Start at the origin. Move left 3 units since the x-coordinate. Start at the origin. Since the x-coordinate is 0, the point

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1 Answers (Lesson -) Lesson - - Stud Guide and Intervention The Coordinate Plane Identif Points In the diagram at the right, points are located in reference to two perpendicular number lines called aes. The horizontal number line is the -ais, and the vertical number line is the -ais. The plane containing the - and -aes is called the coordinate plane. Points in the coordinate plane are named b ordered pairs of the form (, ). The first number, or -coordinate corresponds to a number on the -ais. The second number, or -coordinate, corresponds to a number on the -ais. The aes divide the coordinate plane into Quadrants I, II, III, and IV, as shown. The point where the aes intersect is called the origin. The origin has coordinates (, ). Quadrant II Quadrant I R P Quadrant III Q Quadrant IV Eample Eample Write an ordered pair for point R above. The -coordinate is and the -coordinate is. Thus the ordered pair for R is (, ). Write ordered pairs for points P and Q above. Then name the quadrant in which each point is located. The -coordinate of P is and the -coordinate is. Thus the ordered pair for P is (, ). P is in Quadrant III. The -coordinate of Q is and the -coordinate is. Thus the ordered pair for Q is (, ). Q is in Quadrant IV. Eercises Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. R Q V. N (, ), none. P (, ), III. Q (, 5), none. R (, ), II 5. S (5, ), IV 6. T ( 5, ), none W U T Z N S P A B 7. U (, ), I 8. V (5, ), I 9. W (, ), II. Z (, ), none. A (, ), IV. B (, ), none. Write the ordered pair that describes a point units down from and units to the right of the origin. (, ). Write the ordered pair that is 8 units to the left of the origin and lies on the -ais. ( 8, ) Glencoe/McGraw-Hill Glencoe Algebra - Stud Guide and Intervention (continued) The Coordinate Plane Graph Points To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair. To graph an ordered pair (, ), begin at the origin. Move left or right units. From there, move up or down units. Draw a dot at that point. Eample Plot each point on a coordinate plane. a. R(, ) Start at the origin. Move left units since the -coordinate is. Move up units since the -coordinate is. Draw a dot and label it R. R b. S(, ) Start at the origin. Since the -coordinate is, the point will be located on the -ais. Move down units since the -coordinate is. Draw a dot and label it S. S Eercises Plot each point on the coordinate plane at the right.. A(, ). B(, ) 6. See graph.. C(, ). D(, ) 5. E(, ) 6. F(, ) H A M Q D J B C E L F K P G N I 7. G(5, ) 8. H(, ) 9. I(, 5). J(, ). K(, ). L(, ). M(, ). N(5, ) 5. P(, 5) 6. Q( 5, ) Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

2 Answers (Lesson -) Lesson - - Skills Practice The Coordinate Plane Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. D. A (, ); III. B (, ); I. C (, ); IV. D (, ); II B E C F A 5. E (, ); none 6. F (, ); IV Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. 7. G (, ); II 8. H (, ); IV 9. J (, ); I. K (, ); III G K M L H J. L (, ); I. M (, ); none Plot each point on the coordinate plane at the right.. M(, ). N(, ) 5. P(, ) 6. Q(, ) 7. R(, ) 8. S(, ) M Q R S P N Plot each point on the coordinate plane at the right. 9. T(, ). U(, ) X U T. W(, ). X(, ) Y. Y(, ). Z(, ) W Z Glencoe/McGraw-Hill 5 Glencoe Algebra Answers - Practice (Average) The Coordinate Plane Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located.. A (, ); none. B ( 5, 5); II B F J E L. C ( 5, ); III. D (5, 5); IV 5. E (, ); I 6. F (, ); II I A C K G H D 7. G (, ); IV 8. H (, ); III 9. I (, ); none. J (, 5); I. K (5, ); IV. L (, ); I Plot each point on the coordinate plane at the right.. M(, ). N(, ) 5. P(5, ) 6. Q(, ) 7. R(, 5) 8. S(, ) 9. T( 5, ). V(, 5). W(, ). X(, ). Y(, ). Z(, ) M T Q Z S X R W N V Y P 5. CHESS Letters and numbers are used to show the positions of chess pieces and to describe their moves. For eample, in the diagram at the right, a white pawn is located at f5. Name the positions of each of the remaining chess pieces. white pawns: a7, c6; black pawns: b, d7; white king: g; black king: e a b c d e f g h King Pawn ARCHAELGY For Eercises 6 and 7, use the grid at the right that shows the location of arrowheads ecavated at a midden a place where people in the past dumped trash, food remains, and other discarded items. Meters 6. Write the coordinates of each arrowhead. (, ), (, ), (, ) 7. Suppose an archaeologist discovers two other arrowheads located at (, ) and (, ). Draw an arrowhead at each of these locations on the grid. Meters Glencoe/McGraw-Hill 6 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

