1. Analyzing Functions, Curve Fitting (9-9)
Objective: Identify linear, quadratic, and exponential functions from given data. Write equations that model that data.

2. Identify Functions
You can use linear functions, quadratic functions, and exponential functions to model data.
The general form of the equations and a graph of each function type are listed below.

3. Identify Functions Linear Function Quadratic Function
Exponential Function
y = mx + b
y = ax2 + bx + c
y = a ∙ bx, when b > 0

4. Example 1
Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.
(1, 2), (2, 5), (3, 6), (4, 5), (5, 2)
Quadratic Function

5. Example 1
Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.
(-1, 6), (0, 2), (1, 2/3) , (2, 2/9)
Exponential Function

6. Check Your Progress Choose the best answer for the following.
Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (-2, -6), (0, -3), (2, 0) , (4, 3)
Linear
Quadratic
Exponential

7. Check Your Progress Choose the best answer for the following.
Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (-2, 0), (-1, -3), (0, -4) , (1, -3), (2, 0)
Linear
Quadratic
Exponential

8. Choose a Model Using Differences or Ratios
Another way to determine which model best describes data is to use patterns.
The differences of successive y-values are called first differences.
The differences of successive first differences are called second differences.
If the differences of successive y-values are all equal, the data represents a linear function.
If the second differences are all equal, but the first differences are not equal, the data represents a quadratic function.
If the ratios of successive y-values are all equal and r ≠ 1, the data represents an exponential function.

9. Example 2
Look for a pattern in each table of values to determine which kind of model best describes the data. .
Linear Function X -2 -1 1 2 Y 3 5 7 +2 +2 +2 +2

10. Example 2
Look for a pattern in each table of values to determine which kind of model best describes the data. .
Exponential Function
= 0.3 X -2 -1 1 2 Y 36 12 4
4/3
4/9
= 0.3
-24 -8
-2.7
-0.9
= 0.3
+16
+5.3
+1.8
= 0.3

11. Check Your Progress Choose the best answer for the following.
Look for a pattern in the table of values to determine which kind of model best describes the data.
Linear
Quadratic
Exponential
None of the above X 1 2 3 4 5 Y 9 17 33 57 89 +8
+16
+24
+32 +8 +8 +8

12. Check Your Progress Choose the best answer for the following.
Look for a pattern in the table of values to determine which kind of model best describes the data.
Linear
Quadratic
Exponential
None of the above X 1 2 3 4 5 Y 96 24 6
3/2
3/8
-72
-18
-4.5
-1.125
+54
+13.5
+3.375
= 0.25
= 0.25
= 0.25
= 0.25

13. Write Equations
Once you find the model that best describes the data, you can write an equation for the function.
For a quadratic function in this lesson, the equation will have the form y = ax2.

14. Example 3
Determine which model best describes the data. Then write an equation for the function that models the data.
Exponential Function
y = a ∙ bx
y = 1 ∙ 8x X 1 2 3 4 Y 8 64
512
4096 +7
+56
+448
+3584

15. Check Your Progress Choose the best answer for the following.
Determine which model best describes the data. Then write an equation for the function that models the data.
Quadratic; y = 3x2
Linear; y = 6x
Exponential; y = 3x
Linear; y = 3x X 1 2 3 4 Y 12 27 48 +3 +9
+15
+21

16. Example 4
The table shows the number of children enrolled in a beginner’s karate class for four consecutive years. Determine which model best represents the data. Then write a function that models that data.
Linear Function
y = ax + b
y = 3x + 8
Time (years) 1 2 3 4
Number enrolled 8 11 14 17 20 +3 +3 +3 +3

17. Check Your Progress Choose the best answer for the following.
The table shows the growth of prairie dogs in a colony over the years. Determine which model best represents the data. Then write a function that models the data.
Linear; y = 4x + 4
Quadratic; y = 8x2
Exponential; y = 2 ∙ 4x
Exponential; y = 4 ∙ 2x
Time (years) 1 2 3 4
Prairie Dogs 8 16 32 64 +4 +8
+16
+32