P1. Plot the following points on the real. P2. Determine which of the following are solutions

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1 Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in the Rectangular Coordinate Sstem Determine Whether an Ordered Pair Is a Point on the Graph of an Equation Graph an Equation Using the Point-Plotting Method Identif the Intercepts from the Graph of an Equation 5 Interpret Graphs Preparing for Rectangular Coordinates and Graphs of Equations Before getting started, take this readiness quiz. If ou get a problem wrong, go back to the section cited and review the material. P1. Plot the following points on the real number line: -,, 0, 1. [Section R., pp. 15 1] P. Determine which of the following are solutions to the equation =. (a) = 0 = - (c) = -7 [Section 1.1, pp. 8 9] P. Evaluate the epression for the given values of the variable. (a) = 0 = (c) = - [Section R.5, pp. 0 1] P. Solve the equation + = 8 for. [Section 1., pp. 7 7] P5. Evaluate ƒ -ƒ. [Section R., pp. 19 0] 1 Plot Points in the Rectangular Coordinate Sstem Recall from Section R. that we locate a point on the real number line b assigning it a single real number, called the coordinate of the point. See Preparing for Problem P1. When we graph a point on the real number line, we are working in one dimension. When we wish to work in two dimensions, we locate a point using two real numbers. We begin b drawing two real number lines that intersect at right (90 ) angles. One of the real number lines is drawn horizontal, while the other is drawn vertical. We call the horizontal real number line the -ais, and the vertical real number line is the -ais. The point where the -ais and -ais intersect is called the origin, O. See Figure. Figure 1 1 O 1 1 Preparing for...answers P _ 1 5 P. (a) No No (c) Yes P. (a) 1 (c) 8 P. =- P5. + The origin O has a value of 0 on the -ais and the -ais. Points on the -ais to the right of O are positive real numbers; points on the -ais to the left of O are negative real numbers. Points on the -ais that are above O are positive real numbers; points on the -ais that are below O are negative real numbers. In Figure we label the -ais and the -ais. Notice that an arrow is used at the end of each ais to denote the positive direction. We do not use an arrow to denote the negative direction. The coordinate sstem presented in Figure is called a rectangular or Cartesian coordinate sstem, named after René Descartes ( ), a French mathematician, philosopher, and theologian. The plane formed b the -ais and -ais is often referred to as the -plane, and the -ais and -ais are called the coordinate aes. We can represent an point P in the rectangular coordinate sstem b using an ordered pair (, ) of real numbers. If 7 0, we travel units to the right of the -ais;

2 9 CHAPTER 1 Linear Equations and Inequalities if 0, we travel ƒƒ units to the left of the -ais. If 7 0, we travel units above the -ais; if 0, we travel ƒƒ units below the -ais. The ordered pair (, ) is also called the coordinates of P. The origin O has coordinates. An point on the -ais has coordinates of the form (, 0), and an point on the -ais has coordinates of the form (0, ). If (, ) are the coordinates of a point P, then is called the -coordinate, or abscissa, of P and is called the -coordinate or ordinate, of P. If ou look back at Figure, ou should notice that the - and -aes divide the plane into four separate regions or quadrants. In quadrant I, both the - and -coordinate are positive; in quadrant II, is negative and is positive; in quadrant III, both and are negative; and in quadrant IV, is positive and is negative. Points on the coordinate aes do not belong to a quadrant. See Figure. Figure Quadrant II 0, 0 1 Quadrant I 0, 0 1 O 1 1 Quadrant III 0, 0 Quadrant IV 0, 0 EXAMPLE 1 Plotting Points in the Rectangular Coordinate Sstem and Determining the Quadrant in which the Point Lies Plot the points in the -plane. Tell which quadrant each point is in. (a) A1, B1-, (c) C1-1, - (d) D1, - (e) E1-, 0 Before we plot the points, we draw a rectangular or Cartesian coordinate sstem. See Figure 5(a). We now plot the points. (a) To plot point A1,, from the origin O, we travel units to the right and then units up. Label the point A. See Figure 5. Point A is in quadrant I because both and are positive. Figure 5 B (, ) 1 E (, 0) 1 A (, ) 1 O O 1 1 C ( 1, ) D (, ) (a)

