“ �������������������������������������������������������������������������������������������������������������������������������������������������������������� ���������������� ������������������������ ������������ ������������ �������� ���� �������������������� �������� ������ ������������ �������� �������������� ��������������������������������������������������������”�������������������������������������������������������������������������������������������������������������������������� SECTION 3.1 The Coordinate System, Graphing Equations, and the Midpoint Formula In Chapter 2, we explored linear equations and inequalities in one variable. In Chapter 4, we will examine linear equations and inequalities in two variables. This chapter will provide a foundation for studying the equations in Chapter 4. We will learn how to visualize solutions of equations in two variables. The difference between a relation and a function and some important properties of functions are also presented. This chart shows the number of hybrid electric vehicles (HEVs) sold for the years 1999 through 2010. Each year corresponds to a specific number of cars. This pairing of numbers provides points that we can graph on a coordinate system. Graphing these points enables us to visualize data so that trends and patterns in these data can be observed. In Example 6, we will graph these data and use the graph to observe trends in the sales of HEVs. (Source: http://www.afdc.energy.gov/afdc/data/vehicles.html and http:// www.hybridcars.com) In this section, we will learn how to graph points and solutions of equations and to apply these skills to data from the real world. The Rectangular Coordinate System Each pair of numbers in the table showing the total sales for HEVs can be expressed as an ordered pair. For example, in 2010 the total number of HEVs sold was 274,763. So, we say the year 2010 corresponds to sales of 274,763, which we express as (2010, 274,763). This pairing of numbers is called an ordered pair and is denoted by (x, y). An important skill involving ordered pairs is the ability to plot them so we can visualize them. To plot ordered pairs, we use a rectangular coordinate system or the Cartesian coordinate system. René Descartes, a seventeenth-century French mathematician, is credited with inventing the Cartesian coordinate system. A rectangular coordinate system consists of two real number lines intersecting at right angles. The horizontal number line is usually referred to as the x-axis and the vertical number line is usually referred to as the y-axis. The point where the two number lines intersect is called the origin. Every point on the coordinate system is associated with an ordered pair, (x, y). • The value x is called the first coordinate or x-coordinate. The x-coordinate tells us how far left (if x is negative) or right (if x is positive) to move from the origin. • The value y is called the second coordinate or y-coordinate. The y-coordinate tells us how far up (if y is positive) or down (if y is negative) to move from the x-axis. For example, in the ordered pair (4, 5), 4 is the x-coordinate, which tells us to move 4 units right from the origin. The number 5 is the y-coordinate, which tells us to move 5 units up from the x-axis. ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Plot points and identify their location on a rectangular coordinate system. 2. Determine algebraically if an ordered pair is a solution of an equation. 3. Graph an equation in two variables. 4. Determine graphically if an ordered pair is a solution of an equation. 5. Determine the midpoint of a line segment. 6. Solve application problems. 7. Troubleshoot common errors. Year Sales 1999 17 2000 9,350 2001 20,282 2002 36,035 2003 47,600 2004 84,199 2005 209,711 2006 252,636 2007 352,274 2008 312,386 2009 290,271 2010 274,763 Objective 1 ▶ Plot points and identify their location on a rectangular coordinate system. x y 2 2 4 4 –2 –4 –2 –4 5 t t 4 t x y 2 2 4 4 –2 –4 –2 –4 (4 5) 148
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