1. Lesson 2.8 Quadratic Models
Essential Question: How do you use scatter plots to model data and make predictions?

2. Before we start…
Is there a relationship between the amount of money an employee earns and the numbers of hours the employee works?

3. Classifying Scatter Plots
In real life, many relationships between two variables are parabolic. A scatter plot can be used to give you an idea of which type of model will best fit a set of data.

4. Decide whether each set of data could be better modeled by a linear model, 𝑦=𝑎𝑥+𝑏, a quadratic model, 𝑦=𝑎𝑥2+𝑏𝑥+𝑐, or neither. (0.9, 1.7), (1.2, 2.0), (1.3, 1.9), (1.4, 2.1), (1.6, 2.5), (1.8, 2.8), (2.1, 3.0), (2.5, 3.4), (2.9, 3.7), (3.2, 3.9), (3.3, 4.1), (3.6, 4.4), (4.0, 4.7), (4.2, 4.8), (4.3, 5.0)

5. Decide whether each set of data could be better modeled by a linear model, 𝑦=𝑎𝑥+𝑏, a quadratic model, 𝑦=𝑎𝑥2+𝑏𝑥+𝑐, or neither. (0.9, 3.2), (1.2, 4.0), (1.3, 4.1), (1.4, 4.4), (1.6, 5.1), (1.8, 6.0), (2.1, 7.6), (2.5, 9.8), (2.9, 2.4),(3.2,14.3),(3.3, 15.2), (3.6, 18.1), (4.0, 22.7), (4.2, 24.9), (4.3, 27.2)

6. Decide whether each set of data could be better modeled by a linear model, 𝑦=𝑎𝑥+𝑏, a quadratic model, 𝑦=𝑎𝑥2+𝑏𝑥+𝑐, or neither. (0.9, 1.2), (1.2, 6.5), (1.3, 9.3), (1.4, 11.6), (1.6, 15.2), (1.8, 16.9), (2.1, 14.7), (2.5, 8.1), (2.9, 3.7), (3.2, 5.8), (3.3, 7.1), (3.6, 11.5), (4.0, 20.2), (4.2, 23.7), (4.3, 26.9)

7. Fitting a Quadratic Model to Data
You can use a procedure to find a model for nonlinear data. Once you have used a scatter plot to determine the type of model that would best fit a set of data, there are several ways that you can actually find the model. Each method is best used with a computer or calculator, rather than with and calculations.

8. How do you use scatter plots to model data and make predictions?
The regression feature of a calculator will model the data so you have an equation that you can evaluate to make predictions.

9. a. Use a graphing utility to create a scatter plot of the data.
A study was done to compare the speed x (in miles per hour) with the mileage y (in miles per gallon) of an automobile. The results are shown in the table.
a. Use a graphing utility to create a scatter plot of the data.
b. Use the regression feature of the graphing utility to find model that best fits the data.
c. Approximate the speed at which the mileage is the greatest.
Speed, x
Mileage, y 15
22.3 20
25.5 25
27.5 30
29.0 35
28.7 40
29.9 45
30.4 50
30.2 55
30.0 60
28.8 65
27.4 70
25.3 75
23.3

10. Time, x
Height, y
32.11
0.1
31.8
0.2
31.1
0.3
30.8
0.4
29.85
0.5
28.9
0.6
28.1
0.7
27.65
0.8
26.6
0.9
25.1 1 24
1.1
22.87
1.2
22.21
1.3
20.9
1.4
19.54
1.5
17.99
1.6
17.12
An object is dropped form a height of about 32 feet. The table shows the height of the object at various times. Use a graphing calculator to find a model that best fits the data. Use the model to predict when the object will hit the ground.

11. Use a graphing utility to find a linear model and a quadratic model for the data. Determine which model better fits the data. Let x represents the year, with x = 1 corresponding to 2001.
Year
Number of
recalls made by
a manufacturer
2001 10
2002 11
2003 14
2004
2005 15
2006
2007 16
2008

12. How do you use scatter plots to model data and make predictions?

13. Ticket Out the Door
Decide whether each set of data could be better modeled by a linear model, y = ax + b, a quadratic model, y = ax2 + bx + c, or neither. (0.2, 1.2), (0.6, 1.36), (0.65, 1.39), (0.9, 1.4), (1.2, 1.44), (1.24, 1.6), (1.5, 2.25), (1.8, 3.24)