Graphs and Functions in the Cartesian Coordinate System

Transcription

1 C H A P T E R Graphs and Functions in the Cartesian Coordinate Sstem List price (in thousands of dollars) (99, 0,) (98,,67) (00, 7,) he first self-propelled automobile to carr passengers was built in T 80 b the British inventor Richard Trevithick. B 9 about 600,000 automobiles were operated in the United States alone. Some were powered b steam and some b electricit, but most were powered b gasoline. In 9, to meet the ever growing demand, Henr Ford increased production b introducing a moving assembl line to carr automobile parts. Toda the United States is a nation of cars. Over million automobiles are produced here annuall, and total car registrations number over million. Henr Ford s earl model T sold for $80. In 99 the average price of a new car was $8,000, with some selling for more than $0,000. Prices for new cars rise ever ear, but, unfortunatel for the buer, the moment a new automobile is bought, its value begins to decrease. Much of the behavior of automobile prices can be modeled b linear equations. In Eercises and of Section. ou will use linear equations to find increasing new car prices and depreciating used car prices Year

2 (-) Chapter Graphs and Functions in the Cartesian Coordinate Sstem In this section Ordered Pairs Plotting Points Graphing a Linear Equation Using Intercepts for Graphing Applications Function Notation. GRAPHING LINES IN THE COORDINATE PLANE In Chapter we graphed numbers on a number line. In Chapter we used number lines to illustrate the solution sets to inequalities. In this section we graph pairs of numbers in a coordinate sstem made from two number lines to illustrate solution sets. Ordered Pairs An equation in two variables, such as, is satisfied onl if we find a value for and a value for that make it true. For eample, if and, then the equation becomes, which is a true statement. We write and as the ordered pair (, ). The order of the numbers in an ordered pair is important. For eample, (, ) also satisfies because using for and for gives. However, (, ) does not satisf this equation because. In an ordered pair the value for, the -coordinate, is alwas written first and the value for, the -coordinate, is second: (, ). E X A M P L E helpful hint In this chapter ou will be doing a lot of graphing. Using graph paper will help ou to understand the concepts and to recognize errors. For our convenience, a page of graph paper can be found at the end of this chapter. Make as man copies of it as ou wish. Writing ordered pairs Complete the following ordered pairs so that each ordered pair satisfies the equation. a) (, ) b) (, ) a) Replace with in because the -coordinate is : ( ) 8 So the -coordinate is and the ordered pair is (, ). b) Replace with in because the -coordinate is : So the -coordinate is and the ordered pair is,. Plotting Points To graph ordered pairs of real numbers, we need a new coordinate sstem. The rectangular or Cartesian coordinate sstem consists of a horizontal number line, the -ais, and a vertical number line, the -ais, as shown in Fig... The intersection of the aes is the origin. The aes divide the coordinate plane, or the -plane, into four regions called quadrants. The quadrants are numbered as shown in Fig.., and the do not include an points on the aes.

3 . Graphing Lines in the Coordinate Plane (-) -ais 6 Quadrant II Quadrant I Origin 6 Quadrant III Quadrant IV 6 6 -ais (, ) (, ) Origin FIGURE. FIGURE. E X A M P L E (, ) (, ) E X A M P L E stud (0, ) tip (, ) (, ) FIGURE. It is a good idea to work with others, but don t be misled. Working a problem with help is not the same as working a problem on our own. In the end, mathematics is personal. Make sure that ou can do it. Just as ever real number corresponds to a point on the number line, ever pair of real numbers corresponds to a point in the rectangular coordinate sstem. For eample, the pair (, ) corresponds to the point that lies two units to the right of the origin and five units up. See Fig... To locate (, ), start at the origin and move three units to the left and four units upward, as in Fig... Locating a point in the rectangular coordinate sstem is referred to as plotting or graphing the point. Graphing ordered pairs Plot the points (, ), (, ), (, ), (0, ), and (, ). To plot (, ), start at the origin, move two units to the right, then up three units. To graph (, ), start at the origin, move two units to the left, then down three units. All five points are shown in Fig... Graphing a Linear Equation The solution set to an equation in two variables consists of all ordered pairs that satisf the equation. For eample, the solution set to can be written in set notation as (, ). However, this set notation does not shed an light on the solution set to. We can get a better understanding of the solution set with a visual image or graph of the solution set. Graphing a linear equation Graph the solution set to. If we arbitraril choose, then is determined b the equation : ( ) So the ordered pair (, ) satisfies the equation. In this manner we can make the following table of ordered pairs: 0

