3.6.4 Practice: Modeling: Linear, Quadratic, and Exponential Functions Practice Algebra 1 Sem 2 Points Possible: 20 Name: Date: 10/30/2018 YOUR ASSIGNMENT: National Park Animal Populations You are studying the populations of three animals in a national park. Below are the data sets for the animal populations. Circle the park that you chose: Yellowstone x Year y1 Population 1, Grizzly Bears (per square mile) y2 Population 2, Pocket Gophers (per square mile) y3 Population 3, Osprey (per square mile) x y1 y2 y3 0 4.0 4.0 4.0 1 4.5 4.5 5.8 2 5.0 5.3 7.2 3 5.5 6.4 8.2 4 6.0 8.1 8.8 5 6.5 10.6 9.0 8 8.0 28.6 7.2 10 9.0 60.7 4.0 Everglades x Year y1 Population 1, American Alligator (per square mile) y2 Population 2, Marsh Rabbit (per square mile) y3 Population 3, Florida Panther (per square mile) x y1 y2 y3 0 6.0 6.0 6.0 1 6.5 6.5 7.8 2 7.0 7.3 9.0 3 7.5 8.4 9.8 4 8.0 10.1 10.0 5 8.5 12.6 9.8 8 10.0 30.6 6.0 10 11.0 62.7 1.0 Make Sense of the Problem What do you want to find out? What type of function best models the population growth for each species: linear, quadratic, or exponential? What is the estimated population for each animal at year 12? Analyze the Data Population 1 (y1): american alligator (Write the animal species' name.) Answer the following questions about the growth function of population 1: 1. Is population 1 increasing or decreasing? (0.5 point) population 1 is increasing. 2. What is the rate of change between year 0 and year 1 for population 1? What is it between year 5 and year 10? Include calculations in your answer. (2 points) The the rate of change between year 0 and year 1 is 0.5. The difference between year 5 and year 10 is 2.5 because if you subtract 8.5 from 11.0 you will get 2.5 and that becomes the difference between the to years. 3. Predict the average rate of change from year 5 to year 6 for population 1. Use the average rate of change you found in question 2 in your prediction. (1 point) The average rate of change from year 5 to year 6 in population 1 is 0.5 still because if you add 0.5 to 8.5 you will get 7.0 because the only thing you're doing is adding 0.5 to 8.5. 4. What type of function best models the growth for population 1? Give a reason for your answer. (1 point) Population 2 (y2): marsh rabbit (Write the animal species' name.) Answer the following questions about the growth function of population 2: 5. What is the maximum population per square mile during the first 10 years for population 2? (0.5 point) the maximum population increase per square mile during the first 10 years for population 2 is 62.7. 6. What is the average rate of change for population 2 between years 5 and 10 (x = 5 to x = 10)? (1 point) 7. The average rate of change for population 2 changes by a common ratio (multiplication) of 1.5 each year. What type of function best models this growth? (1 point) 8. Estimate the average rate of change from year 5 to year 6 for population 2. Remember, the average rate of change for this population changes by a ratio of 1.5 each year. Show your work. HINT: First find the average rate of change from year 4 to year 5. (3 points) Population 3 (y3): ______________________________ (Write the animal species' name.) Answer the following questions about the growth function of population 3: 9. What is the maximum population per square mile during the first 10 years for population 3? In what year did this occur? (1 point) 10. What is the average rate of change for population 3 between years 5 and 10 (x = 5 to x = 10)? Show your work. Identify the change as an increase or a decrease. (2 points) 11. What type of function best models the growth for population 3? Give a reason for your answer. (1 point) 12. Use the graph provided. When do all three populations contain the same number of animals? (1 point) 13. Use the following graphs to verify your work on this question. A. Estimate the average rate of change from year 5 to year 8 for population 3. Show your work. (2 points) B. Estimate the average rate of change from year 8 to year 10 for population 3. Show your work. (2 points) C. Based on the answers to Parts A and B, estimate the number of animals in population 3 in year 15, and give a reason for your estimate. (1 point)

2. What is the rate of change between year 0 and year 1 for population 1? What is it between year 5 and year 10? Include calculations in your answer. (2 points)

The rate of change between year 0 and year 1 is .5 and the rate of change between year 5 and 10 is also .5

3. Predict the average rate of change from year 5 to year 6 for population 1. Use the average rate of change you found in question 2 in your prediction. (1 point)

I think the average rate of change from year 5 to year 6 is .5 this is because each year the population is increasing by .5 according to question 2.

4. What type of function best models the growth for population 1? Give a reason for your answer. (1 point)

A linear function would model population 1 the best. This is because the population is increasing by .5 each year. It increases as a steady rate so it will be a straight line.

Population 2 pocket gophers

5. What is the maximum population per square mile during the first 10 years for population 2? (0.5 point)

The maximum population per square mile during the first 10 years is 60.7.

6. What is the average rate of change for population 2 between years 5 and 10 (x = 5 to x = 10)? (1 point)

The average rate of change is 10.02

7. The average rate of change for population 2 changes by a common ratio (multiplication) of 1.5 each year. What type of function best models this growth? (1 point)

A exponential function would best model this growth.

8. Estimate the average rate of change from year 5 to year 6 for population 2. Remember, the average rate of change for this population changes by a ratio of 1.5 each year. Show your work. HINT: First find the average rate of change from year 4 to year 5. (3 points)

The rate of change from year 5 to year 6 is about 3.75 since population 2 changes by a ratio of 1.5 each year. The rate of change from year 4 to 5 is 2.5 so 2.5*1.5 is 3.75.

Population 3 Osprey

9. What is the maximum population per square mile during the first 10 years for population 3? In what year did this occur? (1 point)

The maximum population per square mile during the first 10 years is 9.0 this occured in year 5.

10. What is the average rate of change for population 3 between years 5 and 10 (x = 5 to x = 10)? Show your work. Identify the change as an increase or a decrease. (2 points)

The change is a decrease.

11. What type of function best models the growth for population 3? Give a reason for your answer. (1 point)

A quadratic function best models the growth for population 3 because at first the population increases but then it decreases.

12. Use the graph provided. When do all three populations contain the same number of animals? (1 point)

In year 0 all three populations contained the same amount of animals.

13. Use the following graphs to verify your work on this question.

A. Estimate the average rate of change from year 5 to year 8 for population 3. Show your work. (2 points)

7.2-9/8-5 = -1.8/3 = -0.6

B. Estimate the average rate of change from year 8 to year 10 for population 3. Show your work. (2 points)

4-7.2/10-8 = -3.2/2 = -1.6

C. Based on the answers to Parts A and B, estimate the number of animals in population 3 in year 15, and give a reason for your estimate. (1 point)

The number of animals in year 15 will be 0 this is because the number of animals keeps steadily going down and you cannot have a negative number of animals.