3 Section 1.5 Rectangular Coordinates and Graphs of Equations 95 To plot point B1-,, from the origin O, we travel units to the left and then units up. Label the point B. See Figure 5. Point B is in quadrant II. (c) See Figure 5. Point C is in quadrant III. (d) See Figure 5. Point D is in quadrant IV. (e) See Figure 5. Point E is not in a quadrant because it lies on the -ais. Quick 1. The point where the -ais and -ais intersect in the Cartesian coordinate sstem is called the.. True or False: If a point lies in quadrant III of the Cartesian coordinate sstem, then both and are negative. In Problems and, plot each point in the -plane. Tell in which quadrant or on which coordinate ais each point lies.. (a) (c) A15, B1, - C10, -. (a) (c) A1-, B1-, 0 C1, - (d) D1-, - (d) D1, 1 Determine Whether an Ordered Pair Is a Point on the Graph of an Equation In Section 1.1, we solved linear equations in one variable. The solution was either a single value of the variable, the empt set, or all real numbers. We will now look at equations in two variables. Our goal is to learn a method for representing the solution to an equation in two variables. DEFINITION An equation in two variables, sa and, is a statement in which the algebraic epressions involving and are equal.the epressions are called sides of the equation. Since an equation is a statement, it ma be true or false, depending upon the values of the variables. An values of the variable that make the equation a true statement are said to satisf the equation. For eample, the following are all equations in two variables. = + + = = The first equation = + is satisfied when = and = 7 since = 7 +. It is also satisfied when = - and =. In fact, there are infinitel man choices of and that satisf the equation = +. However, there are some choices of and that do not satisf the equation = +. For eample, = and = does not satisf the equation because Z + (that is, 9 Z ). When we find a value of and that satisfies an equation, it means that the ordered pair (, ) represents a point on the graph of the equation. In Words The graph of an equation is a geometric wa of representing the set of all points that make the equation a true statement. DEFINITION The graph of an equation in two variables and is the set of all points whose coordinates, (, ), in the -plane satisf the equation.

4 9 CHAPTER 1 Linear Equations and Inequalities EXAMPLE Determining Whether a Point Is on the Graph of an Equation Determine if the following coordinates represent points that are on the graph of - =. (a) (, 0) 11, - (c) a 1, - 9 b (a) For (, 0), we check to see if =, = 0 satisfies the equation - =. Let =, = 0: 1-0 = True The statement is true when = and = 0, so the point whose coordinates are (, 0) is on the graph. For 11, -, we have Let = 1, = -: False The statement 5 = is false, so the point whose coordinates are 11, - is not on the graph. (c) For a 1 we have, - 9 b, Let = 1, =-9 : The statement is true when = 1 and =- 9 so the point whose coordinates, are a 1 is on the graph., - 9 b - = - = = - = a 1 b - a - 9 b = True Quick 5. True or False: The graph of an equation in two variables and is the set of all points whose coordinates, (, ), in the Cartesian plane satisf the equation.. Determine if the following coordinates represent points that are on the graph of - = 1. (a) 1, - 1, - (c) a, - 9 b 7. Determine if the following coordinates represent points that are on the graph of = +. (a) (1, ) 1-, -1 (c) 1-, 1 For the remainder of the course we will sa the point (, ) rather than the point whose coordinates are (, ) for the sake of brevit.

