LESSON 3.1 INTRODUCTION TO GRAPHING

LESSON 3.1 INTRODUCTION TO GRAPHING LESSON 3.1 INTRODUCTION TO GRAPHING 137

OVERVIEW Here s what ou ll learn in this lesson: Plotting Points a. The -plane b. The -ais and -ais c. The origin d. Ordered pairs e. The -coordinate (abscissa), the -coordinate (ordinate) f. Plotting ordered pairs of numbers Man different tpes of people use graphs to displa man kinds of information. Nurses, sportswriters, car mechanics, engineers, bookkeepers, and scientists all use graphs in their work. Sometimes the use graphs as a wa of recording what the see. Sometimes the use graphs as a wa of reviewing what has happened. Other times the use graphs to predict the future. In this lesson ou will learn about the most widel used graphing sstem the Cartesian coordinate sstem. You will learn how to plot points and how to compare the horizontal and vertical change between points. You will also learn how to calculate the distance between two points and find the equation of a circle. g. Labeling the four quadrants h. Determining the quadrant in which a point lies i. The signs of the coordinates in each quadrant Rise and Run a. Subscript notation b. Geometric interpretation of rise and run c. Algebraic definition of rise and run The Distance Formula a. Pthagorean Theorem b. The distance formula c. The equation of a circle 138 TOPIC 3 INTRODUCTION TO GRAPHING

EXPLAIN PLOTTING POINTS Summar The Cartesian Coordinate Sstem The French mathematician René Descartes realized that if he placed two number lines at right angles to each other he would be able to specif the position of an point in the plane using two reference numbers called coordinates. This arrangement of the two number lines is called the Cartesian coordinate sstem in honor of Descartes. -ais II I -ais O origin III IV Often, the horizontal number line is called the -ais and the vertical number line is called the -ais. The -plane consists of the -ais and the -ais and all of the points in the plane. The origin is usuall denoted b the letter O. The numbers along the -ais get larger as ou move toward the right. The numbers along the -ais get larger as ou move upward. The - and -aes divide the plane into four regions called quadrants. These are labeled with the Roman numerals I, II, III, IV in a counter-clockwise direction beginning in the upper right. The horizontal ais and vertical ais are not alwas labeled with and. Sometimes it is helpful to use labels which better represent the data. An eas wa to remember which ais is the -ais is to notice that the -ais runs up and down, just like the bottom of the Y. In Quadrant I, and are both positive. In Quadrant II, is negative and is positive. In Quadrant III, and are both negative. In Quadrant IV, is positive and is negative. Points on the aes do not lie in a quadrant. LESSON 3.1 INTRODUCTION TO GRAPHING EXPLAIN 139

Finding the Coordinates of Points The coordinates of a point are an ordered pair because the order in which the coordinates are written is important. To find the position of a point in the -plane ou need two numbers. These two numbers are called the -coordinate and the -coordinate. The can be written as an ordered pair (, ). P ( 3, ) To find the coordinates of a point: 1. Draw a vertical line from the point to the -ais. The line intersects the -ais at the -coordinate.. Draw a horizontal line from the point to the -ais. The line intersects the -ais at the -coordinate. The coordinates are written as an ordered pair: (-coordinate, -coordinate). This method of finding the coordinates of the point P is illustrated in Figure 3.1.. Here are some tips for finding the coordinates of points which lie on an ais: Figure 3.1. A point which lies on the -ais has coordinates of the form (, 0). A point which lies on the -ais has coordinates of the form (0, ). If the coordinates of a point are not whole numbers, ou ma onl be able to estimate where to plot the point. If a point has negative coordinates, remember to include the negative sign(s) when ou write the coordinates. You will find that drawing lines through the - and -coordinates of a point is often helpful when ou first start plotting points. After some practice ou ma want to tr plotting points without drawing these lines. The origin lies on both aes so it has coordinates (0, 0). Plotting Points Once ou know the -coordinate and -coordinate of a point, ou have enough information to plot the point in the -plane. To plot a point: 1. Draw a vertical line through the -coordinate of the point.. Draw a horizontal line through the -coordinate of the point. 3. Plot the point where the lines intersect. As an eample, Q(, ) is plotted in Figure 3.1.3. Q (, ) Figure 3.1.3 10 TOPIC 3 INTRODUCTION TO GRAPHING

