1. Modeling Real-World Data
Essential Questions
How do we apply functions to problem situations?
How do we use mathematical models to make predictions?
Holt McDougal Algebra 2
Holt Algebra 2

2. Much of the data that you encounter in the real world may form a pattern. Many times the pattern of the data can be modeled by one of the functions you have studied. You can then use the functions to analyze trends and make predictions. Recall some of the parent functions that you have studied so far.

4. Because the square-root function is the inverse of the quadratic function, the constant differences for x- and y-values are switched.
Helpful Hint

5. Identifying Models by Using Constant Differences or Ratios
Use constant differences or ratios to determine which parent function would best model the given data set.
The x-values increase by a constant 5. Find the differences of the y-values. 1.
Time (yr) 5 10 15 20 25
Height (in.) 58 93
128
163
198
First differences: 35 35 35 35
The first differences are constant A linear polynomial best describes the data.

6. Identifying Models by Using Constant Differences or Ratios
Use constant differences or ratios to determine which parent function would best model the given data set.
The x-values increase by a constant 4. Find the differences of the y-values. 2.
Time (yr) 4 8 12 16 20
Population
10,000
9,600
9,216
8,847
8,493
First differences:
-400
-384
-369
-354
Second differences: 16 15 15
Ratio:
The ratios are constant An exponential function best describes the data.

7. Identifying Models by Using Constant Differences or Ratios
Use constant differences or ratios to determine which parent function would best model the given data set.
The x-values increase by a constant 1. Find the differences of the y-values. 3.
Time (s) 1 2 3 4 5
Height (m)
132
165
154 99
First differences: 33
-11
-55
-99
Second differences:
-44
-44
-44
The second differences are constant A quadratic polynomial best describes the data.

8. Identifying Models by Using Constant Differences or Ratios
Use constant differences or ratios to determine which parent function would best model the given data set.
The y-values increase by a constant 10. Find the differences of the x-values. 4. x 12 48
108
192
300 y 10 20 30 40 50
First differences: 36 60 84
108
Second differences: 24 24 24
The second x-value differences are constant A square root function best describes the data.

9. Identifying Models by Using Constant Differences or Ratios
Use constant differences or ratios to determine which parent function would best model the given data set.
The x-values increase by a constant 1. Find the differences of the y-values. 5. x 21 22 23 24 y
243
324
432
576
First differences: 81
108
144
Second differences: 27 36
Ratio:
The ratios are constant An exponential function best describes the data.

10. Lesson 15.3 Practice A