


Relations and rectangular coordinate system
RELATIONS AND THE RECTANGULAR COORDINATE SYSTEM Many things in daily life are related. For example, a student’s grade in a course usually is related to the amount of time spent studying, while the number of miles per gallon of gas used on a car trip depends on the speed of the car. Driving at 55 mph might give 31 miles per gallon, while driving at 65 mph might reduce the gas mileage to 28 miles per gallon. Pairs of related numbers, such as 55 and 31 or 65 and 28 in the gas mileage illustration, can be written as ordered pairs. An ordered pair of numbers consists of two numbers, written inside parentheses, in which the sequence of the numbers is important. For example, (4,2) and (2,4) are different ordered pairs because the order of the numbers is different. Notation such as (3,4) has already been used in this book to show an interval on the number line. Now the same notation is used to indicate an ordered pair of numbers. In virtually every case, the intended use will be clear from the context of the discussion. RELATIONS A set of ordered pairs is called a relation. The domain of a relation is the set of first elements in the ordered pairs, and the range of the relation is the set of all possible second elements. In the driving example above, the domain is the set of all possible speeds and the range is the set of resulting miles per gallon. In this text, we confine domains and ranges to real number values. Ordered pairs are used to express the solutions of equations in two variables. For example, we say that (1,2) is a solution of 2 xy=0, since substituting 1 for x and 2 for y in the equation gives 2 (1)2=0 a true statement. When an ordered pair represents the solution of an equation with the variables x and y, the yvalue is written first. Although any set of ordered pairs is a relation, in mathematics we are most interested in those relations that are solution sets of equations. We may say that an equation defines a relation, or that it is the equation of the relation. For simplicity, we often refer to equations such as y=3 x+5or x^{2}+y^{2}=16 Example 1 FINDING ORDERED PAIRS, DOMAINS, AND RANGES For each relation defined below, give three ordered pairs that belong to the relation, and state the domain and the range of the relation. Three ordered pairs from the relation are any three of the five ordered pairs in the set. The domain is the set of first elements, (b) y=4 x1 To find an ordered pair of the relation, choose any number for x or y and substitute in the equation to get the corresponding value of the other variable. For ex y=4 (− 2)1=− 9, giving the ordered pair(− 2,− 9). If y=3, then 3=4 x1 4=4 x 1=x, (c) x=√y1 Verify that the ordered pairs (1,2), (0,1), and (2,5) belong to the relation. Since x equals the principal square root of y1, the domain is restricted to (0,∞). Also, only nonnegative numbers have a real Square root, so the range is determined by the inequality y1≥0 y≥1, giving (1,∞) as the range. THE RECTANGULAR COORDINATE SYSTEM Since the study of relations often involves looking at their graphs, this section includes a brief review of the coordinate plane. As mentioned in Chapter 1, each real number corresponds to a point on a number line. This correspondence is set up by establishing a coordinate system for the line. This idea is extended to the two dimensions of a plane by drawing two perpendicular lines, one horizontal and one vertical. These lines intersect at a point o called the origin. The horizontal line is called the xaxis, and the vertical line is called the yaxis. Starting at the origin, the xaxis can be made into a number line by placing positive numbers to the right and negative numbers to the left. The yaxis can be made into a number line with positive numbers going up and negative numbers going down. The xaxis and yaxis 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 xyplane. The xaxis and yaxis divide the plane into four regions, or is quadrants, labeled as shown in Figure 3.1. The points on the xaxis and yaxis belong to no quadrant.
Each point P in the xyplane corresponds to a unique ordered pair (a,b) of real numbers. The numbers a and b are the coordinates of point P. To locate on the xyplane the point corresponding to the ordered pair(3,4), for example, draw a vertical line through 3 on the xaxis and a horizontal line through 4 on the yaxis. These two lines cross at point A in Figure 3.2. Point A corresponds to the ordered pair (3,4). Also in Figure 3.2, B corresponds to the ordered pair (− 5,6),C to (− 2,− 4),D to (4,− 3), and E to (− 3,0). The point Pcorresponding to the ordered pair (a,b) often is written as P (a,b) as in Figure 3.1 and referred to as “the point (a,b).”
As we shall see later in this chapter the graph of a relation is the set of points in the plane that corresponds to the ordered pairs of the relation. Two formulas, the distance and midpoint formulas, will be useful in our study of relations in this chapter. THE DISTANCE FORMULA By using the Pythagorean theorem, we can develop a formula to find the distance between any two points in a plane. For example, Figure 3.3 shows the points P (− 4,3) and R (8,− 2).
