Problem 28
Question
In Exercises, use a graphing utility to graph the function. $$ y=2^{-x^{2}} $$
Step-by-Step Solution
Verified Answer
As a result of graphing the function \(y=2^{-x^{2}}\), an inverted bell-shaped curve symmetric about the y-axis is obtained. The graph is above the x-axis but never touches it no matter how far along the x-axis you go.
1Step 1: Understand the Function
The given function is an exponential function where the base is 2 and the exponent is \(-x^2\). Note that the exponent \(-x^2\) is a parabola function with a downward orientation due to the negative sign.
2Step 2: Set up the graphing utility
Enter the function \(y=2^{-x^{2}}\) into the graphing utility. Make sure the settings allow to see the full shape of the function.
3Step 3: Graph the function
Plot the function. It is an inverted bell-shaped curve symmetric about the y-axis. The graph is above the x-axis and gets infinitely close to it as x moves away from 0.
Key Concepts
Graphing UtilitiesParabolaSymmetry about the Y-axis
Graphing Utilities
When using graphing utilities to visualize mathematical functions, like exponential functions, it's essential to know how to make the best use of these tools. Graphing calculators or software allow you to plot functions quickly and accurately, helping you understand the behavior of complex equations.
If you're new to graphing utilities, start by familiarizing yourself with the interface. Most utilities have specific steps to input equations. For instance, to graph the function \(y=2^{-x^{2}}\):
If you're new to graphing utilities, start by familiarizing yourself with the interface. Most utilities have specific steps to input equations. For instance, to graph the function \(y=2^{-x^{2}}\):
- Locate the function input area or menu.
- Enter the function as instructed, making sure to use correct syntax for exponents and negative signs.
- Adjust the viewing window to ensure a clear view of the graph's important features.
Parabola
A parabola is a symmetrical curve formed by a quadratic function, like \(y=-x^2\). In the context of the function \(y=2^{-x^{2}}\), the term \(-x^2\) indicates that there is a parabolic element influencing the shape of the graph.
The negative coefficient before \(x^2\) signifies a downward opening parabola. This is critical because it causes the exponential function to have an inverted bell shape. The parabola's vertex, found at the origin \((0,0)\), is the highest point on the graph of \(y=2^{-x^{2}}\). From this vertex, the function value decreases as \(x\) moves away from zero in both the positive and negative directions.
This interplay between exponential growth and a quadratic polynomial offers rich analysis potential and is a great example of how different mathematical concepts can merge to define the attributes of a graph.
The negative coefficient before \(x^2\) signifies a downward opening parabola. This is critical because it causes the exponential function to have an inverted bell shape. The parabola's vertex, found at the origin \((0,0)\), is the highest point on the graph of \(y=2^{-x^{2}}\). From this vertex, the function value decreases as \(x\) moves away from zero in both the positive and negative directions.
This interplay between exponential growth and a quadratic polynomial offers rich analysis potential and is a great example of how different mathematical concepts can merge to define the attributes of a graph.
Symmetry about the Y-axis
Symmetry, especially about the y-axis, is an essential characteristic of many functions, including the graph of the function \(y=2^{-x^{2}}\). This type of symmetry means that if you fold the graph along the y-axis, both halves of the graph will coincide.
The reason behind this symmetry is that the exponent \(-x^2\) evaluates to the same value whether \(x\) is positive or negative, i.e., \((-x)^2 = x^2\). This inherent property of squaring transforms results in a symmetrical look, as changes in sign on the x-variable do not affect the outcome of \(-x^2\).
Understanding symmetry helps predict and understand the shape and behavior of graphs without plotting every point. It simplifies complex analysis by reducing redundancy in calculations and explains why certain functions behave identically over symmetric domains.
The reason behind this symmetry is that the exponent \(-x^2\) evaluates to the same value whether \(x\) is positive or negative, i.e., \((-x)^2 = x^2\). This inherent property of squaring transforms results in a symmetrical look, as changes in sign on the x-variable do not affect the outcome of \(-x^2\).
Understanding symmetry helps predict and understand the shape and behavior of graphs without plotting every point. It simplifies complex analysis by reducing redundancy in calculations and explains why certain functions behave identically over symmetric domains.
Other exercises in this chapter
Problem 28
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