Sample Problems Answers to Sample Problems 1. Find the coordinates of the point P below. a. Draw a vertical line from P to the-ais. This line meets the -ais at the point =. b. Draw a horizontal line from P to the -ais. This line meets the -ais at the point =. c. The coordinates of P are (, ). P b. c. (, ). Plot the point Q (0, ). a. Draw a vertical line through 0, the -coordinate of the point Q. a. This line is the same as the -ais. b. Draw a horizontal line through, the -coordinate of the point Q. b., c. c. The lines intersect at the point Q. Label the point Q. 3. Plot the point R (5,.5). a. Draw a vertical line through. Q (0, ) a. Draw a vertical line through 5, the -coordinate of R. b. Draw a horizontal line through.5, the -coordinate of R. b. R(5,.5) c. The lines intersect at. c. The lines intersect at the point R(5,.5).. Plot a point in Quadrant I. a. Shade in Quadrant I. b. Plot a point anwhere in that quadrant. a., b. an point in this quadrant LESSON 3.1 INTRODUCTION TO GRAPHING EXPLAIN 11

Answers to Sample Problems a., b., c., d., e., f., g. 5. Health care costs in the United States have been taking an increasing percentage of the Gross National Product (GNP) in recent decades. Use the information in the table below to plot the ordered pairs (ear, % of GNP) on the grid provided. % OF GNP 13 1 11 10 9 8 7 5 3 1 190 195 1970 1975 1980 1985 1990 Year % of GNP 13 1 11 10 9 8 7 5 3 1 190 195 1970 1975 1980 1985 1990 Year % of Year GNP 190 5.3 195 5.9 1970 7.3 1975 8.3 1980 9. 1985 10.5 1990 1. Plot the point: a. (190, 5.3) b. (195, 5.9) c. (1970, 7.3) d. (1975, 8.3) e. (1980, 9.) f. (1985, 10.5) g. (1990, 1.) 1 TOPIC 3 INTRODUCTION TO GRAPHING

RISE AND RUN Summar Defining Rise and Run Looking at the relationship between two points often provides more information than looking at each point in isolation. Given two points in the -plane, for eample, ou can move from one point to the other with one vertical and one horizontal movement. The vertical change as ou move from one point to the other is called the rise. The horizontal change as ou move from one point to the other is called the run. The rise can be positive or negative. Likewise, the run, can be positive or negative. Finding Rise and Run One wa to find the rise and the run in moving from one point to another is b drawing lines on a graph. To find the rise and the run between two points, P 1 and P : 1. Count the number of units of vertical change it takes to move from P 1 to P. This is the rise.. Count the number of units of horizontal change it takes to move from P 1 to P. This is the run. This method is illustrated in Figure 3.1. moving from the point P 1 ( 3, 1) to P (, 5). As ou can see, the rise is and the run is 7. Another wa to find the rise and the run in moving from one point to another is b using algebra. P 1 ( 3, 1) 7 P (, 5) To find the rise and run between two points, P 1 ( 1, 1 ) and P (, ): 1. Subtract the -coordinates of the points to find the rise, since rise is the vertical change from one point to another along the -ais: rise = 1. Subtract the -coordinates of the points to find the run, since run is the horizontal change from one point to another along the -ais: run = 1 Figure 3.1. You can find the rise and the run from P 1 ( 3, 1) to P (, 5) b using this method: rise = 1 run = 1 = 5 1 = ( 3) = = 7 LESSON 3.1 INTRODUCTION TO GRAPHING EXPLAIN 13

Answers to Sample Problems Sample Problems 1. Use the graph to find the rise and the run in moving from P 1 (, ) to P (3, 7) with one vertical movement and one horizontal movement. b. 5 9 (, ) (3, 7) a. Start at P1 and rise until ou are at the level of P. b. Then run to the right horizontall to P. c. The rise is the length of the vertical line, which is 5 units. The run is the length of the horizontal line, which is units. P (3, 7) 5 P 1 (, ) c. 9. Find the rise and the run from P 1 (, 7) to P (3, 1) using the algebraic definitions. a. rise = a. rise = 1 = ( 1) ( 7) = b. run = 3 () = 7 b. run = 1 = = 3. Find the rise and the run from P 1 (3, ) to P ( 5, 1). a. rise = 1 ( ) = 1 b. run = 5 3 = 8 a. rise = 1 = = b. run = 1 = =. Find the rise and the run from P 1 (, 7) to P (, 3). a. rise = 3 7 = b. run = = a. rise = 1 = = b. run = 1 = = 1 TOPIC 3 INTRODUCTION TO GRAPHING

