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Chapter 5: Problem 10

Each table has values representing either linear or exponential functions. Find the equation for each function. $$ \begin{array}{cccccc} \hline x & -2 & -1 & 0 & 1 & 2 \\ h(x) & 160 & 180 & 200 & 220 & 240 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{cccccc} \hline x & 0 & 10 & 20 & 30 & 40 \\ j(x) & 200 & 230 & 264.5 & 304.17 & 349.8 \\ \hline \end{array} \end{aligned} $$

Short Answer

Expert verified
h(x) = 20x + 200; j(x) = 200b^{x}, where b eq 1.

Step by step solution

01

Identify the type of function for h(x)

Analyze the values in the table to determine if h(x) is linear or exponential. Notice that as x increases by 1, h(x) increases by a consistent amount of 20.
02

Determine the slope and intercept of h(x)

Since h(x) is linear, we can use the slope-intercept form of a linear equation: \( h(x) = mx + b \). The common difference (rate of change) is 20, so the slope m is 20. Use the point (0, 200) to find the y-intercept b: \( h(x) = 20x + 200 \).
03

Check the equation for h(x)

Verify the equation fits all points by substituting the x-values from the table. Example: For \( x = 1 \), \( h(1) = 20(1) + 200 = 220 \), which matches the table.
04

Identify the type of function for j(x)

For j(x), calculate the ratio of consecutive terms. Notice that the ratios are not consistent, hence the function is not linear but exponential. Calculate the ratios: \( \frac{230}{200} eq \frac{264.5}{230} eq \frac{304.17}{264.5} eq \frac{349.8}{304.17} \).
05

Determine the exponential function j(x)

An exponential function can be written as \( j(x) = ab^{x} \). Use known points to solve for a and b. At \( x = 0 \), \( j(0) = 200 \), so \( a = 200 \). Use another point, say \( (10, 230) \), to find b: \( 230 = 200b^{10} \). Solving for b: \( b = \frac{230}{200}^{\frac{1}{10}} \). Calculate b and verify with other points.
06

Check the equation for j(x)

Verify the equation fits all points by substituting the x-values from the table. Example: For \( x = 20 \), compute \( j(20) \) and compare with the table to confirm correctness.

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Key Concepts

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

linear functions
A linear function is a type of function that forms a straight line when graphed. The general form of a linear equation is \( y = mx + b \). Here, 'm' represents the slope of the line, and 'b' is the y-intercept, which is the point where the line crosses the y-axis.

In the given problem, the function h(x) changes by a constant amount of 20 as x increases by 1. This consistent rate of change indicates that h(x) is a linear function. By identifying the slope and y-intercept, we can write the equation as \( h(x) = 20x + 200 \).

Understanding linear functions is essential for solving many real-world problems, such as predicting costs, calculating distances, and more.
exponential functions
Unlike linear functions, exponential functions grow by a fixed percentage rather than a fixed amount. The general form of an exponential function is \( y = ab^{x} \), where 'a' is the starting value (when x = 0), and 'b' is the growth factor.

In the problem, the function j(x) doesn't increase by a constant amount but instead by varying ratios. This indicates that j(x) is an exponential function. By calculating the initial value and the growth factor 'b,' you can create the equation: \( j(x) = 200b^{x} \. \).

Understanding exponential functions is crucial for modeling phenomena like population growth, radioactive decay, and interest calculations.
slope-intercept form
The slope-intercept form is a way to express linear equations. It's given by the equation \( y = mx + b \), where 'm' represents the slope and 'b' is the y-intercept.

In the exercise, we identified that the function h(x) is linear. By determining the slope as 20 and using the point (0, 200), we can write the slope-intercept form: \( h(x) = 20x + 200 \).

This form makes it easy to quickly identify the slope and y-intercept in any linear equation, allowing for straightforward graphing and analysis.
rate of change
The rate of change refers to how a quantity changes over time. In the context of linear functions, it is represented by the slope 'm' in the equation \( y = mx + b \). The rate of change in exponential functions is captured by the growth factor 'b'.

For h(x), the rate of change is constant at 20, showing a linear relationship. For j(x), the rate of change varies, fitting an exponential pattern.

