202 CHAPTER 2 Graphs and Functions Ordered Pairs The idea of pairing one quantity with another is often encountered in everyday life. • A numerical score in a mathematics course is paired with a corresponding letter grade. • The number of gallons of gasoline pumped into a tank is paired with the amount of money needed to purchase it. • Expense categories are paired with dollars spent by the average American household in 2017. (See the table in the margin.) 2.1 Rectangular Coordinates and Graphs ■ Ordered Pairs ■ The Rectangular Coordinate System ■ The Distance Formula ■ The Midpoint Formula ■ Equations inTwo Variables Pairs of related quantities, such as a 96 determining a grade of A, 3 gallons of gasoline costing $10.50, and 2017 spending on food of $7729, can be expressed as ordered pairs: 196, A2, 13, $10.502, 1food, $77292. An ordered pair consists of two components, written inside parentheses. Category Amount Spent food $ 7729 housing $19,884 transportation $ 9576 health care $ 4928 apparel and services $ 1833 entertainment $ 3203 Data from U.S. Bureau of Labor Statistics. In mathematics, we are most often interested in ordered pairs whose components are numbers. The ordered pairs 1a, b2 and 1c, d2 are equal provided that a = c and b = d. EXAMPLE 1 Writing Ordered Pairs Use the table to write ordered pairs to express the relationship between each category and the amount spent on it. (a) housing (b) entertainment SOLUTION (a) Use the data in the second row: (housing, $19,884). (b) Use the data in the last row: (entertainment, $3203). S Now Try Exercise 13. The Rectangular Coordinate System Each real number corresponds to a point on a number line. This idea is extended to ordered pairs of real numbers by using two perpendicular number lines, one horizontal and one vertical, that intersect at their zero-points. The point of intersection is the origin. The horizontal line is the x-axis, and the vertical line is the y-axis. The x-axis and y-axis together make up a rectangular coordinate system, or Cartesian coordinate system (named for one of its coinventors, René Descartes. The other coinventor was Pierre de Fermat). The plane into which the coordinate system is introduced is the coordinate plane, or xy-plane. See Figure 1. The x-axis and y-axis divide the plane into four regions, or quadrants, labeled as shown. The points on the x-axis or the y-axis belong to no quadrant. Each point P in the xy-plane corresponds to a unique ordered pair 1a, b2 of real numbers. The point P corresponding to the ordered pair 1a, b2 often is written P1a, b2 as in Figure 1 and referred to as “the point 1a, b2.” The numbers a and b are the coordinates of point P. 0 b a x-axis y-axis Quadrant II Quadrant I Quadrant III Quadrant IV Origin P(a, b) Figure 1 Rectangular (Cartesian) Coordinate System NOTE Notation such as 12, 42 is used to show an interval on a number line, and the same notation is used to indicate an ordered pair of numbers. The intended use is usually clear from the context of the discussion.
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