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

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.

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?

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

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.

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.

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 100.000 75.036

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.

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?

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 180.671 1995 263.044 1965 194.303 1998 270.561 1970 205.052 2000 281.422 1975 215.973 2003 294.043 1980 227.726 2025 358.030 1985 238.466 2050 408.695 1990 249.948

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 180.671 1995 263.044 1965 194.303 1998 270.561 1970 205.052 2000 281.422 1975 215.973 2003 294.043 1980 227.726 2025 358.030 1985 238.466 2050 408.695 1990 249.948

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.

Population (millions) If we wanted use the model to calculate any population between 1960-1950, that would be interpolation. Otherwise, all years outside the data would be extrapolation. Year Population (millions) 1960 180.671 1995 263.044 1965 194.303 1998 270.561 1970 205.052 2000 281.422 1975 215.973 2003 294.043 1980 227.726 2025 358.030 1985 238.466 2050 408.695 1990 249.948