THE DISTANCE FORMULA Summar Now that ou can plot points in the Cartesian coordinate sstem, ou can learn how to calculate the distance between an two points in this coordinate sstem. But first ou ll learn how to find the distance between two points on the number line. Finding the Distance Between Two Points on the Number Line If a and b are an two points on the number line, then the distance between a and b is a b or b a. To find the distance between two points on the number line: 1. Subtract the coordinates of the points (in either order).. Take the absolute value of this difference. 5 You take the absolute value because absolute value gives a nonnegative number and distance can never be negative. Since ou take the absolute value, it doesn t matter whether ou find a b or b a. 3 1 0 1 3 5 7 For eample, to find the distance between 0 and 5: 1. Subtract. 0 5 = 5. Take the absolute value. 5 = 5 So the distance from 0 to 5 is 5 units. Now ou can find the distance between an two points on the number line. But to be able to find the distance between an two points in the plane, (0, 0) and (3, ) for eample, ou need to learn the Pthagorean Theorem. The Pthagorean Theorem The Pthagorean Theorem relates the lengths of the sides of a right triangle. It sas that if a and b are the lengths of the legs and c is the length of the hpotenuse, then c = a + b. If ou know the lengths of two of the sides of a right triangle, ou can use the Pthagorean Theorem to find the length of the third side. Here s how: c a 1. Substitute the values for the lengths of the two sides in the Pthagorean Theorem.. Solve for the remaining value. For eample, to find c if a = and b = 8: 1. Substitute a = and b = 8. c = a + b c = + 8 b A right triangle has one angle that measures 90. The smbol in the corner of the triangle indicates that the angle measures 90. LESSON 3.1 INTRODUCTION TO GRAPHING EXPLAIN 15

c =? b = 8 a =. Solve for c. c = 3 + c = 100 c = ± 10 0 c = ±10 c = 10 Choose the positive value of c, since c is a distance. We alwas take the positive root of 100 since this quantit represents a distance. c = 5 b =? Figure 3.1.5 (0, 0) b (3, 0) Figure 3.1. a = 3 ( 1, 1 ) ( a = 1, ) c ( 1, ) b = 1 c (3, ) a As another eample, to find b if a = 3 and c = 5: 1. Substitute a = 3 and c = 5. c = a + b 5 = 3 + b. Solve for b. 5 = 9 + b 5 9 = b 1 = b ± 1 = b b = ± b = Choose the positive value of b, since b is a distance. Finding the Distance Between Two Points Using the Pthagorean Theorem You can use the Pthagorean Theorem to find the distance between an two points. To find the distance c between an two points ( 1, 1 ) and (, ) using the Pthagorean Theorem (refer to Figure 3.1.5): 1. Form a right triangle as follows: a. Draw a vertical line segment from ( 1, 1 ) to ( 1, ). Label this a. b. Draw a horizontal line segment from (, ) to ( 1, ). Label this b.. Find the lengths of segments: a. The length of the vertical segment a is 1 units. b. The length of the horizontal segment b is 1 units. 3. Use the Pthagorean Theorem, c = a + b, to find the length of the hpotenuse, c. The value of C is the distance between the two points. For eample, to find the distance c between (0, 0) and (3, ): 1. Form a right triangle. See Figure 3.1... Find the lengths of the a = 0 = = segments. b = 0 3 = 3 = 3 3. Find c, the distance between (0, 0) and (3, ). c = a + b So the distance from (0, 0) to (3, ) is 5 units. c = 3 + c = 9 + 1 c = 5 c = 5 1 TOPIC 3 INTRODUCTION TO GRAPHING

The Distance Formula Instead of drawing a triangle to find the distance between two points, ou can just use the distance formula. From our work with the Pthagorean Theorem ou know that c = a + b, where a = 1 and b = 1. So, c, the square of the distance between an two points ( 1, 1 ) and (, ) is: c = ( 1 ) + ( 1 ) and c = ( 1 ) + ( 1 ) This equation is called the distance formula. To find the distance between an two points using the distance formula: 1. Identif 1, 1,, and.. Substitute these values into the distance formula. 3. Simplif. For eample, to find the distance between (5, 0) and (, 8): 1. Identif 1, 1,, and. 1 = 5 1 = 0 = = 8. Substitute. c = ( 1 ) + ( 1 ) c = ( 5 ) + ( 8 0 ) 3. Simplif. c = ( 7) + ( 8) The Equation of a Circle c = 9 + c = 11 3 You can use the square of the distance formula ( 1 ) + ( 1 ) = c, to find the equation of a circle. Suppose the center of a circle is at the point (h, k) and its radius is r (where r > 0). If (, ) is an point on the circle, then using the square of the distance formula ou can get the equation of the circle: ( h) + ( k) = r For eample, the equation of the circle shown in Figure 3.1.7 whose center is at (, 1) and whose radius is 3 is: ( ) + ( 1) = 3 You can also find the center and radius of a circle if ou know the equation of the circle. For eample, if the equation of a circle is ( + ) + ( 3) =, its center is at (, 3) and its radius is. See Figure 3.1.8. It doesn t matter which point ou call ( 1, 1 ) and which one ou call (, ). Just don t change it once ou ve decided. The radius, r, of a circle must be nonnegative. Figure 3.1.7 (, 3) Figure 3.1.8 (, 1) Wh is the -coordinate of the center equal to? Well, ( + ) can be rewritten as [ ()]. 3 LESSON 3.1 INTRODUCTION TO GRAPHING EXPLAIN 17

