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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

Mute swans were imported from Europe in the nineteenth century to grace ponds. Now there is concern that their population is growing too rapidly, edging out native species. Their population along the Atlantic coast has grown from 5800 in 1986 to 14,313 in 2002 . The increase is most acute in the mid-Atlantic region, but Massachusetts has also seen a jump, with 2939 mute swans counted in 2002 as compared with 585 in 1986 . a. Compare the growth factor in the mute swan population for the entire Atlantic coast with that for Massachusetts. b. Compare the average rate of change in the mute swan population for the entire Atlantic coast with that for Massachusetts. c. Construct both a linear and an exponential model for the mute swan population in Massachusetts since 1986 . d. Compare the projected populations of mute swans in Massachusetts by the year 2010 as predicted by your linear and exponential models.

Given the following exponential decay functions, identify the decay rate in percentage form. a. \(Q=400(0.95)^{t}\) b. \(A=600(0.82)^{\mathrm{r}}\) c. \(P=70,000(0.45)^{t}\) d. \(y=200(0.655)^{x}\) e. \(A=10(0.996)^{T}\) f. \(N=82(0.725)^{T}\)

Which function has the steepest graph? $$ \begin{array}{l} F(x)=100(1.2)^{x} \\ G(x)=100(0.8)^{x} \\ H(x)=100(1.2)^{-x} \end{array} $$

Two cities each have a population of 1.2 million people. City A is growing by a factor of 1.15 every 10 years, while city \(\mathbf{B}\) is decaying by a factor of 0.85 every 10 years. a. Write an exponential function for each city's population \(P_{A}(t)\) and \(P_{B}(t)\) after \(t\) years. b. For each city's population function generate a table of values for \(x=0\) to \(x=50,\) using 10 -year intervals, then sketch a graph of each town's population on the same grid.

Identify and interpret the decay factor for each of the following functions: a. \(P=450(0.43)^{t}\) b. \(f(t)=3500(0.95)^{t}\) c. \(y=21(3)^{-x}\)
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