4 (-) Chapter Graphs and Functions in the Cartesian Coordinate Sstem helpful hint The graph of a linear equation is a straight line that eists in our minds. The straight line in our minds has no thickness, is perfectl straight, and etends infinitel. All attempts to draw it on paper fall short. The best we can do is to use a sharp pencil to keep it thin, a ruler to make it straight, and arrows to indicate that it is infinite. Plot these ordered pairs as shown in Fig... Of course, there are infinitel man ordered pairs that satisf, but the all lie along the line in Fig... The arrows on the ends of the line indicate that it etends without bound in both directions. The line in Fig.. is a graph of the solution set to. In an equation such as the value of depends on the value of. So is the independent variable and is the dependent variable. Because the graph of is a line, the equation is a linear equation. = + FIGURE. E X A M P L E Graphing a linear equation Graph. Plot at least four points. = + FIGURE. If we write in terms of, we get. Now arbitraril select four values for and calculate the corresponding values for : 0 Plot these points and draw a line through them as shown in Fig... If the coefficient of a variable in a linear equation is 0, then that variable is usuall omitted from the equation. For eample, the equation 0 is written as. Because is multiplied b 0, an value of can be used as long as is. Because the -coordinates are all the same, the graph is a horizontal line. calculator close-up To graph with a graphing calculator, first press Y and enter. Xmin 0, X ma 0, Xscl, Ymin 0, Yma 0, Yscl These settings are referred to as the standard window. Press GRAPH to draw the graph. Even though the calculator does not draw a ver good straight line, it supports our conclusion that Fig.. is the graph of Net press WINDOW to set the viewing window as follows: 0

5 . Graphing Lines in the Coordinate Plane (-) E X A M P L E = FIGURE.6 Graphing a horizontal line Graph. Plot at least four points. The following table gives four points that satisf, or 0. Note that it is eas to determine in this case because is alwas. 0 0 The horizontal line through these points is shown in Fig..6. If the coefficient of is 0 in a linear equation, then the graph is a vertical line. E X A M P L E 6 Graphing a vertical line Graph. Plot at least four points. = We can think of the equation as 0. Because is multiplied b 0, the equation is satisfied b an ordered pair with an -coordinate of. 0 0 The vertical line through these points is shown in Fig..7. FIGURE.7 Using Intercepts for Graphing The -intercept of a line is the point where the line crosses the -ais. The -intercept has a -coordinate of 0. Similarl, the -intercept of a line is the point where the line crosses the -ais. The -intercept has an -coordinate of 0. If a line has distinct - and -intercepts, then these intercepts can be used as two points that determine the location of the line. (Horizontal lines, vertical lines, and lines through the origin do not have two distinct intercepts.) E X A M P L E 7 helpful hint You can find the intercepts for 6 using the coverup method. Cover up with our pencil, then solve 6 mentall to get and an -intercept of (, 0). Now cover up and solve 6 to get and a -intercept of (0, ). Using intercepts to graph Use the intercepts to graph the line 6. Let 0in 6to find the -intercept: (0) 6 6 Let 0in 6to find the -intercept: (0) 6 6

6 6 (-6) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Cost (in thousands of dollars) E X A M P L E 8 C (0, ) = 6 (, 0) FIGURE Number of pairs FIGURE.9 n The -intercept is 0,, and the -intercept is (, 0). The line through the intercepts is shown in Fig..8. To check, find another point that satisfies the equation. The point (, ) satisfies the equation and is on the line in Fig..8. CAUTION Even though two points determine the location of a line, finding at least three points will help ou to avoid errors. Applications In applications we often use variables such as C for cost, R for revenue, and n for the number of items so that it is easier to remember what the variables represent. In this case we rename the aes. Which ais is labeled with which variable is somewhat arbitrar. However, when one variable depends on, or is determined b another, the dependent variable is usuall on the vertical ais and the independent variable is on the horizontal ais. Graphing a linear equation in an application The cost per week C (in dollars) of producing n pairs of shoes for the Reebop Shoe Compan is given b the linear equation C n Graph the equation for n between 0 and 800 inclusive (0 n 800). Make a table of values for n and C as follows: n C n Graph the line as shown in Fig..9. Because C n 8000 epresses C in terms of n, C is the dependent variable and the vertical ais is labeled C. To accommodate the large numbers, we let each unit on the n-ais represent 00 pairs and each unit on the C-ais represent $,000. Note how the aes in Fig..9 are labeled. The n-ais starts at 0 and each unit represents 00 pairs. Because all of the costs were between $8,000 and $9,600 we omitted the tick marks for through 7 and put a wave in the C-ais to indicate that some numbers are missing. Omitting the numbers from through 7 makes the difference between the $8,000 and $9,600 costs look greater. WARM-UPS True or false? Eplain our answer.. The point (, ) satisfies the equation. False. The vertical ais is usuall called the -ais. False. The point (0, 0) is in quadrant I. False. The point (0, ) is on the -ais. True. The graph of 7 is a vertical line. True 6. The graph of 8 0 is a horizontal line. True 7. The -intercept for the line is (0, ). True 8. If C n, then C 0 when n. True 9. If P and P, then 6. False 0. The vertical ais should be A when graphing A r. True