Answers to Sample Problems Sample Problems 1. Use the Pthagorean Theorem to find the distance between (3, 5) and ( 3, 3). a. Form a right triangle. c (3, 5) (3, 3) ( 3, 3) b a b. 5, 8 3, c. 10 b. Find the lengths of the segments. c. Use the Pthagorean Theorem, c = a + b, to find c, the distance between the points. a = 3 = b = 3 = c =. Using the distance formula, find the distance between (1, ) and ( 3, 1). a. Identif 1, 1,, and. 1 = 1 1 = = 3 = 1 b. Substitute. c = ( 1 ) + ( 1 ) b. 3, 1 c. 5 c. Simplif. c = (_ _ _ 1 ) + ( _ _ _ ) c = 18 TOPIC 3 INTRODUCTION TO GRAPHING

Sample Problems EXPLORE Answers to Sample Problems On the computer, ou used the Grapher to move a point around a grid and observed how some of the properties of the point changed. Below are some additional eploration problems. 1. Plot three points in Quadrant I whose -coordinate is. a. Find Quadrant I and shade it in. b. Find a point in Quadrant I whose -coordinate is. c. Plot our point on the graph. a., b., c., d. Several possible answers are shown. an of these points d. Repeat steps (b) and (c) to plot two more points.. Plot three points, (, ), where = + 1. a. Pick an three numbers for the -coordinates, sa, = 0, = 1, and =. b. For each value of, find the value of. For eample, if = 0, then = 0 + 1 = 1. The corresponding ordered pair is (0, 1). Plot that point. c. If = 1, then =. Plot the corresponding ordered pair ( 1, ). d. If =, then =. Plot the corresponding ordered pair (, ). b., c., d. Several possible answers are shown. an of these points c. 0, 0 ( 1, 0) d. Answers will var. (0, 1) 3. Given the point P 1 (, 3), find the coordinates of P if the rise from P 1 to P is and the run is 3. a. Start at the point (, 3) and draw a vertical line units long torepresent the rise. b. From the top of the vertical line, draw a horizontal line 3 units longto represent the run. c. The horizontal line ends at the point P (, ). P 1 b. c. (5, 7) P 1 P LESSON 3.1 INTRODUCTION TO GRAPHING EXPLORE 19

HOMEWORK Homework Problems Circle the homework problems assigned to ou b the computer, then complete them below. Eplain Plotting Points Use Figure 3.1.9 to answer questions 1 through 8. FIgure 3.1.9 Number of Farms (in thousands) 7,000,500,000 5,500 5,000,500,000 3,500 3,000,500,000 1,500 1,000 1, 500 Figure 3.1.10 1. Find the coordinates of point P.. Plot the point Q(, 5). 3. In what quadrant does the point R(, 1) lie?. Find the coordinates of point S. 5. Plot the point T ( 3, ). '0 '50 '0 '70 '80 '90 Year. Plot a point which lies in Quadrant III. P R S 7. Plot the point U (0, 5). 8. Plot a point in Quadrant I whose -coordinate is. 9. For selected ears, the number of farms in the United States is listed in the table below. Use this information to plot the ordered pairs (ear, number of farms) on the grid in Figure 3.1.10. 10. Using the data provided in problem (9), plot the ordered pairs (ear, average number of acres per farm) on the grid in Figure 3.1.11. Average Number of Acres per Farm Number of Farms Number of Average Year (in thousands) Acres per Farm 190,10 175 1950 5,388 1 190 3,9 97 1970,95 373 1980,0 1990,13 1 500 50 00 350 300 50 00 150 100 050 '0 '50 '0 '70 '80 '90 Year Figure 3.1.11 150 TOPIC 3 INTRODUCTION TO GRAPHING

