 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 1.4 Equations of Lines and Modeling

Slope-Intercept Equation Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley A line has slope and y-intercept (0, 16). Find an equation of the line.

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley A line has slope and contains the point (–3, 6). Find an equation of the line. Using the point (  3, 6), we substitute –3 for x and 6 for y, then solve for b.

Point-Slope Equation Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley The point-slope equation of the line with slope m passing through (x 1, y 1 ) is y  y 1 = m(x  x 1 ).

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Find the equation of the line containing the points (2, 3) and (1,  4).

Parallel Lines Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Vertical lines are parallel. Non-vertical lines are parallel if and only if they have the same slope and different y-intercepts.

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 or m 2 = -1/m 1 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Perpendicular Lines Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. a) y + 2 = 5x, 5y + x =  15

Example (continued) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. b) 2y + 4x = 8, 5 + 2x = –y

Example (continued) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Determine whether each of the following pairs of lines is parallel, perpendicular, or neither. c) 2x +1 = y, y + 3x = 4

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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).

Mathematical Modeling Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.

Curve Fitting Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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.

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 below on the U.S. Gross Domestic Product with a linear function. Then estimate the GDP in Find the line of best fit for the scatterplot.

Modeling Linear Regression Credit-Card Debt. Model the data given in the table below with a linear function, and estimate the average credit-card debt per U.S. household in 2005 and in Use 1990 as the year the beginning of the model. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley YearCC Debt 1992 $3, $6, $8, $9, $10, f(x) = x