1. Section 1.4 Equations of Lines and Modeling Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

2. Objectives  Determine equations of lines.  Given the equations of two lines, determine whether their graphs are parallel or perpendicular.  Model a set of data with a linear function.  Fit a regression line to a set of data; then use the linear model to make predictions.

3. Slope-Intercept Equation Recall the slope-intercept equation y = mx + b or f (x) = mx + b. If we know the slope and the y-intercept of a line, we can find an equation of the line using the slope-intercept equation.

4. Example A line has slope and y-intercept (0, 16). Find an equation of the line. We use the slope-intercept equation and substitute for m and 16 for b:

5. Example A line has slope and contains the point (–3, 6). Find an equation of the line. We use the slope-intercept equation and substitute for m: Using the point (  3, 6), we substitute –3 for x and 6 for y, then solve for b. Equation of the line is

6. Point-Slope Equation The point-slope equation of the line with slope m passing through (x 1, y 1 ) is y  y 1 = m(x  x 1 ).

7. Example Find the equation of the line containing the points (2, 3) and (1,  4). First determine the slope Using the point-slope equation, substitute 7 for m and either of the points for (x 1, y 1 ):

8. Parallel Lines Vertical lines are parallel. Nonvertical lines are parallel if and only if they have the same slope and different y- intercepts.

9. Perpendicular Lines Two lines with slopes m 1 and m 2 are perpendicular if and only if the product of their slopes is  1: m 1 m 2 =  1.

10. Perpendicular Lines Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b).

11. Example Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. a) y + 2 = 5x, 5y + x =  15 Solve each equation for y. The slopes are negative reciprocals. The lines are perpendicular.

12. Example (continued) Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. b) 2y + 4x = 8, 5 + 2x = –y Solve each equation for y: The slopes are the same. The lines are parallel.

13. Example (continued) Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. c) 2x +1 = y, y + 3x = 4 Solve each equation for y. The slopes are not the same and their product is not –1. They are neither parallel nor perpendicular.

14. Example Write equations of the lines (a) parallel and (b) perpendicular to the graph of the line 4y – x = 20 and containing the point (2,  3). Solve the equation for y: So the slope of this line is

15. Example (continued) The line parallel to the given line will have the same slope, 1/4. We use either the slope-intercept or point- slope equation for the line. Substitute 1/4 for m and use the point (2,  3) and solve the equation for y.

16. Example (continued) (b)The slope of the perpendicular line is the negative reciprocal of 1/4, or – 4. Use the point-slope equation, substitute – 4 for m and use the point (2, –3) and solve the equation.

17. Mathematical Modeling When a real-world problem can be described in a mathematical language, we have a mathematical model. The mathematical model gives results that allow one to predict what will happen in that real-world situation. If the predictions are inaccurate or the results of experimentation do not conform to the model, the model must be changed or discarded. Mathematical modeling can be an ongoing process.

18. Curve Fitting In general, we try to find a function that fits, as well as possible, observations (data), theoretical reasoning, and common sense. We call this curve fitting, it is one aspect of mathematical modeling. In this chapter, we will explore linear relationships. Let’s examine some data and related graphs, or scatter plots and determine whether a linear function seems to fit the data.

19. Example The gross domestic product (GDP) of a country is the market value of final goods and services produced. Market value depends on the quantity of goods and services and their price. Model the data in the table with a linear function. Then estimate the GDP in 2014.

20. Example (continued) Choose any two data points to determine the equation. We’ll use (5, 4.2) and (25, 12.6). First, find the slope. Substitute 0.42 for m and use either point, we’ll use (5, 4.2) in the point-slope equation. x is number of years after 1980, y is in trillions of dollars.

21. Example (continued) Now we can estimate the GDP in 2014 by substituting 34 (2014 – 1980) for x in the model We estimate that the GDP will be $16.38 trillion in 2014.

22. Linear Regression Linear regression is a procedure that can be used to model a set of data using a linear function. We use the data on the number of U.S. households with cable television. We can fit a regression line of the form y = mx + b to the data using the LINEAR REGRESSION feature on a graphing calculator.

23. Example Fit a regression line to the data given in the table. Use the function to function to estimate the GDP in 2014. Enter the data in lists on the calculator. The independent variables or x values are entered into List 1, the dependent variables or y values into List 2. The calculator can then create a scatterplot.

24. Example (continued) Here are screen shots of the calculator showing Lists 1 and 2 and the scatterplot of the data.

25. Example (continued) Here are screen shots of selecting the LINEAR REGRESSION feature from the STAT CALC menu. From the screen in the middle, we find the linear equation that best models the data is

26. Example (continued) Substitute 34 into the regression equation. The GDP is estimated to be $15.91 trillion.