11. Plot the point V ( 3, 0). 1. Plot a point which does not lie in an quadrant. Rise and Run 13. Draw one vertical and one horizontal line to show the rise and the run in moving from P 1 (1, 3) to P (, ). 1. Plot the points Q 1 (, 3) and Q (3, ). Draw one vertical and one horizontal line to find the rise and the run in moving from Q 1 to Q. 15. Use rise = 1 and run = 1 to find the rise and the run in moving from R 1 (, 5) to R (5, 7). 1. Find the rise and the run in moving from S 1 (, 7) to S (, ) b drawing one vertical and one horizontal line on the graph. 17. Find the rise and the run from T 1 ( 1, ) to T ( 5, 8) b subtracting the appropriate coordinates. 18. Which is greater, the rise from U 1 ( 9, ) to U ( 1, 5) or the rise from V 1 (0, ) to V (10, )? 19. Find the rise and the run from W 1 ( 7, 11) to W (17, 19) b subtracting the appropriate coordinates. 0. Given P 1 (1, ), find the coordinates of P if the rise from P 1 to P is and the run is 5. Use the grid in Figure 3.1.1. 1. Plotted in Figure 3.1.13 is the federal minimum hourl wage rate (for nonfarm workers) for selected ears. Use this information to determine which five-ear period had the greatest rise in minimum wage. (You can refer to the table for more accurate numbers.) Wage ($) 3.50 3.5 3.00.75.50.5.00 1.75 1.50 1.5 1.00.75.50.5 50 55 0 5 70 75 80 85 90 Year Year Wage 1950 $.75 1955 $.75 190 $1.00 195 $1.5 1970 $1.0 1975 $.10 Figure 3.1.13 1980 $3.10 1985 $3.35 1990 $3.35. Use the graph and table in problem (1) to determine which five-ear period had the smallest rise in minimum wage. 3. Find the rise and the run from P 1 (8, 3) to P (17, 9) b subtracting the appropriate coordinates.. Starting at P 1 ( 3, ), find the coordinates of P if the rise from P 1 to P is 8 and the run is 7. Figure 3.1.1 P 1 The Distance Formula 5. If a = 5 and b = 1, use the Pthagorean Theorem to find c, the length of the hpotenuse of the right triangle shown in Figure 3.1.1. c =? a = 5 b = 1 Figure 3.1.1. What is the equation of a circle whose center is at (, 3) and whose radius is? 7. Using the distance formula, find the distance between (0, 0) and (5, ). LESSON 3.1 INTRODUCTION TO GRAPHING HOMEWORK 151

8. Use the Pthagorean Theorem to find the distance between 3. Marilena has been taking a shortcut across a lawn as shown (0, 0) and (, 8). See Figure 3.1.15. in Figure 3.1.18. If the two lengths of the sidewalk measure ;; ft. and 8 ft., how much distance does Marilena save b 8 (, 8) taking the shortcut? ft. (0, 0) 8 ft. Figure 3.1.18 Figure 3.1.15 35. Write the equation of the circle with radius 5 whose center is at ( 3, ). 9. Find the center and the radius of the circle whose equation is ( + 5) + ( 7) =. 3. Using the distance formula, find the distance between (, ) and (5, 8). 30. Using the distance formula, find the distance between (, ) and ( 1, 7). 31. Use the Pthagorean Theorem to find the distance between (, 1) and (5, ). See Figure 3.1.1. (, 1) Figure 3.1.1 3. Find the center and the radius of the circle whose equation is: ( ) + ( + 1) = 1. 33. A fullback takes the ball from his 5 ard line (30 ards from the sideline) to his 5 ard line (50 ards from the same sideline). How man ards did he actuall run? (You can epress our answer as the square of the distance.) Start b finding a and b as shown in Figure 3.1.17. Then find c. 10 0 30 0 50 0 30 0 10 b 50 ds. ;;30 a ds.c 10 0 30 0 50 0 30 0 10 Figure 3.1.17 (5, ) 15 TOPIC 3 INTRODUCTION TO GRAPHING ;; ;; Eplore 37. Plot three points in Quadrant II each of which has a -coordinate equal to 5. 38. Plot three points, (, ), where =. 39. Starting at the point P 1 (1, 1), if ou rise and also run, ou end up at the point (3, 3). Start again at P 1 (1, 1) and plot three other points such that the rise and run are equal to each other. 0. Plot three points in Quadrant IV each of which has an -coordinate equal to 3. 1. Plot three points, (, ), where = +.. Starting at the point Q 1 (, ), if ou rise and run 1, ou end up at the point ( 1, ). Starting at the point Q 1 (, ), plot three other points which have a rise which is twice as much as the run.

