Choosing a Model ALGEBRA 1 LESSON 8-5 Graph each set of points. Which model is most appropriate for each set? a. (–2, 2.25), (0, 3), (1, 4) (2, 6) b. (–2, –2), (0, 2), (1, 4), (2, 6) c. (–1, 5), (2, 11), (0, 3), (1, 5), (–2, 11) exponential model linear model quadratic model 8-5

There is a common second difference, 2.8. Choosing a Model ALGEBRA 1 LESSON 8-5 a. Which kind of function best models the data below? Write an equation to model the data. Step 1: Graph the data Step 2: The data appear to be quadratic. Test for a common second difference. x y 0 0 1 1.4 2 5.6 3 12.6 4 22.4 +1 + 1.4 + 4.2 + 7.0 + 9.8 + 2.8 x y 0 0 1 1.4 2 5.6 3 12.6 4 22.4 There is a common second difference, 2.8. 8-5

Step 3: Write a quadratic model. y = ax2 Choosing a Model ALGEBRA 1 LESSON 8-5 a. (continued) Step 3: Write a quadratic model. y = ax2 5.6 = a(2)2 Use a point other than(0, 0) to find a. 5.6 = 4a Simplify. 1.4 = a Divide each side by 4. y = 1.4x2 Write a quadratic function. Step 4: Test two points other than (2, 5.6) and (0, 0) y = 1.4(1)2 y = 1.4(3)2 y = 1.4 • 1 y = 1.4 • 9 y = 1.4 y = 12.6 (1, 1.4) and (3, 12.6) are both data points. The equation y = 1.4x2 models the data. 8-5

Choosing a Model ALGEBRA 1 LESSON 8-5 b. Which kind of function best models the data below? Write an equation to model the data. Step 1: Graph the data Step 2: The data appear to suggest an exponential model. Test for a common ratio. x y –1 4 0 2 1 1 2 0.5 3 0.25 +1 2  4 = 0.5 1  2 = 0.5 0.5  1 = 0.5 0.25  0.5 = 0.5 x y –1 4 0 2 1 1 2 0.5 3 0.25 8-5

Step 3: Write an exponential model. Choosing a Model ALGEBRA 1 LESSON 8-5 b. (continued) Step 3: Write an exponential model. Relate: y = a • bx Define: Let a = the initial value, 2. Let b = the decay factor, 0.5. Write: y = 2 • 0.5x Step 4: Test two points other than (0, 2). y = 2 • 0.51 y = 2 • 0.52 y = 2 • 0.5 y = 2 • 0.25 y = 1.0 y = 0.5 (1, 1) and (2, 0.5) are both data points. The equation y = 2 • 0.5x models the data. 8-5

Step 1: Graph the data to decide which model is most appropriate. Choosing a Model ALGEBRA 1 LESSON 8-5 Suppose you are studying deer that live in an area. The data in the table was collected by a local conservation organization. It indicates the number of deer estimated to be living in the area over a five-year period. Determine which kind of function best models the data. Write an equation to model the data. Year Estimated Population 0 90 1 69 2 52 3 40 4 31 Step 1: Graph the data to decide which model is most appropriate. The graph curves, and it does not look quadratic. It may be exponential. 8-5

Step 2: Test for a common ratio. Choosing a Model ALGEBRA 1 LESSON 8-5 (continued) Step 2: Test for a common ratio. Year Estimated Population 0 90 1 69 2 52 3 40 4 31 +1 69  90 0.766 52  69 0.753 40  52 0.769 31  40 0.775 The common ratio is roughly 0.77. The population of deer is roughly 0.77 times its value the previous year. 8-5

Step 3: Write an exponential model. Choosing a Model ALGEBRA 1 LESSON 8-5 (continued) Step 3: Write an exponential model. Relate: y = a • bx Define: Let a = the initial value, 90. Let b = the decay factor, 0.77. Write: y = 90 • 0.77x Step 4:  Test two points other than (0, 90). y = 90 • 0.771 y = 90 • 0.772 y 69 y 53 The predicted value (1, 69) matches the corresponding data point. The point (2, 53) is close to the data point (2, 52). The equation y = 90 • 0.77x models the data. 8-5

Choosing a Model ALGEBRA 1 LESSON 8-5 Which kind of function best models the data in each table? Write an equation to model the data. x y –1 15 0 3 1 0.6 2 0.12 3 0.024 1. x y –1 –5 0 –3 1 –1 2 1 3 3 2. x y –1 2.2 0 0 1 2.2 2 8.8 3 19.8 3. exponential; y = 3 • 0.2x linear; y = 2x – 3 quadratic; y = 2.2x2 8-5