1. MAT 150 Algebra 1-7 Modeling Linear Functions
Topics:
Find exact Linear Models for data
Determine if a set of data can be modeled exactly or approximately
Scatter Plots
Linear Model = good fit?
Solve Problems using Linear Models

2. Exact and Approximate Linear Models
We can determine that the data points fit exactly on a line by determining that the changes in output values are equal for equal changes in the input values. In this case, we say that the inputs are uniform and the first differences are constant.
If the first differences of data outputs are constant for uniform inputs, the rate of change is constant and linear function can be found that fits the data exactly.
If the first differences are “nearly constant,” a linear function can be found that is an approximate fit for the data.

3. Fitting Lines to Data Ex 1: Linear Correlation
Which graph(s) has
Positive Linear Correlation?
Constant Correlation?
Negative Linear Correlation?
No Correlation?
Correlation that is not Linear?

4. Construct a scatter plot of the data in the table
Determine if the points plotted fit exactly or only approximately by a linear function.
Create a linear model for the data in the table
Use the rounded function y = f(x) that was found above to evaluate f(8) and f(11). X 1 2 3 5 7 9 4 Y 6 12

5. Discrete Function vs. Continuous Function
Discrete Function – Describe data or function that is presented in the form of a table or scatter plot. Continuous Function – A function or graph when the inputs can be any real number or any real number between two specified values.

6. Technology Note
After a model for a data set has been found, it can be rounded for reporting purposes. However, do not use a rounded model in calculations, and do not round answers during the calculation process unless instructed to do so. When the model is used to find numerical answers, the answers, would be rounded in the way that agrees with the context of the problem.

7. Earnings & Gender Education Attainment
Average Annual Earnings of Males ($ in thousands)
Average Annual Earnings of Females
($ in thousands)
Less than ninth grade
21.659
17.659
Some high school
26.277
19.162
High School Graduate
35.725
26.029
Some College
41.875
30.816
Associate’s degree
44.404
33.481
Bachelor’s degree
57.220
41.681
Master’s degree
71.530
51.316
Doctorate degree
82.401
68.875
Professional degree
75.036

8. Let x represent the earnings for the males, let y represent earnings for females, and create a scatterplot on your graphing calculator of the data.
Decide if there is a linear correlation. If so, create a linear model that expresses the female earnings (y) as a function of male (x) annual earnings.
Graph the linear function and the data points on the graphing calculator.

9. According to this model, what amount would a female make if the average male with the same education made $90, 568?
According to this model, what amount would a male make if the average female with the same education made $64,910?

10. Population (millions)
U.S. Population
The total U.S. population for selected years beginning in 1960 and projected to 2050 is shown in the table below, with the population given in millions.
Year
Population (millions)
1960
1995
1965
1998
1970
2000
1975
2003
1980
2025
1985
2050
1990

11. Population (millions)
Align the data to represent the number of years after 1960, and draw a scatter plot of the data.
Create the linear equation that is the best fit for these data, where y is in millions and x is the number of years after 1960.
Graph the equation of the linear model on the same graph with the scatter plot and discuss how well the model fits the data.
Use the model to estimate the population in 2015 and in 2025.
Year
Population (millions)
1960
1995
1965
1998
1970
2000
1975
2003
1980
2025
1985
2050
1990

12. Applying Models
Interpolation – Using a model to find an output for an input between two given data points. Extrapolation – A model is evaluated for prediction using input(s) outside the given data points.

13. Population (millions)
If we wanted use the model to calculate any population between , that would be interpolation. Otherwise, all years outside the data would be extrapolation.
Year
Population (millions)
1960
1995
1965
1998
1970
2000
1975
2003
1980
2025
1985
2050
1990