Understanding rate of change helps in making predictions and understanding relationships between variables in various fields such as economics, physics, and biology.

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Most popular questions from this chapter

Find \(C\) and \(a\) such that the function \(f(x)=C a^{x}\) satisfies the given conditions. a. \(f(0)=6\) and for each unit increase in \(x,\) the output is multiplied by 1.2 . b. \(f(0)=10\) and for each unit increase in \(x,\) the output is multiplied by 2.5

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. \(y=21(3)^{-x}\)

In a chain letter one person writes a letter to a number of other people, \(N,\) who are each requested to send the letter to \(N\) other people, and so on. In a simple case with \(N=2\), let's assume person Al starts the process. Al sends to \(\mathrm{B} 1\) and \(\mathrm{B} 2 ; \mathrm{B} 1\) sends to \(\mathrm{C} 1\) and \(\mathrm{C} 2 ; \mathrm{B} 2\) sends to \(\mathrm{C} 3\) and \(\mathrm{C} 4\); and so on. A typical letter has listed in order the chain of senders who sent the letters. So \(\mathrm{D} 7\) receives a letter that has \(\mathrm{A} 1, \mathrm{~B} 2\), and \(\mathrm{C} 4\) listed. If these letters request money, they are illegal. A typical request looks like this: \(\cdot\) When you receive this letter, send \(\$ 10\) to the person on the top of the list. \(\cdot\) Copy this letter, but add your name to the bottom of the list and leave off the name at the top of the list. \(\cdot\) Send a copy to two friends within 3 days. For this problem, assume that all of the above conditions hold. a. Construct a mathematical model for the number of new people receiving letters at each level \(L,\) assuming \(N=2\) as shown in the above tree. b. If the chain is not broken, how much money should an individual receive? c. Suppose A 1 sent out letters with two additional phony names on the list (say Ala and Alb) with P.O. box addresses she owns. So both \(\mathrm{B} 1\) and \(\mathrm{B} 2\) would receive a letter with the list \(\mathrm{A} 1, \mathrm{~A} 1 \mathrm{a},\) Alb. If the chain isn't broken, how much money would Al receive? d. If the chain continued as described in part (a), how many new people would receive letters at level \(25 ?\) e. Internet search: Chain letters are an example of a "pyramid growth" scheme. A similar business strategy is multilevel marketing. This marketing method uses the customers to sell the product by giving them a financial incentive to promote the product to potential customers or potential salespeople for the product. (See Exercise \(31 .)\) Sometimes the distinction between multilevel marketing and chain letters gets blurred. Search the U.S. Postal Service website (www.usps.gov) for "pyramid schemes" to find information about what is legal and what is not. Report what you find.

(Graphing program recommended.) Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react with nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide (via plants). As long as a plant or animal is alive, carbon-14 is maintained in the organism at a constant level. Once the organism dies, however, carbon-14 decays exponentially into carbon-12. By comparing the amount of carbon- 14 to the amount of carbon-12, one can determine approximately how long ago the organism died. Willard Libby won a Nobel Prize for developing this technique for use in dating archaeological specimens. The half-life of carbon-14 is about 5730 years. In answering the following questions, assume that the initial quantity of carbon- 14 is 500 milligrams. a. Construct an exponential function that describes the relationship between \(A,\) the amount of carbon- 14 in milligrams, and \(t,\) the number of 5730 -year time periods. b. Generate a table of values and plot the function. Choose a reasonable set of values for the domain. Remember that the objects we are dating may be up to 50,000 years old. c. From your graph or table, estimate how many milligrams are left after 15,000 years and after 45,000 years. d. Now construct an exponential function that describes the relationship between \(A\) and \(T,\) where \(T\) is measured in years. What is the annual decay factor? The annual decay rate? e. Use your function in part (d) to calculate the number of milligrams that would be left after 15,000 years and after 45,000 years.

Estimate the doubling time using the rule of 70 when: a. \(P=2.1(1.0475)^{t}\), where \(t\) is in years b. \(Q=2.1(1.00475)^{T}\), where \(T\) is in years
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