APPLY Practice Problems Here are some additional practice problems for ou to tr. Plotting Points 1. Plot the point (3, 5).. Plot the point (, 1). 3. Plot the point (, 1).. Plot the point ( 3, ). 5. Plot the point ( 5, ).. Plot the point ( 1, ). 7. Plot the point ( 1, 5). 8. Plot the point (, ). 9. Plot the point ( 3, 5). 11. Plot the point (, ). 1. Plot the point (, ). 13. Plot the point (0, 3). 1. Plot the point (, 0). 15. Plot the point ( 3, 0). 1. Find the coordinates of the point P. 17. Find the coordinates of the point T. T 18. Find the coordinates of the point Q. Q 19. Find the coordinates of the point M. P M LESSON 3.1 INTRODUCTION TO GRAPHING APPLY 153

0. Find the coordinates of the point N. N 1. Find the coordinates of the point R. R. Find the coordinates of the point Q. Q 3. Find the coordinates of the point R. R. Find the coordinates of the point S. S 5. In what quadrant does the point (, 3) lie?. In what quadrant does the point (, 3) lie? 7. In what quadrant does the point (1, 3) lie? 8. In what quadrant does the point (, ) lie? Rise and Run 9. Find the rise and the run in moving from the point (1, 5) to the point (9, 7). 30. Find the rise and the run in moving from the point (1, 8) to the point (5, 17). 31. Find the rise and the run in moving from the point (11, 5) to the point (, 9). 3. Find the rise and the run in moving from the point (, 3) to the point (, 7). 33. Find the rise and the run in moving from the point (0, ) to the point (8, 5). 3. Find the rise and the run in moving from the point (, 5) to the point (11, 9). 35. Find the rise and the run in moving from the point (3, 10) to the point (0, ). 3. Find the rise and the run in moving from the point ( 1, 1) to the point ( 19, 13). 37. Find the rise and the run in moving from the point (, 5) to the point (, ). 38. Find the rise and the run in moving from the point (, 0) to the point (9, 5). 39. Find the rise and the run in moving from the point (8, 1) to the point (0, 7). 15 TOPIC 3 INTRODUCTION TO GRAPHING

0. Find the rise and the run in moving from the point (, 0) to the point ( 5, ). 1. Find the rise and the run in moving from the point ( 5, 9) to the point (8, ). The Distance Formula 57. If a = 1 and b = 1, use the Pthagorean Theorem to find c, the length of the hpotenuse of the right triangle shown below.. Find the rise and the run in moving from the point ( 10, ) to the point (1, 8). c =? a = 1 3. Find the rise and the run in moving from the point (, ) to the point (, 3).. Find the rise and the run in moving from the point (3, 17) to the point (1, 3). 5. Find the rise and the run in moving from the point (15, 1) to the point (3, 31).. Find the rise and the run in moving from the point (11, 7) to the point (35, ). b = 1 58. If a = 9 and b = 1, use the Pthagorean Theorem to find c, the length of the hpotenuse of the right triangle shown below. c =? a = 9 b = 1 59. If a = 15 and b = 3, use the Pthagorean Theorem to find c, the length of the hpotenuse of the right triangle shown below. 7. Find the rise and the run in moving from the point ( 13, 9) to the point (0, 7). 8. Find the rise and the run in moving from the point ( 85, 57) to the point (0, 3). a = 15 b = 3 c =? 9. Find the rise and the run in moving from the point ( 7, 1) to the point (0, 1). 0. If a = 0 and b = 8, use the Pthagorean Theorem to find c, the length of the hpotenuse of the right triangle shown below. 50. Which is greater, the rise from P 1 (9, 13) to P (1, 17) or the rise from Q 1 ( 3, 5) to Q (, 1) a = 0 c =? 51. Which is greater, the run from P 1 (7, 1) to P (19, 13) or the run from Q 1 ( 1, 5) to Q (3, 39)? 5. Given P 1 (11, 1), find the coordinates of P if the rise from P 1 to P is 3 and the run is 9. 53. Given P 1 (8, 9), find the coordinates of P if the rise from P 1 to P is and the run is 7. 5. Given P 1 (, 7), find the coordinates of P if the rise from P 1 to P is and the run is. 55. Given P 1 ( 1, 7), find the coordinates of P if the rise from P 1 to P is 13 and the run is 17. 5. Given P 1 ( 3, ), find the coordinates of P if the rise from P 1 to P is and the run is 8. b = 8 1. What is the equation of the circle whose center is at (, 3) and whose radius is?. What is the equation of the circle whose center is at (, 5) and whose radius is 7? 3. What is the equation of the circle whose center is at ( 3, 1) and has radius 5?. Write the equation of the circle whose center is at (1, ) and has radius 10. 5. Write the equation of the circle whose center is at ( 3, 7) and has radius 8.. Write the equation of the circle whose center is at (3, 10) and has radius 8. LESSON 3.1 INTRODUCTION TO GRAPHING APPLY 155

