Chapter 1: Linear Functions, Equations, and Inequalities 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Linear Models 1.5 Linear Equations and Inequalities 1.6 Applications of Linear Functions 1.1 Real Numbers and the Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Inequalities 1.5 Linear Equations and Inequalities 1.6 Applications of Linear Functions

1.4 Equations of Lines and Linear Models Point-Slope Form The line with slope m passing through the point (x1, y1) has equation

1.4 Examples Using Point-Slope Example 1 Using the Point-Slope Form Find the slope-intercept form of the line passing through the points shown. (1, 7) and (3, 3) Solution

1.4 Examples Using Point-Slope Example 2 Using the Point-Slope Form The table below shows a list of points found on the line Find the equation of the line. Solution

1.4 Standard Form of the Equation of a Line A linear equation written in the form where A, B, and C are real numbers, is said to be in standard form.

1.4 Equations of Lines in Ax + By = C Form Graph Analytic Solution x-intercept: (2,0) y-intercept: (0,3) Graphing Calculator Solution

1.4 Parallel Lines Parallel Lines Two distinct nonvertical lines are parallel if and only if they have the same slope.

1.4 Parallel Lines Example Solution Find the equation of the line that passes through the point (3,5) and is parallel to the line with equation Graph both lines in the standard viewing window. Solution Solve for y in terms of x.

1.4 Parallel Lines 10 -10 10 -10

1.4 Perpendicular Lines Perpendicular Lines Two lines, neither of which is vertical, are perpendicular if and only if their slopes have product –1.

1.4 Perpendicular Lines Example Find the equation of the line that passes through the point (3,5) and is perpendicular to the line with equation Graph both lines in the standard viewing window. Use slope from the previous example. The slope of a perpendicular line is

1.4 Perpendicular Lines 10 -15 15 -10

1.4 Modeling Medicare Costs Linear Models and Regression Discrete data points can be plotted and the graph is called a scatter diagram. Useful when analyzing trends in data. e.g. Estimates for Medicare costs (in billions) x (Year) y (Cost) 2002 264 2003 281 2004 299 2005 318 2006 336 2007 354

1.4 Modeling Medicare Costs Scatter diagram where x = 0 corresponds to 2002, x = 1 to 2003, etc. Data points (0, 264), (1, 281), (2, 299), (3, 318), (4, 336) and (5, 354) b) Linear model – pick 2 points, (0, 264) and (3, 318) c) Predict cost of Medicare in 2010.

1.4 The Least-Squares Regression Line Enter data into lists L1 (x list) and L2 (y list) Least-squares regression line: LinReg in STAT/CALC menu

1.4 Correlation Coefficient Correlation Coefficient r Determines if a linear model is appropriate range of r: r near +1, low x-values correspond to low y-values and high x-values correspond to high y-values. r near –1, low x-values correspond to high y-values and high x-values correspond to low y-values. means there is little or no correlation. To calculate r using the TI-83, turn Diagnostic On in the Catalog menu.

1.4 Application of Least-Squares Regression Example Predicting the Number of Airline Passengers Airline Passengers (millions) Airport 1992 2005 Harrisburg International 0.7 1.4 Dayton International 1.1 2.4 Austin Robert Mueller 2.2 4.7 Milwaukee General Mitchell 4.4 Sacramento Metropolitan 2.6 5.0 Fort Lauderdale-Hollywood 4.1 8.1 Washington Dulles 5.3 10.9 Greater Cincinnati 5.8 12.3

1.4 Application of Least-Squares Regression Scatter Diagram Linear Regression: Prediction for 2005 at Raleigh-Durham International using this model: FAA’s prediction: 10.3 million 14 6 14 6