7 . Graphing Lines in the Coordinate Plane (-7) 7. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the point called at the intersection of the - and -ais? The origin is the point where the -ais and -ais intersect.. What is an ordered pair? An ordered pair is a pair of real numbers in which there is a first number and a second one.. What are the - and -intercepts? Intercepts are points where a graph crosses the aes.. What tpe of equation has a graph that is a horizontal line? The graph of an equation of the tpe k where k is a fied number is a horizontal line.. What tpe of equation has a graph that is a vertical line? The graph of an equation of the tpe k where k is a fied number is a vertical line. 6. Which variable usuall goes on the vertical ais? The dependent variable usuall goes on the vertical ais. Complete the given ordered pairs so that each ordered pair satisfies the given equation. See Eample. 7. (, ), (, ), 6 (, 0), (, ) 8. (, ), (, ),,, (, )., 0 -ais. (, ) I. (0, ) -ais. (, ) IV Graph each linear equation. Plot four points for each line. See Eamples (, ), (, 6), 9 (, ), (, 6) 0. (, ), (, ),,, (, ) Plot the following points in a rectangular coordinate sstem. For each point, name the quadrant in which it lies or the ais on which it lies. See Eample.. (, ) I. (, ) II , III. (, 6) III. (0, ) -ais 6. (0, ) -ais 7. (, ) IV 8., 0 -ais 9. (, ) II 0. (0, ) -ais..

8 8 (-8) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Find the - and -intercepts for each line and use them to graph the line. See Eample

9 . Graphing Lines in the Coordinate Plane (-9) 9 Solve the following. See Eample 8.. Camaro inflation. The rising list price P (in dollars) for a new Camaro Z8 Coupe can be modeled b the equation P 9,66 69n, where n is the number of ears since 99 (Edmund s New Car Prices, a) What will be the list price for a new Z8 in 00? b) What is the annual increase in list price? c) Sketch a graph of this equation for 0 n 0. a) $, b) $69 c) applicant s abilit to repa, a higher rating indicating higher risk. The interest rate, r, is then determined b the formula r 0.0t 0.. If our rating were 8, then what would be our interest rate? Sketch the graph of the equation for t ranging from 0 to 0. %. Camaro Z8 depreciation. The 998 average retail price P (in dollars) for an n-ear-old Camaro Z8 Coupe can be modeled b the equation P 8,67,960n, where n (Edmund s Used Car Prices, www. edmunds.com). a) What was the average retail price of a -ear-old Z8 in 998? b) How much does this model depreciate each ear? c) Sketch a graph of this equation for n. a) $0,8 b) $960 c) 7. Little Chicago Pizza. The equation C 0.0t 8.9 gives the customer s cost in dollars for a pan pizza, where t is the number of toppings. a) Find the cost of a five-topping pizza. b) Find t if C. and interpret our result. a) $. is the cost of a five-topping pizza. b) is the number of toppings on a $. pizza. 8. Long distance charges. The formula L 0.0n.9 gives the monthl bill in dollars for AT&T s one rate plan, where n is the number of minutes of long distance used during the month. Answer each question and interpret the result. a) Find the bill for 0 minutes. $6.9 b) Find n if L.. 8 c) What are the coordinates for the L-intercept shown on the accompaning graph? (0,.9) d) What are the coordinates of the n-intercept? ( 9., 0) AT&T s One Rate Plan Monthl bill (in dollars) 0 0. Rental cost. For a one-da car rental the X-press Car Compan charges C dollars, where C is determined b the formula C 0.6m and m is the number of miles driven. What is the charge for a car driven 00 miles? Sketch a graph of the equation for m ranging from 0 to 000. $6 6. Measuring risk. The Friendl Bob Loan Compan gives each applicant a rating, t, from 0 to 0 according to the Time (minutes) FIGURE FOR EXERCISE 8 9. Cost, Revenue, and Profit. Hillar sells roses at a bus Los Angeles intersection. The formulas C 0. 0, R.0, and P give her weekl cost, revenue, and profit in terms of, where is the number of roses that she sells in one week. a) Find C, R, and P if 80. Interpret our results. b) Find if P 99 and interpret our result.