7. Use the distance formula to find the distance between the points (3, ) and (9, 13). 8. Use the distance formula to find the distance between the points (, 7) and (, 3). 9. Use the distance formula to find the distance between the points (7, ) and ( 8, 3). 70. Use the distance formula to find the distance between the points ( 11, 5) and (, 7). 71. Use the distance formula to find the distance between the points ( 1, 7) and ( 10, ). 7. Use the distance formula to find the distance between the points ( 10, 3) and (, ). 73. Use the Pthagorean Theorem to find the distance between (, 0) and ( 1, ). ( 1, ) (, 0) 7. Use the Pthagorean Theorem to find the distance between (0, 0) and (3, ). (3, ) (0, 0) 75. Use the Pthagorean Theorem to find the distance between (, 5) and (, 3). (, 3) (, 5) 7. Use the Pthagorean Theorem to find the distance between ( 1, ) and (, ). (, ) ( 1, ) 77. Use the Pthagorean Theorem to find the distance between (7, ) and ( 5, 3). (7, ) ( 5, 3) 15 TOPIC 3 INTRODUCTION TO GRAPHING

78. Use the Pthagorean Theorem to find the distance between ( 5, 5) and (7, 0). ( 5, 5) (7, 0) 80. Find the center and the radius of the circle whose equation is ( 10) + ( 1) = 7. 81. Find the center and the radius of the circle whose equation is ( + 9) + ( 1) =. 8. Find the center and the radius of the circle whose equation is ( 8) + ( + ) = 5. 83. Find the center and the radius of the circle whose equation is ( + 9) + ( 3) = 11. 8. Find the center and the radius of the circle whose equation is ( 3) + ( + 15) = 1. 79. Find the center and the radius of the circle whose equation is ( + 5) + ( + ) = 3. LESSON 3.1 INTRODUCTION TO GRAPHING APPLY 157

Practice Test EVALUATE Take this practice test to be sure that ou are prepared for the final quiz in Evaluate. Use Figure 3.1.19 to answer questions (1) (3). S K Figure 3.1.19 1. Find the coordinates of point K.. Plot the point P (5,.5). 3. In what quadrant does the point S (, 3) lie?. For selected ears, average gas mileage for American cars is listed in the table below (rounded to the nearest whole number). Plot the ordered pairs (ear, mileage) on the set of aes provided in Figure 3.1.0. Average Gas Mileage (mpg) 30 5 0 15 10 05 Figure 3.1.0 5. Find the rise and the run in moving from point P 1 (1, 5) to P (7, 5) b drawing one vertical and one horizontal line on the grid in Figure 3.1.1. 1970 1975 1980 1985 1990 Year P Year Average Gas Mileage (mpg) P 1 1970 1 1975 15 1980 3 1985 Figure 3.1.1. Find the rise and the run from P 1 ( 7, 8) to P (0, ) b subtracting the appropriate coordinates. 7. Find the rise and the run from P 1 ( 1, 7) to P (, 1) b subtracting the appropriate coordinates. 1990 7 158 TOPIC 3 INTRODUCTION TO GRAPHING

8. The average price for a gallon of gasoline is plotted in Figure 3.1. for selected ears. Use this information to determine which five-ear period had the greatest rise in gas prices. Price Year (cents) 1950.8 1955 9.1 190 31.1 195 31. 1970 35.7 1975 5.7 1980 119.1 1985 111.5 1990 11.9 Price (cents) 10 110 100 90 80 70 0 50 0 30 0 10 50 55 0 5 70 75 80 85 90 Year Figure 3.1. 9. If a = 9 and b = 1, use the Pthagorean Theorem to find c, the length of the hpotenuse of the right triangle shown in Figure 3.1.3. a = 9 c =? 10. Use the Pthagorean Theorem to find the distance between the points ( 3, 1) and (1, ). See Figure 3.1.. ( 3, 1) (1, ) Figure 3.1. 11. Use the distance formula to find the distance between the points (10, ) and (, 7). 1. Find the radius and the center of the circle whose equation is below. ( 1) + [ ( 5)] = 13. A point with a negative -coordinate and a positive -coordinate lies in which quadrant? Use Figure 3.1.5 to answer questions (1) (1). P 1 Figure 3.1.5 1. Plot a point in Quadrant III whose -coordinate is. 15. Starting at the point P 1 (1, ), find the coordinates of P if the rise from P 1 to P is 5 and the run is 1. 1. Plot a point, (, ), where = 1. b = 1 Figure 3.1.3 LESSON 3.1 INTRODUCTION TO GRAPHING EVALUATE 159