3 Answers (Lesson -) Lesson - - Reading to Learn Mathematics The Coordinate Plane Pre-Activit How do archaeologists use coordinate sstems? Read the introduction to Lesson - at the top of page 9 in our tetbook. What do the terms grid sstem, grid, and coordinate sstem mean to ou? See students work. Reading the Lesson. Use the coordinate plane shown at the right. a. Label the origin. b. Label the -ais. c. Label the -ais.. Eplain wh the coordinates of the origin are (, ). Sample answer: The origin is the intersection of two number lines at their common zero point.. Use the ordered pair (, ). a. Eplain how to identif the - and -coordinates. The -coordinate is the first number, the -coordinate is the second number. b. Name the - and -coordinates. The -coordinate is, the -coordinate is. c. Describe the steps ou would use to locate the point for (, ) on the coordinate plane. Start at the origin. Move left units. Then move up units.. What does the term quadrant mean? Sample answer: one of the four regions in the coordinate plane Helping You Remember 5. Eplain how the wa the aes are labeled on the coordinate plane can help ou remember how to plot the point for an ordered pair. Sample answer: The right side of the horizontal ais is labeled with the letter. This is the side of the horizontal number line where ou find positive numbers. The top end of the vertical number line is labeled with the letter. This is the part of the vertical number line where ou find positive numbers. Glencoe/McGraw-Hill 7 Glencoe Algebra - Enrichment Midpoint The midpoint of a line segment is the point that lies eactl halfwa between the two endpoints of the segment. The coordinates of the midpoint of a line segment whose endpoints are (, ) and (, ) are given b,. Find the midpoint of each line segment with the given endpoints.. (7, ) and (, ). (5, ) and (9, 8) (, ) (7, 5). (, ) and (, ). (, 6) and (, 5) (, ) ( 6.5,.5) Plot each segment in the coordinate plane. Then find the coordinates of the midpoint. 5. J K with J(5, ) and K(, ) 6. P Q with P(, ) and Q(, ) (, ) P (5, ) J (,.5) (.5, ) Q (, ) (, ) K (.5, ) (,.5) You are given the coordinates of one endpoint of a line segment and the midpoint M. Find the coordinates of the other endpoint. 7. A(, ) and M( 6, 7) 8. D(, ) and M(, 6) (, ) (7, 6) Glencoe/McGraw-Hill 8 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

