Problem 33 - Algebra A Combined Function

Question

Solve. See Example 4. The table shows the domestic box office (in billions of dollars) for the U.S. movie industry during the years shown. (Source: Motion Picture Association of America) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Box Office (in billions of dollars) } \\ \hline 2003 & 9.17 \\ \hline 2004 & 9.22 \\ \hline 2005 & 8.83 \\ \hline 2006 & 9.14 \\ \hline 2007 & 9.63 \\ \hline 2008 & 9.79 \\ \hline \end{array} $$ a. Write this paired data as a set of ordered pairs of the form (year, box office). b. In your own words, write the meaning of the ordered pair (2006,9.14) c. Create a scatter diagram of the paired data. Be sure to label the axes appropriately. d. What trend in the paired data does the scatter diagram show?

Step-by-Step Solution

Answer

The ordered pair (2006, 9.14) indicates $9.14 billion in 2006. The scatter diagram shows an overall increase from 2005 to 2008.

Step 1: Writing Ordered Pairs

To convert the data into ordered pairs, each year will be paired with its corresponding box office value. The ordered pairs are: \((2003, 9.17), (2004, 9.22), (2005, 8.83), (2006, 9.14), (2007, 9.63), (2008, 9.79)\).

Step 2: Interpret the Ordered Pair (2006, 9.14)

The ordered pair \((2006, 9.14)\) means that in the year 2006, the domestic box office revenue in the U.S. was 9.14 billion dollars.

Step 3: Creating a Scatter Diagram

Draw a scatter plot on a graph with the x-axis labeled as "Year" and the y-axis labeled as "Box Office (in billions of dollars)". Plot each ordered pair on the graph to visually represent the data.

Step 4: Analyzing Trends in the Scatter Diagram

Review the scatter plot to identify any visible trends. The scatter diagram shows a slight increase in box office revenue from 2003 to 2004, followed by a decrease in 2005, and then a gradual increase from 2005 through 2008.

Key Concepts

Ordered Pairs

Understanding ordered pairs is crucial when concerning scatter plots and data analysis. An ordered pair is expressed in the form (x, y), where the first element, x, denotes the horizontal axis, and the second element, y, denotes the vertical axis. For example, in our movie box office data, the ordered pair

• (2003, 9.17)
• (2004, 9.22)
• (2005, 8.83)
• (2006, 9.14)
• (2007, 9.63)
• (2008, 9.79)

indicates each year and its respective box office earnings in billions of dollars. Each ordered pair tells a story of how the industry's earnings varied over these years. Ordered pairs serve as the building blocks for plotting points on a graph or scatter plot. They allow us to represent data visually, making it simpler to observe patterns or trends as we graph the points for analysis.

Data Interpretation

Data interpretation involves explaining or assigning meaning to data and evaluating it to derive conclusions. This is especially important when working with ordered pairs and scatter plots. For instance, the ordered pair (2006, 9.14) signifies that in 2006, the domestic box office revenue was $9.14 billion. Interpretations like this help us understand the numerical relationships within the data.
With ordered pairs, we can explore relationships between two variables, like years and box office earnings, and make sense of any underlying connections. Proper interpretation allows for informed conclusions, which can guide decision-making or further research. We use both qualitative and quantitative methods in data interpretation to ensure accuracy and reliability of the findings. The visual element of scatter plots complements numerical discussions by making data interpretation more intuitive.

Trend Analysis

Trend analysis focuses on identifying patterns or directions in data over time. In the context of our box office data, scatter plots help reveal these trends by displaying how data points (ordered pairs) align or shift over a range of years.
A careful look at the scatter plot of our data might show a slight rise from 2003 to 2004, a decrease in 2005, and then a steady recovery up to 2008.
Trend analysis is not just about noting these movements; it also involves understanding the reasons behind them and projecting future trends. It allows businesses and analysts to make forecasts based on historical data patterns. Understanding trends in data helps to reveal insights such as seasonality, cyclical patterns, or irregular fluctuations. In the movie industry, for instance, such trends can indicate factors influencing box office revenues, such as release of blockbuster movies or global events affecting cinema attendance.