function before graphing it. It involves selecting specific \(x\) values, plugging them into the function, and calculating the corresponding \(y\) values. This method helps in plotting points on the graph to sketch the function accurately. For \(y=2^{-x^{2}}\), starting with a range of \(x\) from \(-3\) to \(3\) gives a comprehensive view of the function's behaviors.

• Select \(x\) values: for instance, \(-3, -2, -1, 0, 1, 2, 3\).
• Calculate \(y\) for each \(x\): \(y = 2^{-x^{2}}\).
• Form pairs: \((x, y) = (x, 2^{-x^{2}})\).

Using this method not only aids in plotting the function manually but also helps better comprehend the relationship between \(x\) and \(y\). It sets the stage for recognizing patterns like symmetry or changes in rate, which are crucial for understanding the graph's overall shape.