5 Section 1.5 Rectangular Coordinates and Graphs of Equations 97 Graph an Equation Using the Point-Plotting Method One of the most elementar methods for graphing an equation is the point-plotting method. With this method, we choose values for one of the variables and use the equation to determine the corresponding values of the remaining variable. If and are the variables in the equation, it does not matter whether we choose values of and use the equation to find the corresponding or choose and find. Convenience will determine which wa we go. EXAMPLE How to Graph an Equation b Plotting Points Graph the equation = - + b plotting points. Step-b-Step Step 1: We want to find all points (, ) that satisf the equation. To determine these points we choose values of (do ou see wh?) and use the equation to determine the corresponding values of. See Table 9. Table 9 = - + (, ) (-) + = 10 -(-) + = 8 -(-1) + = (-, 10) (-, 8) (-1, ) 0 -(0) + = (0, ) 1 -(1) + = (1, ) -() + = 0 (, 0) -() + = - (, -) Step : We plot the ordered pairs listed in the third column of Table 9 as shown in Figure (a). Now connect the points to obtain the graph of the equation (a line) as shown in Figure. Figure (, 10) (, 8) 8 ( 1, ) (0, ) (1, ) (, 0) (, ) (a) The graph of the equation shown in Figure does not show all the points that satisf = - +. For eample, in Figure the point 18, -1 is part of the graph of = - +, but it is not shown. Since the graph of = - + can be etended as far as we please, we use arrows on the ends of the graph to indicate that the pattern shown continues. It is important to show enough of the graph so that anone who is looking at it will see the rest of it as an obvious continuation of what is there. This is called a complete graph.

6 98 CHAPTER 1 Linear Equations and Inequalities EXAMPLE Graphing an Equation b Plotting Points Graph the equation = b plotting points. Table 10 shows several points on the graph. Table 10 = (, ) = (-) = 1 = (-) = 9 = (-) = = (-1) = 1 (-, 1) (-, 9) (-, ) (-1, 1) 0 = (0) = 0 1 = (1) = 1 (1, 1) = () = (, ) = () = 9 (, 9) = () = 1 (, 1) In Figure 7(a), we plot the ordered pairs listed in Table 10. In Figure 7, we connect the points in a smooth curve. Figure (, 1) 1 (, 1) (, 1) 1 (, 1) (, 9) 1 8 (, 9) (, 9) 1 8 (, 9) (, ) ( 1, 1) (, ) (1, 1) (, ) ( 1, 1) (, ) (1, 1) (a) Work Smart Notice we use a different scale on the - and -ais in Figure 7. Work Smart Eperience will pla a huge role in determining which -values to choose in creating a table of values. For the time being, start b choosing values of around = 0 as in Table 10. Two questions that ou might be asking ourself right now are How do I know how man points are sufficient? and How do I know which -values (or -values) I should choose in order to obtain points on the graph? Often, the tpe of equation we wish to graph indicates the number of points that are necessar. For eample, we will learn in the net section that if the equation is of the form = m + b, then its graph is a line and onl two points are required to obtain the graph (as in Eample ). Other times, more points are required. At this stage in our math career, ou will need to plot quite a few points to obtain a complete graph. However, as our eperience and knowledge grow, ou will learn to be more efficient in obtaining complete graphs. Quick In Problems 8 10, graph each equation using the point-plotting method. 8. = = = +

7 Section 1.5 Rectangular Coordinates and Graphs of Equations 99 EXAMPLE 5 Graphing the Equation Graph the equation = b plotting points. Because the equation is solved for, we will choose values of and use the equation to find the corresponding values of. See Table 11. We plot the ordered pairs listed in Table 11 and connect the points in a smooth curve. See Figure 8. Table 11 (, ) (-) = 9 (-) = (-1) = 1 (9, -) (, -) (1, -1) 0 0 = = 1 (1, 1) = (, ) = 9 (9, ) Figure 8 (1, 1) (, ) (9, ) (1, 1) (, ) (9, ) 8 10 Quick In Problems 11 and 1, graph each equation using the point-plotting method. 11. = + 1. = 1-1 Work Smart In order for a graph to be complete, all of its intercepts must be displaed. Figure 9 touches -intercepts crosses -intercept Identif the Intercepts from the Graph of an Equation One of the ke components that should be displaed in a complete graph is the intercepts of the graph. DEFINITION The intercepts are the coordinates of the points, if an, where a graph crosses or touches the coordinate aes. The -coordinate of a point at which the graph crosses or touches the -ais is an -intercept, and the -coordinate of a point at which the graph crosses or touches the -ais is a -intercept. See Figure 9 for an illustration. Notice that an -intercept eists when = 0 and a -intercept eists when = 0. EXAMPLE Finding Intercepts from a Graph Find the intercepts of the graph shown in Figure 0. What are the -intercepts? What are the -intercepts? Figure 0 5 (0, ) (.8, 0) 5 (, 0) (1, 0) 5 5