4 Answers (Lesson -) Lesson - - Stud Guide and Intervention Transformations on the Coordinate Plane Transform Figures Transformations are movements of geometric figures. The preimage is the position of the figure before the transformation, and the image is the position of the figure after the transformation. Reflection A figure is flipped over a line. Translation A figure is slid horizontall, verticall, or both. Dilation A figure is enlarged or reduced. Rotation A figure is turned around a point. Eample Determine whether each transformation is a reflection, translation, dilation, or rotation. a. The figure has been flipped over a line, so this is a reflection. b. The figure has been turned around a point, so this is a rotation. c. The figure has been reduced in size, so this is a dilation. d. The figure has been shifted horizontall to the right, so this is a translation. Eercises Determine whether each transformation is a reflection, translation, dilation, or rotation.. reflection. dilation. rotation. translation 5. dilation 6. rotation Glencoe/McGraw-Hill 9 Glencoe Algebra Answers - Stud Guide and Intervention (continued) Transformations on the Coordinate Plane Transform Figures on the Coordinate Plane You can perform transformations on a coordinate plane b changing the coordinates of each verte. The vertices of the image of the transformed figure are indicated b the smbol, which is read prime. Reflection over -ais (, ) (, ) Reflection over -ais (, ) (, ) Translation (, ) ( a, b) Dilation (, ) (k, k) Rotation 9 counterclockwise (, ) (, ) Rotation 8 (, ) (, ) Eample A triangle has vertices A(, ), B(, ), and C(, ). Find the coordinates of the vertices of each image below. a. reflection over the -ais To reflect a point over the -ais, multipl the -coordinate b. A(, ) A (, ) B(, ) B (, ) C(, ) C (, ) The coordinates of the image vertices are A (, ), B (, ), and C (, ). b. dilation with a scale factor of Find the coordinates of the dilated figure b multipling the coordinates b. A(, ) A (, ) B(, ) B (, 8) C(, ) C (6, ) The coordinates of the image vertices are A (, ), B (, 8), and C (6, ). Eercises Find the coordinates of the vertices of each figure after the given transformation is performed.. triangle RST with R(, ), S(, ), T(, ) reflected over the -ais R (, ), S (, ), T (, ). triangle ABC with A(, ), B(, ), C(, ) rotated about the origin 8 A (, ), B (, ), C (, ). parallelogram ABCD with A(, ), B(, ), C(, ), D(, ) translated units down A (, ), B (, ), C (, ), D (, ). quadrilateral RSTU with R(, ), S(, ), T(, ), U(, ) dilated b a factor of R (, ), S (, ), T (, ), U (, ) 5. triangle ABC with A(, ), B(, ), C(, ) rotated counterclockwise 9 A (, ), B (, ), C (, ) 6. heagon ABCDEF with A(, ), B(, ), C(, ), D(, ), E(, ), F(, ) translated units up and unit to the left A (, ), B (, 5), C (, 6), D (, 6), E (, ), F (, ) Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A5 Glencoe Algebra

5 Answers (Lesson -) Lesson - - Skills Practice Transformations on the Coordinate Plane Identif each transformation as a reflection, translation, dilation, or rotation.... rotation reflection translation dilation reflection rotation For Eercises 7, complete parts a and b. a. Find the coordinates of the vertices of each figure after the given transformation is performed. b. Graph the preimage and its image. 7. triangle ABC with A(, ), B(, ), and 8. parallelogram PQRS with P(, ), C(, ) reflected over the -ais Q(, ), R(, ), and S(, ) A (, ), B (, ), and translated units up P (, ), C (, ) Q (, ), R (, ), and S (, ) A A P Q B C R S Q C B P S R 9. trapezoid JKLM with J(, ), K(, ),. triangle STU with S(, ), T(5, ), and L(, ), and M(, ) dilated b a U(, ) rotated 9 counterclockwise scale factor of J (, ), K (, ), about the origin L (, ), and M (, ) S (, ), T (, 5), U (, ) T J K S S J K U U M L M L T Glencoe/McGraw-Hill Glencoe Algebra - Practice (Average) Transformations on the Coordinate Plane Identif each transformation as a reflection, translation, dilation, or rotation.... reflection translation rotation For Eercises 6, complete parts a and b. a. Find the coordinates of the vertices of each figure after the given transformation is performed. b. Graph the preimage and its image.. triangle DEF with D(, ), 5. trapezoid EFGH with 6. triangle XYZ with X(, ), E(, ), and F(, ) E(, ), F(, ), Y(, ), and Z(, ) translated units left G(, ), and H(, ) rotated 9 counterclockwise and units down reflected over the -ais about the origin D (, ), E (, ), E (, ), F (, ), X (, ), F (, ) G (, ), H (, ) Y (, ), Z (, ) D F E D F E E F H G H G E F X Z Y Z X Y GRAPHICS For Eercises 7 9, use the diagram at the right and the following information. A designer wants to dilate the rocket b a scale factor of, and then translate it 5 units up. 7. Write the coordinates for the vertices of the rocket. A(, ), B(, ), C(, 5), D(, 6), E(, 6), F(, 5), G(, ) 8. Find the coordinates of the final position of the rocket. A,, B,, C,, D,, E,, F,, G, E G F A B C D 9. Graph the image on the coordinate plane.. DESIGN Ramona transformed figure ABCDEF to design a pattern for a quilt. Name two different sets of transformations she could have used to design the pattern. Sample answer: reflection over the -ais, 9 counterclockwise rotation, and then reflection over the -ais; three 9 counterclockwise rotations F E A D B C Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A6 Glencoe Algebra

