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Chapter 4: Problem 28

Use a graphing utility to graph the function. \(y=2^{-x^{2}}\)

Short Answer

Expert verified
To graph the function \(y=2^{-x^{2}}\) using a graphing utility, the function is entered into the graphing utility without changes, the software is directed to graph, and then the resulting graph is analyzed.

Step by step solution

01

Identify the Function

The function given is \(y=2^{-x^{2}}\). This function is a transformation of the base function \(2^{-x}\). Here, the exponent is modified as \(-x^{2}\), indicating that for every value of \(x\), a squared operation is performed first before negating and subsequently using it as an exponent.
02

Input the Function into the Graphing Utility

Using a graphing utility, input the function as it is. Ensure that the syntax matches the requirements of the graphing utility. With most graphing software or calculators, this would involve simply typing '2^(-x^2)' into the function field.
03

Ordering the Graph

After inputting the function, order the graphing utility to plot the function. This is usually done by pressing 'enter' or clicking an 'OK' or 'Graph' button.
04

Interpret the Graph

Examine the graph produced by the graphing utility. The graph plotted represents the function \(y=2^{-x^{2}}\), giving a visual representation of how \(y\) changes with respect to \(x\). Usually, the graph of \(2^{-x^{2}}\) will be a curve that opens downward and increases as \(x\) approaches zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponentiation
Exponentiation is a fundamental concept in mathematics, involving the raising of a base number to the power of an exponent. In our function, we see this in the form of the expression \(2^{-x^2}\). Here, 2 is the base and \(-x^2\) is the exponent.
  • The base 2 indicates that our function's growth is exponential, meaning it changes multiplicatively with the exponent.
  • The exponent \(-x^2\) indicates two operations. First, the variable \(x\) is squared, and then it is negated.
Squaring \(x\) before applying the negative sign leads to symmetry around the y-axis in the graph because squaring any positive or negative value of \(x\) results in the same value (i.e., \((-x)^2 = x^2\)). This symmetry simplifies the behavior of the function significantly.
Transformation of Functions
Transformation of functions is a technique used to modify the basic appearance and position of a graph. In our function \(y=2^{-x^2}\), there are specific transformations affecting its behavior.
  • The inclusion of \(-x^2\) as an exponent transforms the base function \(y=2^{-x}\).
  • This transformation involves a modification of the rate at which the function decreases. Due to the quadratic term \(x^2\), the decay doubles as the value of \(x\) moves away from zero.
These changes mean that the graph will appear more compressed in comparison to its base function and will resemble an inverted parabola symmetric about the y-axis, instead of a simple exponential decline.
Graph Interpretation
Graph interpretation involves analyzing a graph to understand the relationships and behaviors it represents. For the function \(y=2^{-x^2}\), interpreting the graph reveals several important features:
  • The curve is symmetric around the y-axis due to the exponent \(x^2\), meaning the function is even.
  • The graph approaches \(y = 0\) as \(|x|\) increases, because the term \(2^{-x^2}\) tends towards zero.
  • At \(x = 0\), the graph achieves its maximum value of \(y = 1\), since \(2^{0} = 1\).
Understanding these properties helps in visualizing the function and recognizing how small changes in \(x\) can lead to significant changes in \(y\), especially as \(x\) moves through positive and negative extremes.

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Most popular questions from this chapter

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