INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences  2011 Pearson Education, Inc. Chapter 3 Lines, Parabolas, and Systems

 2011 Pearson Education, Inc. To develop the notion of slope and different forms of equations of lines. To develop the notion of demand and supply curves and to introduce linear functions. Chapter 3: Lines, Parabolas and Systems Chapter Objectives

 2011 Pearson Education, Inc. Lines Applications and Linear Functions 3.1) 3.2) Chapter 3: Lines, Parabolas and Systems Chapter Outline

 2011 Pearson Education, Inc. Slope of a Line The slope of the line is for two different points (x 1, y 1 ) and (x 2, y 2 ) is Chapter 3: Lines, Parabolas and Systems 3.1 Lines

 2011 Pearson Education, Inc. The line in the figure shows the relationship between the price p of a widget (in dollars) and the quantity q of widgets (in thousands) that consumers will buy at that price. Find and interpret the slope. Chapter 3: Lines, Parabolas and Systems 3.1 Lines Example 1 – Price-Quantity Relationship

 2011 Pearson Education, Inc. Solution: The slope is Chapter 3: Lines, Parabolas and Systems 3.1 Lines Example 1 – Price-Quantity Relationship Equations of lines A point-slope form of an equation of the line through (x 1, y 1 ) with slope m is

 2011 Pearson Education, Inc. Find an equation of the line passing through (−3, 8) and (4, −2). Solution: The line has slope Using a point-slope form with (−3, 8) gives Chapter 3: Lines, Parabolas and Systems 3.1 Lines Example 3 – Determining a Line from Two Points

 2011 Pearson Education, Inc. Chapter 3: Lines, Parabolas and Systems 3.1 Lines Example 5 – Find the Slope and y-intercept of a Line The slope-intercept form of an equation of the line with slope m and y-intercept b is. Find the slope and y-intercept of the line with equation y = 5(3  2x). Solution: Rewrite the equation as The slope is −10 and the y-intercept is 15.

 2011 Pearson Education, Inc. a.Find a general linear form of the line whose slope-intercept form is Solution: By clearing the fractions, we have Chapter 3: Lines, Parabolas and Systems 3.1 Lines Example 7 – Converting Forms of Equations of Lines

 2011 Pearson Education, Inc. b. Find the slope-intercept form of the line having a general linear form Solution: We solve the given equation for y, Chapter 3: Lines, Parabolas and Systems 3.1 Lines Example 7 – Converting Forms of Equations of Lines

 2011 Pearson Education, Inc. Chapter 3: Lines, Parabolas and Systems 3.1 Lines Parallel and Perpendicular Lines Parallel Lines are two lines that have the same slope. Perpendicular Lines are two lines with slopes m 1 and m 2 perpendicular to each other only if

 2011 Pearson Education, Inc. Chapter 3: Lines, Parabolas and Systems 3.1 Lines Example 9 – Parallel and Perpendicular Lines The figure shows two lines passing through (3, −2). One is parallel to the line y = 3x + 1, and the other is perpendicular to it. Find the equations of these lines.

 2011 Pearson Education, Inc. Chapter 3: Lines, Parabolas and Systems 3.1 Lines Example 9 – Parallel and Perpendicular Lines Solution: The line through (3, −2) that is parallel to y = 3x + 1 also has slope 3. For the line perpendicular to y = 3x + 1,

 2011 Pearson Education, Inc. Problem 3.1 Find the slope of the straight line that passes through the given points 1.(-2,10), (15,3) 2.(2,-4),(3,-4) 3.(0,-4), (3,6) 4.(1,-7), (9,0)

 2011 Pearson Education, Inc. Chapter 3: Lines, Parabolas and Systems 3.2 Applications and Linear Functions Example 1 – Production Levels Suppose that a manufacturer uses 100 lb of material to produce products A and B, which require 4 lb and 2 lb of material per unit, respectively. Solution: If x and y denote the number of units produced of A and B, respectively, Solving for y gives

 2011 Pearson Education, Inc. Chapter 3: Lines, Parabolas and Systems 3.2 Applications and Linear Functions Demand and Supply Curves Demand and supply curves have the following trends:

 2011 Pearson Education, Inc. Chapter 3: Lines, Parabolas and Systems 3.2 Applications and Linear Functions Example 3 – Graphing Linear Functions Linear Functions A function f is a linear function which can be written as Graph and. Solution:

 2011 Pearson Education, Inc. Chapter 3: Lines, Parabolas and Systems 3.2 Applications and Linear Functions Example 5 – Determining a Linear Function If y = f(x) is a linear function such that f(−2) = 6 and f(1) = −3, find f(x). Solution: The slope is. Using a point-slope form:

 2011 Pearson Education, Inc. Problem 3.2