6 Answers (Lesson -) Lesson - - Reading to Learn Mathematics Transformations on the Coordinate Plane Pre-Activit How are transformations used in computer graphics? Read the introduction to Lesson - at the top of page 97 in our tetbook. In the sentence, Computer graphic designers can create movement that mimics real-life situations, what phrase indicates the use of transformations? create movement Reading the Lesson. Suppose ou look at a diagram that shows two figures ABCDE and A B C D E. If one figure was obtained from the other b using a transformation, how do ou tell which was the original figure? The letters that have no prime smbols are used for vertices of the original figure.. Write the letter of the term and the Roman numeral of the figure that best matches each statement. a. A figure is flipped over a line. C, I A. dilation I. b. A figure is turned around a point. D, III B. translation II. c. A figure is enlarged or reduced. A, II C. reflection III. d. A figure is slid horizontall, verticall, D. rotation IV. or both. B, IV Helping You Remember. Give eamples of things in everda life that can help ou remember what reflections, dilations, and rotations are. Sample answer: For a reflection, think of looking at ourself in a mirror. For a dilation, think of how our hand looks if ou hold it far from our face and then move it straight in, ver close to our face. For a rotation, think of twisting the top of a jar to open the jar. Glencoe/McGraw-Hill Glencoe Algebra Answers - Enrichment The Legendar Cit of Ur The cit of Ur was founded more than five thousand ears ago in Mesopotamia (modern-da Iraq). It was one of the world s first cities. Between 9 and 9, archeologists discovered man treasures from this ancient cit. A large cemeter from the 6th centur B.C. was found to contain large quantities of gold, silver, bronze, and jewels. The man cultural artifacts that were found, such as musical instruments, weapons, mosaics, and statues, have provided historians with valuable clues about the civilization that eisted in earl Mesopotamia.. Suppose that the ordered pairs below represent the volume (cm ) and mass (grams) of ten artifacts from the cit of Ur. Plot each point on the graph. A(, 5) B(5, 5) C(, 76) D(5, 55) E(, 5) F(, 88) G(, ) H(5, 675) I(, 9) J(5, ) 8 6 A F D J E I H B C 5 5 Volume (cm ) G Mass (grams) 8 6. The equation relating mass, densit, and volume for silver is m.5v. Which of the points in Eercise are solutions for this equation? D and G. Suppose that the equation m 8.8V relates mass, densit, and volume for the kind of bronze used in the ancient cit of Ur. Which of the points in Eercise are solutions for this equation? C, F, J. Eplain wh the graph in Eercise shows onl quadrant. Neither the mass nor the volume can be a negative number. Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A7 Glencoe Algebra

