1. 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
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2. 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.
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3. 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.
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4. 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
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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.
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6. 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.
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7. 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 
52 
40 
31 
The common ratio is roughly 0.77.
The population of deer is roughly 0.77 times its value the previous year.
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8. 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 • y = 90 • 0.772
y 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.
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9. 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 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
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