Problem 51 - Algebra and Trigonometry

Question

Exercises \(47-52\) 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 \(72,\) you will use your graphing utility to obtain these functions.) Hamachiphobia $$ \begin{array}{ccc} {} & {\text { Percentage }} & {\text { Percentage } W h o} \\ {} & {\text { Who Won't }} & {\text { Don't Approve of }} \\ {\text { Generation }} & {\text { Try Sushi }} & {\text { Marriage Equality }} \\\ {\text { Millennials }} & {42} & {36} \\ {\text { Gen } X} & {52} & {36} \\ {\text { Boomers }} & {60} & {49} \\ {\text { Silent/Greatest }} & {72} & {66} \\ {\text { Generation }} \end{array} $$

Step-by-Step Solution

Answer

Two scatter plots need to be created, one for each percentage data. Depending on how the points are distributed, a linear, exponential, or logarithmic function will be chosen as the best fit.

Step 1: Creating the Scatter Plot

Firstly, scatter plots need to be created based on the given data. The generations are taken as independent variables (on the x-axis) and the percentages as dependent variables (on the y-axis). Two scatter plots will be created in this step, one for 'Percentage Who Won't Try Sushi' and another for 'Percentage Who Don't Approve of Marriage Equality'.

Step 2: Analyzing the Scatter Plot

Once the scatter plots are created, the next step involves the analysis of these plots. A curve is fitted to every scatter plot and attention is given to whether the data points are distributed in a straight line, or they represent exponential or logarithmic trends. For example, if the points form a straight line, it can be said that a linear function is the best fit for the data. However, if the points increase or decrease rapidly at first and then level off, an exponential function could be a better fit. On the other hand, if the points rise quickly and then gradually slow down, a logarithmic function could be the ideal fit.

Step 3: Determining the Most Suitable Function

This step requires selection of the most appropriate function - logarithmic, linear, or exponential - based on our analysis from the previous step. It can be determined by analyzing the trends of the scatter plot and correlations of the data sets. It's crucial to determine which function the data best fits, as it provides insights about the underlying relationship between the variables and can further help in prediction or decision making.

Key Concepts

Data Analysis

Data analysis is a crucial step in understanding large sets of information, like the generation and percentage data given in the exercise. It involves carefully examining the data through various methods to draw conclusions. Here, a scatter plot is used as the primary tool for visualizing the relationship between the generations and their respective percentages for two categories: **those who won't try sushi** and **those who don't approve of marriage equality**.
Analyzing data using scatter plots helps identify patterns and trends. By plotting the data, you can visually discern whether there's a correlation or trend between the variables. In this context:

• Each data point represents a generation and their respective percentage for a specific survey response.
• By comparing these points on a graph, you can determine if any particular pattern emerges.

Observations from the scatter plots can guide you in selecting an appropriate function to model the data. Treat data analysis as the detective work of math, pinpointing clues and drawing insights from them.

Linear Function

A linear function is a fundamental concept in mathematics and data modeling. It represents a relationship where the change between variables is consistent. This is evident if, on plotting the data on a scatter plot, the points align to form a straight line.
Linear functions take the form:\[y = mx + c\]where:

• \(m\) is the slope of the line, indicating the rate of change between the two variables.
• \(c\) is the y-intercept, showing where the line intersects the y-axis.

In the given exercise, if the scatter plot of any dataset appears to form a trend that is a straight line, it suggests that a linear function might best describe the relationship. This implies that the percentages change at a constant rate as you move from one generation to another. Identifying such a linear trend can make it easier to predict future data points.

Exponential Function

In contrast to linear functions, exponential functions are suited to data that demonstrates a rapid increase or decrease that eventually levels off or steadies. Such a trend usually appears as a curve that starts steeply and gradually flattens.
An exponential function generally follows this form:\[y = a \cdot e^{bx}\]where:

• \(a\) is the starting value or \(y\)-intercept.
• \(e\) is the base of the natural logarithm.
• \(b\) characterizes the rate of growth or decay.

In the provided exercise, should the scatter plot indicate that percentages change more dramatically with earlier generations but level off with later ones, an exponential model might be a best fit. It is crucial to identify such behavior because it could suggest a natural limit or saturation point. Exponential functions are often used in biological contexts or populations where limits are due to resources or environments.

Logarithmic Function

Logarithmic functions are excellent for data that increases quickly at first but then shows a decreasing rate of increase. They are often useful when data demonstrates rapid growth that slows over time.
The general form of a logarithmic function is:\[y = a + b \cdot \,\log(x)\]where:

• \(a\) is a constant that shifts the graph vertically.
• \(b\) determines the rate at which the function increases.

In the context of the exercise, if the data plotting shows a rapid initial response among younger generations that diminishes with older ones, a logarithmic function might fit best. Recognizing this pattern is essential for predictions in scenarios where early momentum is strong but slows down as time or resources increase. Often, logarithmic functions are used in situations involving learning curves or economics, where initial benefits reduce over time.