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The image shows a graph of immigration trends to the United States from 1905 to 2005, with related questions. Here's a breakdown of how to solve the given questions:

### Data from the graph:
- 1905: 1026 thousand immigrants
- 1925: 309 thousand immigrants
- 1945: ~100 thousand immigrants
- 1955: ~238 thousand immigrants
- 1975: ~385 thousand immigrants
- 1995: ~800 thousand immigrants
- 2005: 1122 thousand immigrants

### Solving the questions:

---

**a. Average annual rate of change (1905–1955):**

- Immigration in 1905 = 1026 thousand
- Immigration in 1955 = 238 thousand
- Time = 50 years

\[
\text{Rate of change} = \frac{238 - 1026}{50} = \frac{-788}{50} = -15.76 \text{ thousand/year}
\]

---

**b. Average annual rate of change (1955–2005):**

- Immigration in 1955 = 238 thousand
- Immigration in 2005 = 1122 thousand
- Time = 50 years

\[
\text{Rate of change} = \frac{1122 - 238}{50} = \frac{884}{50} = 17.68 \text{ thousand/year}
\]

---

**c. Average annual rate of change (1905–2005):**

- Immigration in 1905 = 1026 thousand
- Immigration in 2005 = 1122 thousand
- Time = 100 years

\[
\text{Rate of change} = \frac{1122 - 1026}{100} = \frac{96}{100} = 0.96 \text{ thousand/year}
\]

---

**d. Compare average rates from parts a, b, and c:**

- Part a: -15.76 thousand/year
- Part b: 17.68 thousand/year
- Part c: 0.96 thousand/year

The average of the values from parts a and b is:

\[
\text{Average of a and b} = \frac{-15.76 + 17.68}{2} = \frac{1.92}{2} = 0.96 \text{ thousand/year}
\]

This is the same as the value in part c, which suggests that the rates of change in different time periods balance out when averaged over the entire century.

---

**e. Predict the number of immigrants in 2009:**

Using the rate from part c (0.96 thousand/year), we can predict the number of immigrants in 2009:

\[
\text{Immigrants in 2009} = 1122 + (0.96 \times 4) = 1122 + 3.84 = 1125.84 \text{ thousand}
\]

The actual number of immigrants in 2009 was 1,130.818 thousand, which is very close to our prediction (1125.84 thousand).

---

Would you like more details or have any questions on this solution?

### Further Questions:
1. How would you calculate the standard deviation of the immigration data?
2. Can you determine the immigration trend if the rate of change accelerates after 2005?
3. How would a change in government policy affect the predictions made in part e?
4. What could cause fluctuations between the periods of 1925 and 1945?
5. How would you graph the rate of change of immigration over time?

**Tip**: When calculating average rates of change over long periods, consider major historical events (e.g., wars, economic shifts) that could affect immigration trends.

Find the average annual rate of change in immigration to the United States over different time periods based on the provided graph, and predict the number of immigrants in 2009.

Analyze the average annual rate of change in immigration to the United States from 1905 to 2005 based on a historical graph. Includes step-by-step solutions for calculating the rate of change over various time periods and predicting future immigration levels. Suitable for high school students studying algebra or calculus.

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