2. Copyright © 2009 Pearson Education, Inc. CHAPTER 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling 1.5 Linear Equations, Functions, Zeros and Applications 1.6 Solving Linear Inequalities

3. Copyright © 2009 Pearson Education, Inc. 1.4 Equations of Lines and Modeling  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.

4. Slide 1.4 - 4 Copyright © 2009 Pearson Education, Inc. 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.

5. Slide 1.4 - 5 Copyright © 2009 Pearson Education, Inc. Example A line has slope and y-intercept (0, 16). Find an equation of the line. Solution: We use the slope-intercept equation and substitute for m and 16 for b:

6. Slide 1.4 - 6 Copyright © 2009 Pearson Education, Inc. Example A line has slope and contains the point (–3, 6). Find an equation of the line. Solution: 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

7. Slide 1.4 - 7 Copyright © 2009 Pearson Education, Inc. 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 ).

8. Slide 1.4 - 8 Copyright © 2009 Pearson Education, Inc. Example Find the equation of the line containing the points (2, 3) and (1,  4). Solution: First determine the slope Using the point-slope equation, substitute 7 for m and either of the points for (x 1, y 1 ):

9. Slide 1.4 - 9 Copyright © 2009 Pearson Education, Inc. Parallel Lines Vertical lines are parallel. Nonvertical lines are parallel if and only if they have the same slope and different y-intercepts.

10. Slide 1.4 - 10 Copyright © 2009 Pearson Education, Inc. 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.

11. Slide 1.4 - 11 Copyright © 2009 Pearson Education, Inc. Perpendicular Lines Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b).

12. Slide 1.4 - 12 Copyright © 2009 Pearson Education, Inc. 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.

13. Slide 1.4 - 13 Copyright © 2009 Pearson Education, Inc. 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.

14. Slide 1.4 - 14 Copyright © 2009 Pearson Education, Inc. 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.

15. Slide 1.4 - 15 Copyright © 2009 Pearson Education, Inc. 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). Solution: Solve the equation for y: So the slope of this line is

16. Slide 1.4 - 16 Copyright © 2009 Pearson Education, Inc. Example (continued) (a)The line parallel to the given line will have the same slope,. We use either the slope-intercept or point- slope equation for the line. Substitute for m and use the point (2,  3) and solve the equation for y.

17. Slide 1.4 - 17 Copyright © 2009 Pearson Education, Inc. Example (continued) (b)The slope of the perpendicular line is the negative reciprocal of, or – 4. Use the point- slope equation, substitute – 4 for m and use the point (2, –3) and solve the equation.

18. Slide 1.4 - 18 Copyright © 2009 Pearson Education, Inc. Mathematical Modeling (optional) 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.

19. Slide 1.4 - 19 Copyright © 2009 Pearson Education, Inc. Curve Fitting b (optional) 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.

20. Slide 1.4 - 20 Copyright © 2009 Pearson Education, Inc. Example (optional) Model the data in the table on the number of U.S. households with cable television with a linear function. Then predict the number of cable subscribers in 2010. Scatterplot

21. Slide 1.4 - 21 Copyright © 2009 Pearson Education, Inc. Example (continued) (optional) Choose any two data points to determine the equation. We’ll use (2, 81.5) and (6, 94.0). First, find the slope. Substitute 3.125 for m and use either point, we’ll use (6, 94.0) in the point-slope equation. x is number of years after 1999, y is in millions. Here’s the graph of this line.

22. Slide 1.4 - 22 Copyright © 2009 Pearson Education, Inc. Example (continued) (optional) Now we can estimate the number of cable subscribers in 2010 by substituting 11 for x in the model (2010 – 1999 = 11). We predict there will be about 109.6 million U.S. households with cable television in 2010.

23. Slide 1.4 - 23 Copyright © 2009 Pearson Education, Inc. Linear Regression (optional) 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.

24. Slide 1.4 - 24 Copyright © 2009 Pearson Education, Inc. Example (optional) Fit a regression line to the data given in the table. Use the function to predict the number of cable subscribers in 2010. Solution: 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.

25. Slide 1.4 - 25 Copyright © 2009 Pearson Education, Inc. Example (continued) (optional) Here are screen shots of the calculator showing Lists 1 and 2 and the scatterplot of the data.

26. Slide 1.4 - 26 Copyright © 2009 Pearson Education, Inc. Example (continued) (optional) Here are screen shots of selecting the LINEAR REGRESSION feature from the STAT CALC menu. From the screen on the right, we find the linear equation that best models the data is

27. Slide 1.4 - 27 Copyright © 2009 Pearson Education, Inc. Example (continued) (optional) The calculator can also graph the regression line on the same graph as the scatterplot. Substitute 11 into the regression equation. It predicts 108.2 million cable subscribers in 2010.