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Chapter 5: Problem 5

Determine which of the following functions are exponential. For each exponential function, identify the growth or decay factor and the vertical intercept. a. \(y=5\left(x^{2}\right)\) b. \(y=100 \cdot 2^{-x}\) c. \(P=1000(0.999)\)

Short Answer

Expert verified
Function 2 is exponential with a vertical intercept of 100 and a decay factor of 2.

Step by step solution

01

- Understand Exponential Functions

An exponential function is typically in the form of either \(y = ab^x\) or \(y = ab^{-x}\), where \(a\) is the vertical intercept, and \(b\) is the growth (if \(b > 1\)) or decay factor (if \(0 < b < 1\)). Now let's check each given function to see if it fits this form.
02

- Analyze Function 1

Given: \(y = 5(x^2)\). This function is quadratic rather than exponential because the variable \(x\) is not an exponent. Therefore, it is not an exponential function.
03

- Analyze Function 2

Given: \(y = 100 \cdot 2^{-x}\). This function matches the exponential form \(y = ab^{-x}\) with \(a = 100\) and \(b = 2\). Here, \(a\) (the vertical intercept) is 100, \(b\) is 2, and since \(b > 1\), this is an exponential decay function.
04

- Analyze Function 3

Given: \(P = 1000(0.999)\). This can be simplified to \(P = 999\), which is a constant function rather than an exponential function. Therefore, it is not exponential.

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Key Concepts

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

Exponential Growth
Exponential growth happens when the quantity increases over time at a rate proportional to its current value. We usually describe it with the formula \(y = ab^x\), where \(a\) is the initial amount and \(b\) is the growth factor.
If \(b\) is greater than 1, the function exhibits exponential growth.
Key features of exponential growth:
  • The growth rate is constant and proportional to the value of the quantity.
  • The graph of the function is a continuously increasing curve.
  • Examples include population growth, investment returns, and the spread of diseases.
In our given problems, none of the functions display exponential growth. However, understanding this concept helps in identifying and contrasting different types of exponential behavior.
Always remember - for exponential growth, the base \(b\) must be greater than 1.
Exponential Decay
Exponential decay refers to a situation where a quantity decreases over time at a rate proportional to its current value. It is represented by the formula \(y = ab^{-x}\) or \(y = ab^x\), where \(0 < b < 1\).
In the given example, the function \(y = 100 \times 2^{-x}\) showcases exponential decay:
  • The initial amount \(a\) is 100.
  • The decay factor is \(2\) because the base is greater than 1, and we're using the negative exponent.
  • The graph of this function would show a rapid decline asymptoting towards zero as \(x\) increases.
Examples of exponential decay include radioactive decay, depreciation of assets, and the cooling of hot objects.
Recognizing exponential decay can help in many scientific and financial applications.
Vertical Intercept
The vertical intercept, often simply called the intercept, is the point where the graph of a function crosses the y-axis. This occurs when \(x = 0\).
In exponential functions, it is represented by the coefficient \(a\) in the formulas \(y = ab^x\) or \(y = ab^{-x}\).
For the given function \(y = 100 \times 2^{-x}\), the vertical intercept is 100.
Key points about vertical intercepts:
  • It's where we start the measurement, i.e., the initial value of the function.
  • It provides a reference to understand how the function behaves initially.
  • In exponential functions, changing the vertical intercept shifts the graph up or down without affecting its overall shape.
Understanding the vertical intercept is crucial for graphing and interpreting exponential functions effectively.

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Most popular questions from this chapter

1\. (Graphing program recommended.) Create a table of values for the following functions, then graph the functions. a. \(f(x)=6+1.5 x\) b. \(g(x)=6(1.5)^{x}\) c. \(h(x)=1.5(6)^{x}\)

Given an initial value of 50 units for parts (a)-(d) below, in each case construct a function that represents \(Q\) as a function of time \(t\). Assume that when \(t\) increases by 1: a. \(Q(t)\) doubles b. \(Q(t)\) increases by \(5 \%\) c. \(Q(t)\) increases by ten units d. \(Q(t)\) is multiplied by 2.5

(Requires technology to find a best-fit function.) The accompanying table shows the U.S. international trade in goods and services. $$ \begin{aligned} &\text { U.S }\\\ &\text { International Trade (Billions of Dollars) }\\\ &\begin{array}{crr} \hline & \text { Total } & \text { Total } \\ \text { Year } & \text { Exports } & \text { Imports } \\ \hline 1960 & 25.9 & 22.4 \\ 1965 & 35.3 & 30.6 \\ 1970 & 56.6 & 54.4 \\ 1975 & 132.6 & 120.2 \\ 1980 & 271.8 & 291.2 \\ 1985 & 288.8 & 410.9 \\ 1990 & 537.2 & 618.4 \\ 1995 & 793.5 & 891.0 \\ 2000 & 1070.6 & 1448.2 \\ 2005 & 1275.2 & 1992.0 \\ \hline \end{array} \end{aligned} $$ a. U.S. imports and exports both expanded rapidly between 1960 and \(2005 .\) Use technology to plot the total U.S. exports and total U.S. imports over time on the same graph. b. Now change the vertical axis to a logarithmic scale and generate a semi-log plot of the same data as in part (a). What is the shape of the data now, and what does this suggest would be an appropriate function type to model U.S. exports and imports? c. Construct appropriate function models for total U.S. imports and for total exports. d. The difference between the values of exports and imports is called the trade balance. If the balance is negative, it is called a trade deficit. The balance of trade has been an object of much concern lately. Calculate the trade balance for each year and plot it over time. Describe the overall pattern. e. We have a trade deficit that has been increasing rapidly in recent years. But for quantities that are growing exponentially, the "relative difference" is much more meaningful than the simple difference. In this case the relative difference is \(\frac{\text { exports }-\text { imports }}{\text { exports }}\) This gives the trade balance as a fraction (or if you multiply by 100 , as a percentage) of exports. Calculate the relative difference for each year in the above table and graph it as a function of time. Does this present a more or less worrisome picture? That is, in particular over the last decade, has the relative difference remained stable or is it also rapidly increasing in magnitude?

Represents an exponential function. Construct that function and then identify the corresponding growth or decay rate in percentage form. $$ \begin{array}{ccccc} \hline x & 0 & 1 & 2 & 3 \\ y & 225.00 & 228.38 & 231.80 & 235.28 \\ \hline \end{array} $$

Each of two towns had a population of 12,000 in \(1990 .\) By 2000 the population of town A had increased by \(12 \%\) while the population of town B had decreased by \(12 \%\). Assume these growth and decay rates continued. a. Write two exponential population models \(A(T)\) and \(B(T)\) for towns A and \(\mathrm{B}\), respectively, where \(T\) is the number of decades since 1990 . b. Write two new exponential models \(a(t)\) and \(b(t)\) for towns A and \(\mathrm{B}\), where \(t\) is the number of years since 1990 . c. Now find \(A(2), B(2), a(20)\), and \(b(20)\) and explain what you have found.
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