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Chapter 3: Problem 45

Use a graphing utility to graph the exponential function. $$y=1.08^{-5 x}$$

Short Answer

Expert verified
The graph of the given exponential function \(y=1.08^{-5 x}\) will show an exponential decay pattern, indicating that the function decreases as x increases.

Step by step solution

01

Insert the Function into the Graphing Utility

In a selected graphing utility, input the function \(y=1.08^{-5 x}\), ensuring to equally incorporate all parentheses. This is important to represent the function correctly, especially the exponent of -5x.
02

Graph the Function

After entering the function, proceed to graph it. A well-configured graphing utility will automatically adjust the view port to best feature your function.
03

Interpret the Graph

Examine the resulting graph. The graph of our given exponential function should decrease as x increases. The function exhibits an exponential decay because the base of the exponent is less than 1. As we multiply by the base 1.08 for successively larger negative x values, the y-value keeps diminishing, that's why our exponential function decreases.

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Key Concepts

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

Graphing Utilities
Graphing utilities are powerful tools that help visualize mathematical functions. They take an algebraic representation, like our exponential function, and convert it into a graph.
In this exercise, you would insert the expression into the graphing utility to view its graphical representation.
  • Ensure to input the entire function, including any necessary brackets.
  • Graphing utilities, such as online calculators, software applications, or even specific calculator models, can display graphs of functions with ease.
  • These tools often have features to adjust the viewing window, allowing for a more detailed analysis of specific function behaviors.
The convenience and precision of graphing utilities make them a staple in understanding complex functions.
Exponential Decay
Exponential decay describes a process where quantities diminish at a rate proportional to their current value. In the function \(y=1.08^{-5x}\), the negative exponent is a sign of this decay process.
As you examine its properties, you'll notice:
  • The base of 1.08 indicates how the quantity multiplies as the exponent negative changes.
  • Although the base is greater than 1, the negative in the exponent turns it into a decay situation.
This means as \(x\) becomes larger, the expression \(1.08^{-5x}\) approaches zero, showcasing the core feature of exponential decay. Understanding how exponential functions behave with varying parameters is crucial to mastering their many applications.
Function Graphing
Function graphing is the process of drawing a graph based on a function's algebraic representation.
It assists in exploring the relationship between variables like \(x\) and \(y\). For our function \(y=1.08^{-5x}\), graphing it will:
  • Clearly show how \(y\) decreases sharply as \(x\) increases.
  • Provide a visual representation of the exponential decay mentioned earlier.
  • Allow you to pinpoint key features such as intercepts and asymptotic behavior.
Graphing a function is instrumental in understanding its structure and predicting behavior for other values of \(x\). This hands-on approach is often more intuitive than algebraic manipulation alone.

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Most popular questions from this chapter

A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 1000 animals and that the growth of the pack will be modeled by the logistic curve \(p(t)=\frac{1000}{1+9 e^{-0.1656 t}}\) where \(t\) is measured in months (see figure). (a) Estimate the population after 5 months. (b) After how many months will the population be \(500 ?\) (c) Use a graphing utility to graph the function. Use the graph to determine the horizontal asymptotes, and interpret the meaning of the asymptotes in the context of the problem.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{2}(2 x-3)=\log _{2}(x+4)$$

Complete the table for the time \(t\) (in years) necessary for \(P\) dollars to triple if interest is compounded continuously at rate \(r\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% \\ \hline t & & & & & & \\ \hline \end{array} $$

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln (x+5)=\ln (x-1)-\ln (x+1)$$

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.
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