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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

What is the growth or decay factor for each given time period? a. Weight increases by \(0.2 \%\) every 5 days. b. Mass decreases by \(6.3 \%\) every year. c. Population increases \(23 \%\) per decade. d. Profit increases \(300 \%\) per year. e. Blood alcohol level decreases \(35 \%\) per hour.

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

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. \(P=1000(0.999)\)

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

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.
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