10 0 (-0) Chapter Graphs and Functions in the Cartesian Coordinate Sstem c) Find R C if 00 and interpret our result. a) Her weekl cost, revenue, and profit are $7.0, $,7, and $77.0. b),00. She had a profit of $99 on selling,00 roses. c) 99. The difference between revenue and cost is $99, which is her profit. 60. Velocit of a pop up. A pop up off the bat of Mark McGwire goes straight into the air at 88 feet per second (ft/sec). The formula v t 88 gives the velocit of the ball in feet per second, t seconds after the ball is hit. a) Find the velocit for t and t seconds. What does a negative velocit mean? b) For what value of t is v 0? Where is the ball at this time? c) What are the two intercepts on the accompaning graph? Interpret this answer. d) If the ball takes the same time going up as it does coming down, then what is its velocit as it hits the ground? a) ft/sec, 8 ft/sec, going down b).7 seconds, at maimum height c) (0, 88) indicates that at t 0 second the velocit was 88 ft/sec, (.7, 0) indicates that at t.7 seconds the velocit was 0 ft/sec d) 88 ft/sec c) According to a theorem in geometr, the three medians of an triangle intersect at a single point. Estimate the point of intersection of the three medians in our triangle. a) (, ), (, 0), (, ) c) 8, GRAPHING CALCULATOR EXERCISES Graph each linear equation on a graphing calculator. Choose a viewing window that shows both intercepts. Answers ma var Velocit (feet/second) Time (seconds) FIGURE FOR EXERCISE GETTING MORE INVOLVED 6. Midpoint formula. The point M midwa between (, ) and (, ) can be found using the formula M,. Find the midpoint for each of the following pairs of points and plot the points. a) (, ) and (, 6) (, ) b) (6, ) and (, ) (, ) c), and,, 8 6. Intersecting medians. Using graph paper, draw a triangle with vertices (0, 6), (6, ), and (, ). a) Find the midpoint of each side of our triangle. b) A median is a line segment connecting a verte of a triangle with the midpoint of the opposite side. Draw the three medians for our triangle

11 . Slope of a Line (-) SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail and stud the concept of slope of a line. Slope Using Coordinates to Find Slope Parallel Lines Perpendicular Lines Applications of Slope Slope If a highwa has a 6% grade, then in 00 feet (measured horizontall) the road rises 6 feet (measured verticall). See Fig..0. The ratio of 6 to 00 is 6%. If a roof rises 9 feet in a horizontal distance (or run) of feet, then the roof has a 9 pitch. A roof with a 9 pitch is steeper than a roof with a 6 pitch. The grade of a road and the pitch of a roof are measurements of steepness. In each case the measurement is a ratio of rise (vertical change) to run (horizontal change). 6% GRADE 6 00 SLOW VEHICLES KEEP RIGHT ft run 9 ft rise helpful hint Since the amount of run is arbitrar, we can choose the run to be. In this case slope ri se rise. So the slope is the amount of change in for a change of in the -coordinate.this is wh rates like 0 miles per hour (mph), 8 hours per da, and two people per car are all slopes. FIGURE.0 9 pitch We measure the steepness of a line in the same wa that we measure steepness of a road or a roof. The slope of a line is the ratio of the change in -coordinate, or the rise, to the change in -coordinate, or the run, between two points on the line. Slope Slope change in -coordinate change in -coordinate Consider the line in Fig..(a) on the net page. In going from (0, ) to (, ), there is a change of in the -coordinate and a change of in the -coordinate, rise run