To find the distance between these two points, complete a right triangle as shown in the figure. This right triangle has its 90° angle at (8,3). The horizontal side of the triangle has length 8(− 4)=12, where absolute value is used to make sure that the distance is not negative. The vertical side of the triangle has length To obtain a general formula for the distance between two points on a coordinate plane, let P (x_{1},y_{1}) and R (x_{2},y_{2}) be any two distinct points in a plane, as shown in Figure 3.4. Complete a triangle by locating point Q with coordinates (x_{2},y_{1}). Using the Pythagorean theorem gives the distance between P and R, written d (P,R), as d (P,R)=√(x_{2}x_{1})^{2}+(y_{2}y_{1})^{2}
NOTE The use of absolute value bars is not necessary in this formula, since for all real numbers a and b, ab^{2}=(ab)^{2}. DISTANCE FORMULA Suppose that P (x_{1},y_{1}) and R (x_{2},y_{2}) are two points in a coordinate plane. Then the distance between P and R, written d (P,R), is given by the distance formula d (P,R)=√(x_{2}x_{1})^{2}+(y_{2}y_{1})^{2} Although the proof of the distance formula assumes that P and R are not on a horizontal or vertical line, the result is true for any two points. Example 2 USING THE DISTANCE FORMULA Find the distance between P (− 8,4) and Q (3,− 2). =√11^{2}+(− 6)^{2} =√121+36=√157 Example 3 DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE Are the three points M (− 2,5), N (12,3), and , M (10,− 11) the vertices of a right triangle? By these results, [d (M,Q)]^{2}=[d (M,N)]^{2}+[d (N,Q)]^{2} since 20^{2}=√200+√(200)^{2}, or 400=400, is a true Statement. This proves that the triangle is a right triangle with hypotenuse connecting M and Q. Example 4 DETERMINING WHETHER THREE POINTS ARE COLLINEAR Are the points (− 1,5), (2,− 4), and (4,− 10) collinear? The distance between (− 1,5) and (2,− 4) is √(− 12)^{2}+[5(− 4)]^{2}=√9+81=√90=3 √10The distance between (2,− 4) and (4,− 10) is √(24)^{2}+[− 4(− 10)]^{2}=√4+36=√40=2 √10Finally, the distance between the remaining pair of points, (− 1,5) and (4,− 10)is √(− 14)^{2}+[5(− 10)]^{2}=√25+225=√250=5 √10Because 3 √10+2 √10=5 √10, the three points are collinear. THE MIDPOINT FORMULA The midpoint formula is used to find the coordinates of the midpoint of a line segment. (Recall that the midpoint of a line segment is equidistant from the endpoints of the segment.) To develop the midpoint formula, let (x_{1},y_{1}) and (x_{2},y_{2}) be any two distinct points in a plane. (Although Figure 3.6 shows x_{1}<y_{2}, no particular order is required.) Assume that the two points are not on a horizontal or vertical line. Let (x,y) be the midpoint of the segment connecting (x_{1},y_{1}) and (x_{2},y_{2}). Draw vertical lines from each of the three points to the xaxis, as shown in Figure 3.6. x_{2}x=xx_{1} x_{2}+x_{1}=2 x x=x_{1}+x_{2}2
By this result, the xcoordinate of the midpoint is the average of the coordinates of the endpoints of the segment. In a similar manner, the ycoordinate of the midpoint is y_{1}+y_{2}2, proving the following statement. MIDPOINT FORMULA The midpoint of the line segment with endpoints (x_{1},y_{1})) and (x_{2},y_{2}) is In other words, the midpoint formula says that the coordinates of the midpoint of a segment are found by calculating the average of the xcoordinates and the average of the ycoordinates of the endpoints of the segment. In Exercise 43, you are asked to verify that the coordinates above satisfy the definition of midpoint. Example 5 : USING THE MIDPOINT FORMULA Find the midpoint M of the segment with endpoints(8,− 4) and (− 9,6).
Example 6 USING THE MIDPOINT FORMULA A line segment has an endpoint at (2,− 8)and midpoint at (− 1,− 3)Find the other endpoint of the segment. The formula for the xcoordinate of the midpoint is x_{1}+x_{2}2. Here the xcoordinate of the midpoint is − 1. Letting x_{1}=2 gives − 1=2+x_{2}2 − 2=2+x_{2} − 4=x_{2}