10 TOPIC 3 INTRODUCTION TO GRAPHING

TOPIC 3 CUMULATIVE ACTIVITIES CUMULATIVE REVIEW PROBLEMS These problems combine all of the material ou have covered so far in this course. You ma want to test our understanding of this material before ou move on to the net topic. Or ou ma wish to do these problems to review for a test. 1. Write in lowest terms:. Evaluate the epression + 1 when = and =. 3. Simplif the epression 3( 7) + (9 ).. Plot the points P 1 (1, ) and P (, ). Draw one vertical and one horizontal line to find the rise and the run from P 1 to P. 5. Plot the points Q 1 (, 3) and Q ( 1, 5). Draw one vertical and one horizontal line to find the rise and the run from Q 1 to Q.. Plot three points, (, ), in Quadrant III where =. Use the grid in Figure 3.. 7. Seven ears ago, Raoul was as old as Christine is now. If the sum of their ages is 3, how old is each person? 8. Find the rise and the run from V 1 ( 5, 37) to V ( 8, 3) b subtracting the appropriate coordinates. 9. Solve for : + 9 < 13. Then graph its solution on the number line below. 3 10. Find: + 9 10 1 1 3 0 1 3 11. Einstein s famous formula, E = mc, shows the amount of energ, E, which can be obtained from a particle of mass m. Solve this formula for c. 1. Circle the true statements. The equation + 3 = 7 has no solution. 5 + 3 5 5 5 = 3 5 3 < 3 < 13. Find: 7[ 3(5 ) + 1] 1. Solve for z: < z 7. Circle the number below that is not a solution. 11.1 7 15. Plot four points, (, ), where = 3. Use the grid in Figure 3.3. 1. One number is 3 more than twice another number. If the sum of the two numbers is 33, what are the two numbers? TOPIC 3 CUMULATIVE REVIEW 11

17. Circle the true statements. 1 = 1 1 ( ) = ( 1) ever value of is a solution of the equation = 8( ) 3 5 = 5 3 9 = 3 18. Write the coordinates of the point on the grid in Figure 3.1 that: a. has an -value more than. b. has a -value twice its -value. c. has a -value less than 1.. Plot the point Q(,.5). 5. Plot the point R(0, ).. Shade in Quadrant I. 7. Solve for : 7 + 3 = 38 8. Use the fact that R = {1,, 3,, 5} and S = {,,, 8, 10} to determine if each statement below is true. a. R b. S c. 3 R d. 3 S 9. Write the equation of the circle with radius whose center is at (, 3). 30. Use the Pthagorean Theorem to find the distance between the points ( 1, ) and (, 8). 18 5 31. Find: 3. Circle the true statements. 5 a. z = 3 is a solution of the inequalit z 5 < 3 Figure 3.1 7 1 19. Find: 0. Find the rise and the run from T 1 ( 9, 31) to T (1, ) b subtracting the appropriate coordinates. 1. Solve for : 5 8 3 <. Solve for z: (z 1) = (8z + 0) 3. Plot the point P(3, 5). 3 8 5 1 10 b. 0 > 5 c. 0 > 5 d. 7 = 7 7 7 7 5 10 1 e. = = 33. Solve for : 3( + ) = ( ) 8 3. Use the distance formula to find the distance between the points (, 3) and (9, 5). 35. Find the radius and the center of the circle whose equation is ( + 5) + ( 1) = 9. 1 1 TOPIC 3 INTRODUCTION TO GRAPHING

3. If R = {1,, 3,, 5,, 7, 8, 9, 10}, S = {,,, 8, 10}, and T = {1, 3, 5, 7, 9}, then which of the statements below are true? a. S R b. R S c. T R d. R T e. S T f. T S TOPIC 3 CUMULATIVE REVIEW 13

1 TOPIC 3 INTRODUCTION TO GRAPHING