Problem 77 - Algebra A Combined Function

Question

The graph shows the number of U.S. Starbucks locations from 2002 to \(2008 .\) The height of the graph for each year shown corresponds to the number of Starbucks locations in the United States. Use this graph to answer . (We study graphs such as this further in Section 6.1.) Between which two years did the greatest increase in the number of Starbucks locations occur?

Step-by-Step Solution

Answer

The greatest increase occurred between the years with the largest difference in location counts based on the graph.

Step 1: Understand the Graph

Observe the graph that shows the number of Starbucks locations in the U.S. from 2002 to 2008. Each bar in the graph represents a year and its height corresponds to the number of locations in that year.

Step 2: Compare Year-to-Year Increases

Look at the graph and identify the number of Starbucks locations for each year. Calculate the increase in locations by subtracting the previous year's number from the current year's number. Record this increase for each pair of consecutive years.

Step 3: Identify the Greatest Increase

Compare all the calculated year-to-year increases to find which pair of consecutive years showed the greatest increase in the number of Starbucks locations.

Key Concepts

Graph Analysis

Graph analysis is a crucial skill in data interpretation. It allows us to derive useful information from visual data representations. In a graph showing the number of Starbucks locations over several years, each bar represents the quantity of locations in that specific year. The height of the bar corresponds directly to the number of locations. To perform effective graph analysis, first, ensure you understand what each axis represents. For this type of graph:

• The x-axis typically displays years, indicating a chronological sequence.
• The y-axis usually shows the quantity, such as the number of locations.

Once you identify these elements, you can use the graph to track trends over time. This might involve noticing which bars are taller by checking numerical labels or relative heights, allowing you to recognize patterns of increase, decrease, or stability. Graphs provide an immediate snapshot of data trends, making them a powerful tool for analysis.

Year-to-Year Comparison

A year-to-year comparison involves examining data changes over consecutive years. This is particularly useful in understanding growth patterns or trends. Using our Starbucks example, you would first note the number of locations in each year given on the graph. It’s like tracking progress: see each bar as a milestone over time.
After noting these numbers, calculate the difference between each year's data points to determine how much the number changes annually. This involves simple subtraction. For instance, if there were 1,000 locations in 2003 and 1,200 in 2004, the increase is calculated by subtracting 2003’s number from 2004’s:\[1,200 - 1,000 = 200\]By doing this for each year, you reveal the rate of change, helping you spot years with significant growth, decline, or consistency. This method illuminates patterns that raw numbers don't immediately show.

Increase Calculation

Increase calculation is an essential part of understanding data trends. It helps quantify how much something has grown over a certain period. Using the data from our graph, you can compute the increase by taking one year’s total and subtracting the previous year’s total. This translates visually as the difference in height between two bars.
For example, if one year's bar is significantly taller than the previous one, this indicates a substantial increase in locations. Mathematically, if 2006 had 10,000 locations and 2005 had 8,000, the increase is:\[10,000 - 8,000 = 2,000\]This process is repeatedly applied across the graph data to uncover which time period had the most significant boost. Often, visual inspection combined with these calculations will clearly highlight the greatest increase, giving insights into business growth and operational expansion. By mastering this, you gain more profound insights into historical data trends and their implications.