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Chapter 5: Problem 24

In Exercises 23-26, use a graphing utility to graph the exponential function. $$ y=3^{-|x|} $$

Short Answer

Expert verified
The graph of \(y = 3^{-|x|}\) is a decreasing curve for x > 0 and an increasing curve for x < 0. There is an asymptote at y = 0 and the graph is symmetric along the y-axis.

Step by step solution

01

Understand the Function

First, we need to understand what the function \(y = 3^{-|x|}\) is telling us. The absolute value sign in |-x| indicates the distance of x from zero. Therefore, the value inside the exponential function will always be a non-positive number (0 or negative). This results in a function that decreases rapidly for positive values of x and increases rapidly for negative values of x.
02

Use A Graphing Utility

To graph the given function, we can use an online graphing utility like Desmos, GeoGebra or a graphing calculator. Simply input \(y = 3^{-|x|}\) into the tool. Do not forget to use right syntax for this function on your specific tool.
03

Analyze The Graph

After plotting the graph, analyze it. The graph of \(y = 3^{-|x|}\) will be symmetric along the y-axis because the absolute value makes the function even. As x approaches \(\infty\) or \(-\infty\), the value of |x| will also increase, making the value of the function approach zero. Thus, there is a horizontal asymptote at y = 0.

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Key Concepts

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

Graphing Utility
Graphing utilities are powerful tools that help us visualize complex functions easily. They provide a visual representation of mathematical functions, which is particularly useful for understanding how functions behave. Tools you can use include:
  • Desmos: An interactive graphing tool accessible online.
  • GeoGebra: Offers various mathematical visualization tools.
  • Graphing Calculators: Handheld devices that allow for graphing functionalities, like the TI-84.
To graph the function, simply enter the equation, being careful with syntax, especially when dealing with absolute values and exponents. Once entered, you can see a visual representation that helps you analyze aspects such as symmetry and intercepts.
Absolute Value
The absolute value of a number is its distance from zero on the number line, disregarding direction. This means that both \(x\) and \(-x\) will yield the same outcome in terms of distance from zero.
In the function \(y = 3^{-|x|}\), the absolute value ensures that the input to the exponent is always non-positive.
  • For positive \(x\), the expression becomes \(-x\), converting it to a negative or zero value.
  • For negative \(x\), it simply makes the value less negative or zero.
This property results in a graph that mirrors itself along the y-axis, as both sides of the graph will reflect the symmetric effect initiated by the absolute value.
Asymptotes
Asymptotes refer to lines that a graph gets infinitely close to, but never actually touches. Understanding them provides insight into the behavior of functions at extreme values.
For the function \(y = 3^{-|x|}\), there is a horizontal asymptote at \(y = 0\). This is because as \(x\) moves away from zero in either direction, the exponential expression \(3^{-|x|}\) approaches zero.
  • As \(|x| \) grows larger, \(-|x|\) becomes smaller (more negative), making the exponent a very tiny positive number.
  • This results in a fraction of one, hence the entire expression tends towards zero.
The presence of an asymptote can describe limits and boundaries within the function's behavior.
Even Functions
Even functions are identified by their symmetry about the y-axis. A function \(f(x)\) is considered even if, for all inputs \(x\), the equation \(f(x) = f(-x)\) holds true. This means that the function's graph looks the same to the left and right of the y-axis.
The function \(y = 3^{-|x|}\) is an example of an even function due to its incorporation of the absolute value in \(-|x|\). This causes \(y\) to remain unchanged with inputs of the same magnitude but opposite signs.
  • In simpler terms, the graph's shape is mirrored about the y-axis.
  • This even characteristic assists in understanding the function's overall shape and behavior, as computations for one side apply equally to the other side.
Recognizing even functions at a glance can simplify graphing and analysis, streamlining your approach to such equations.

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Most popular questions from this chapter

The table of values was obtained by evaluating a function. Determine which of the statements may be true and which must be false. $$ \begin{array}{|l|l|l|l|} \hline x & 1 & 2 & 8 \\ \hline y & 0 & 1 & 3 \\ \hline \end{array} $$ (a) \(y\) is an exponential function of \(x\). (b) \(y\) is a logarithmic function of \(x\). (c) \(x\) is an exponential function of \(y\). (d) \(y\) is a linear function of \(x\).

\(f(x)=e^{x}, \quad g(x)=\ln x\)

\(y=\log \left(\frac{x}{5}\right)\)

In Exercises 1-8, write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25\) \(\log _{4} 64=3\)

\(y=\log (-x)\)
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