The accompanying line graph shows how the average salary for baseball players in a certain league has increased over a​ 12-year period. Notice the average salary is given for​ odd-numbered years only. Also note that the left scale does not start at zero. Which​ 2-year period(s) had the smallest​ increase?

Question

2007
2019
2.5
3
3.5
4
4.5
5
Year
Average Salary ($ millions)

Answer 1

To determine the 2-year period(s) with the smallest increase in the average salary, you need to look at the data points on the line graph for each pair of consecutive odd-numbered years and calculate the difference in the average salary for those years.

Let's list the odd-numbered years in your 12-year period, assuming they are consecutive from the starting year:

1. **Year 1 (Y1)**
2. **Year 3 (Y3)**
3. **Year 5 (Y5)**
4. **Year 7 (Y7)**
5. **Year 9 (Y9)**
6. **Year 11 (Y11)**
7. **Year 13 (Y13)**

Now, identify the average salary for each of these years using the graph.

For each 2-year period between these years, calculate the increase:

1. \[ \Delta_{1-3} = \text{Average Salary in Y3} - \text{Average Salary in Y1} \]
2. \[ \Delta_{3-5} = \text{Average Salary in Y5} - \text{Average Salary in Y3} \]
3. \[ \Delta_{5-7} = \text{Average Salary in Y7} - \text{Average Salary in Y5} \]
4. \[ \Delta_{7-9} = \text{Average Salary in Y9} - \text{Average Salary in Y7} \]
5. \[ \Delta_{9-11} = \text{Average Salary in Y11} - \text{Average Salary in Y9} \]
6. \[ \Delta_{11-13} = \text{Average Salary in Y13} - \text{Average Salary in Y11} \]

Find the period with the smallest increase by comparing these differences.

For simplicity, let's say the average salaries given in the graph are:

1. Y1: $2.5 million
2. Y3: $3.0 million
3. Y5: $3.5 million
4. Y7: $4.0 million
5. Y9: $4.5 million
6. Y11: $5.0 million
7. Y13: $5.5 million

The increases would be:

1. \[ \Delta_{1-3} = 3.0 - 2.5 = 0.5 \text{ million} \]
2. \[ \Delta_{3-5} = 3.5 - 3.0 = 0.5 \text{ million} \]
3. \[ \Delta_{5-7} = 4.0 - 3.5 = 0.5 \text{ million} \]
4. \[ \Delta_{7-9} = 4.5 - 4.0 = 0.5 \text{ million} \]
5. \[ \Delta_{9-11} = 5.0 - 4.5 = 0.5 \text{ million} \]
6. \[ \Delta_{11-13} = 5.5 - 5.0 = 0.5 \text{ million} \]

In this example, all the increases are the same (0.5 million), so every pair of consecutive years has the same, smallest increase.

However, please check the actual data points on your graph to make these calculations accurately, as the real values might be different from this illustration.