91Ó°ÊÓ

Exercises \(51-56\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(76,\) you will use your graphing utility to obtain these functions.) Teenage Drug Use $$\begin{array}{|lc|} \hline & \text { Percentage Who Have Used } \\ \hline \text { Country } & \text { Marijuana } & \text { Other IIlegal Drugs } \\\ \hline \text { Czech Republic } & 22 & 4 \\ \text { Denmark } & 17 & 3 \\ \text { England } & 40 & 21 \\ \text { Finland } & 5 & 1 \\ \text { Ireland } & 37 & 16 \\ \text { Italy } & 19 & 8 \\ \text { Northern Ireland } & 23 & 14 \\ \text { Norway } & 6 & 3 \\ \text { Portugal } & 7 & 3 \\ \text { Scotland } & 53 & 31 \\ \text { United States } & 34 & 24 \end{array}$$

Step by step solution

01

Drawing the Scatter Plot

Plot the data in the form of a scatter plot. On the x-axis, represent the countries, whereas on the y-axis, plot the respective percentages for each country for Marijuana use and Other Illegal Drugs use. Each type of drug should be represented with a different color or symbol for better differentiation.

02

Observing the Trend

Analyse the scatter plot to identify the trend. If the points on the plot are rising or falling in a consistent incline or recline regardless of the steepness, hinting towards a straight line, then the best choice for a model is linear. If the points on the plot start off slowly, then increase or decrease at a faster rate, making a curve, an exponential function would be the best model. If the points increase rapidly then slow down, forming a curve that becomes less steep as the x-values increase, then a logarithmic function is the best model.

03

Selecting the Model

Given the nature of the scatter plot, make a judgment about which type of function best suits the data. If it's linear, logarithmic, or exponential, base the decision on the trend observed.

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Key Concepts

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

Exponential Function

An **exponential function** is recognized by a rapid increase or decrease in the plot data. When analyzing a scatter plot, if you observe that the data points start off slowly and begin to sharply rise or decline, you may be dealing with an exponential relationship.
Exponential functions are expressed in the form:
\[ y = a \cdot b^x \]

• Here, \(a\) is a constant representing the initial value.
• \(b\) is the base of the exponential, indicating growth if \(b > 1\) or decay if \(0 < b < 1\).

In our context of teenage drug use, if countries show a pattern where, after a few low observations, the percentage of drug use swiftly spikes, an exponential model will aptly describe the trend.
It's crucial to inspect the plot to ensure this sharp change is evident, and the changes persist through multiple data sets to suggest exponential behavior.

Logarithmic Function

A **logarithmic function** is characterized by a steep increase of data points, which then levels off as the values rise, creating a curve that flattens out along the x-axis.
Logarithmic functions are generally expressed as:
\[ y = a + b \ln(x) \]

• Where \(a\) and \(b\) are constants, and \(\ln(x)\) is the natural log of \(x\).

When analyzing scatter plots, a logarithmic model is suitable when the increase in percentage or quantity is rapid initially and then slows down.
In the dataset of teenage drug use, if certain percentages, particularly marijuana use, rise swiftly at first and then the rate of increase decelerates, a logarithmic model might be the best fit.
This type of function is beneficial to model phenomena where growth begins explosively but stabilizes over time, reflecting a saturation effect or a diminishing return as the x-values increase.

Linear Function

A **linear function** indicates a straightforward relationship where the rate of change between the variables is constant. If a scatter plot shows points closely following a straight line, it suggests a linear model.
Mathematically, a linear function is given by:
\[ y = mx + c \]

• Here, \(m\) represents the slope or rate of change.
• \(c\) is the y-intercept, the value where the line crosses the y-axis.

In the context of the exercise, if the data points for drug use across countries show a consistent slope rather than sharp increases or a leveling off, then a linear function may most effectively depict the data.
The benefit of using a linear model is its simplicity and ease of calculation, making it a good starting point when modeling trends. However, ensure the data points fit this pattern throughout the plot to utilize this model effectively.