8 100 CHAPTER 1 Linear Equations and Inequalities The intercepts of the graph are the points 1-, 0, 10,, 11, 0, and 1.8, 0 The -intercepts are -, 1, and.8. The -intercept is. In Eample, ou should notice the following: If we do not specif the tpe of intercept (- versus -), then we report the intercept as an ordered pair. However, if we specif the tpe of intercept, then we onl need to report the coordinate of the intercept. For -intercepts, we report the -coordinate of the intercept (since it is understood that the -coordinate is 0); for -intercepts, we report the -coordinate of the intercept (since the -coordinate is understood to be 0). Quick 1. The points, if an, at which a graph crosses or touches a coordinate ais are called. 1. True or False:The graph of an equation must have at least one -intercept. 15. Find the intercepts of the graph shown in the figure. What are the -intercepts? What are the -intercepts? ( 5, 0) (1, 0) (.7, 0) (0, 0.9) 5 Interpret Graphs Graphs pla an important role in helping us to visualize relationships that eist between two variables or quantities. We have all heard the epression A picture is worth a thousand words. A graph is a picture that illustrates the relationship between two variables. B visualizing this relationship, we are able to see important information and draw conclusions regarding the relationship between the two variables. EXAMPLE 7 Interpret a Graph The graph in Figure 1 shows the profit P for selling gallons of gasoline in an hour at a gas station. The vertical ais represents the profit and the horizontal ais represents the number of gallons of gasoline sold. Figure 1 P (75, 55) Profit ($) Number of Gallons

9 Section 1.5 Rectangular Coordinates and Graphs of Equations 101 (a) What is the profit if 150 gallons of gasoline are sold? How man gallons of gasoline are sold when profit is highest? What is the highest profit? (c) Identif and interpret the intercepts. (a) Draw a vertical line up from 150 on the horizontal ais until we reach the point on the graph. Then draw a horizontal line from this point to the vertical ais. The point where the horizontal line intersects the vertical ais is the profit when 150 gallons of gasoline are sold. The profit from selling 150 gallons of gasoline is $00. The profit is highest when 75 gallons of gasoline are sold. The highest profit is $55. (c) The intercepts are (0, -00), (100, 0), and (750, 0). For (0, -00): If the price of gasoline is too high, demand for gasoline (in theor) will be 0 gallons. This is the eplanation for selling 0 gallons of gasoline. The negative profit is due to the fact that the compan has $0 in revenue (since it didn t sell an gas), but had hourl epenses of $00. For (100, 0): The compan sells just enough gas to pa its bills. This can be thought of as the break-even point. For (750, 0): The 750 gallons sold represents the maimum number of gallons that the station can pump and still break even. Quick 1. The graph shown represents the cost C (in thousands of dollars) of refining gallons of gasoline per hour (in thousands). The vertical ais represents the cost and the horizontal ais represents the number of gallons of gasoline refined. C 1000 Cost (thousands of $) Number of Gallons (thousands) (a) What is the cost of refining 50 thousand gallons of gasoline per hour? What is the cost of refining 00 thousand gallons of gasoline per hour? (c) In the contet of the problem, eplain the meaning of the graph ending at 700 thousand gallons of gasoline. (d) Identif and interpret the intercept.

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