Problem 34 - Algebra A Combined Function

Question

Solve. See Example 4. The table shows the amount of money (in billions of dollars) that Americans spent on their pets for the years shown. (Source: American Pet Products Manufacturers Association) $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Pet-Related Expenditures } \\ \text { (in billions of dollars) } \end{array} \\ \hline 2005 & 36.3 \\ \hline 2006 & 38.5 \\ \hline 2007 & 41.2 \\ \hline 2008 & 43.4 \\ \hline \end{array} $$ a. Write this paired data as a set of ordered pairs of the form (year, pet- related expenditures). b. In your own words, write the meaning of the ordered pair (2007,41.2) c. Create a scatter diagram of the paired data. \(\mathrm{Be}\) sure to label the axes appropriately. d. What trend in the paired data does the scatter diagram show?

Step-by-Step Solution

Answer

a. (2005, 36.3), (2006, 38.5), (2007, 41.2), (2008, 43.4) b. In 2007, Americans spent $41.2 billion on pets. c. Scatter diagram shows an upward trend.

Step 1: Write Ordered Pairs

To write the data as ordered pairs, pair each year with its corresponding pet-related expenditure. This gives us: (2005, 36.3), (2006, 38.5), (2007, 41.2), (2008, 43.4).

Step 2: Explain an Ordered Pair

The ordered pair (2007, 41.2) means that in the year 2007, Americans spent $41.2 billion on pet-related expenditures.

Step 3: Create a Scatter Diagram

Plot each ordered pair on a coordinate system where the x-axis represents the year and the y-axis represents pet-related expenditures in billions of dollars. Each point on the diagram corresponds to an ordered pair from Step 1.

Step 4: Identify Trend from Scatter Diagram

Observe that as the year increases from 2005 to 2008, the pet-related expenditures also increase. This indicates an upward trend over the years, suggesting that spending on pets has been increasing annually.

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in data analysis. They are used to pair two values, allowing us to represent the relationship between two variables in a clear and structured way. In this exercise, the ordered pairs represent the years and the corresponding pet-related expenditures:

• (2005, 36.3)
• (2006, 38.5)
• (2007, 41.2)
• (2008, 43.4)

By using ordered pairs, we can easily correlate a specific year with its expenditure in billions of dollars.
The ordered pair format (year, expenditure) helps us quickly identify and understand key information. For example, the ordered pair (2007, 41.2) means that in the year 2007, Americans spent $41.2 billion on pet-related expenses. This simple representation makes it easier to perceive patterns in data and sets the foundation for creating visualizations like scatter diagrams.

Scatter Diagram

A scatter diagram, also known as a scatter plot, is a type of graph used to display values for two variables in a data set. It helps to visualize the relationship between the variables, making patterns more accessible to identify. To create a scatter diagram from the ordered pairs, each pair is plotted as a point on a coordinate system:
The x-axis denotes the year, while the y-axis represents pet-related expenditures in billions of dollars.
As each ordered pair corresponds to a unique point, the points collectively illustrate the data's distribution.

• Label the x-axis as "Year"
• Label the y-axis as "Pet Expenditure (in billions)"

A well-labeled scatter diagram offers a clear, visual representation of the paired data, providing insight into how one variable changes over time or in relation to another variable.
In this exercise, the points on the scatter diagram will reveal the pattern of pet-related expenses from 2005 to 2008.

Trend Analysis

Trend analysis involves examining the patterns in a series of data points over time. By analyzing the scatter diagram, we can conduct a trend analysis to discover insights and make predictions. The data points from the exercise show that pet-related expenditures have been increasing annually from 2005 to 2008.
Here's how you can identify trends:

• Look for the direction in which the data points are moving.
• In this case, a clear upward movement indicates a rising trend.

The upward trend suggests that the spending on pets increased each year. This information may imply growing consumer interest in pets or an increase in the pet industry overall. Trend analysis is pivotal in forecasting future trends and planning accordingly, whether for business decisions or academic research.