7 Answers (Lesson -) Lesson - - Stud Guide and Intervention Relations Represent Relations A relation is a set of ordered pairs. A relation can be represented b a set of ordered pairs, a table, a graph, or a mapping. A mapping illustrates how each element of the domain is paired with an element in the range. Eample Eample Epress the relation {(, ), (, ), (, )} as a table, a graph, and a mapping. State the domain and range of the relation. X Y The domain for this relation is {,, }. The range for this relation is {,, }. A person plaing racquetball uses calories per hour for ever pound he or she weighs. a. Make a table to show the relation between weight and calories burned in one hour for people weighing,,, and pounds. Source: The Math Teacher s Book of Lists b. Give the domain and range. domain: {,,, } range: {,, 8, 5} c. Graph the relation. 5 8 Calories 8 5 Weight (pounds) Eercises. Epress the relation {(, ), (, ), (, )} as a table, a graph, and a mapping. Then determine the domain and range. domain: {,, }; range: {, } X Y. The temperature in a house drops for ever hour the air conditioner is on between the hours of 6 A.M. and A.M. Make a graph to show the relationship between time and temperature if the temperature at 6 A.M. was 8 F Temperature ( F) Time (A.M.) Glencoe/McGraw-Hill 5 Glencoe Algebra - Stud Guide and Intervention (continued) Relations Inverse Relations The inverse of an relation is obtained b switching the coordinates in each ordered pair. Eample Epress the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. X Y 6 5 Relation: {(6, 5), (, ), (, ), (, )} Inverse: {(5, 6), (, ), (, ), (, )} Eercises Epress the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of each relation... X Y {(, ), (, ), (, ), (, 5)}; {(, ), (, 8), (, ), (5, )}; {(, ), (, ), (, ), (5, )} {(, ), ( 8, ), (, ), (, 5)}.. X Y {(, 5), (, ), (, ), (, )}; {(, ), (, ), (, 6), (, 7)}; {(5, ), (, ), (, ), (, )} {(, ), (, ), (6, ), (7, )} {(, ), (, ), (, ), (, )}; {(, ), (, ), (, ), (, ), (, )}; {(, ), (, ), (, ), (, )} {(, ), (, ), (, ), (, ), (, )} Glencoe/McGraw-Hill 6 Glencoe Algebra Glencoe/McGraw-Hill A8 Glencoe Algebra

8 Answers (Lesson -) Lesson - - Skills Practice Relations Epress each relation as a table, a graph, and a mapping. Then determine the domain and range.. {(, ), (, ), (, ), (, )} X Y D {,,, }; R {,, }. {(, ), (, ), (, ), (, )} X Y D {,,, }; R {,,, }. {(, ), (, ), (, ), (, )} X Y D {,, }; R {,,, } Epress the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of the relation.. 5. X Y {(, 5), (, ), (7, 6), {(9, 7), ( 5, 6), (, 8), {(, ), (, ), (, ), (, )}; {( 5, ), (, )}; {(7, 9), (, )}; {(, ), (, ), (6, 7), (, )} (6, 5), (8, ), (, )} (, ), (, ), (, )} Glencoe/McGraw-Hill 7 Glencoe Algebra Answers - Practice (Average) Relations Epress each relation as a table, a graph, and a mapping. Then determine the domain and range.. {(, ), (, ), (, ), (, ), (, )} X Y D {,,,, }; R {,,, } Epress the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of the relation... X Y {(, 9), ( 8, ), {(9, 5), (9, ), ( 6, 5), {(, ), (, ), (, 6), (, )}; (, ), (8, 5), (8, 7)}; (, ), (, ), (, ), {(9, ), (, 8), {(5, 9), (, 9), ( 5, 6), (, )}; {(, ), (, ), ( 6, ), (, )} (, ), ( 5, 8), (7, 8)} (, ), (, ), (, ), (, )} BASEBALL For Eercises 5 and 6, use the graph that shows the batting average for Barr Bonds of the San Francisco Giants. Source: 5. Find the domain and estimate the range. D {996, 997, 998, 999,, }; R {.6,.9,.,.6,.8,.8} 6. Which seasons did Bonds have the lowest and highest batting averages? lowest: 999, highest:....8 Barr Bonds Batting Year METERS For Eercises 7 and 8, use the table that shows the number of meteors Ann observed each hour during a meteor shower. 7. What are the domain and range? D {,,,, }; R {5, 6, 8} 8. Graph the relation. 5 5 Meteor Shower Number of Meteors Average Time Number of (A.M.) Meteors Time (A.M.) Glencoe/McGraw-Hill 8 Glencoe Algebra Glencoe/McGraw-Hill A9 Glencoe Algebra

9 Answers (Lesson -) Lesson - - Reading to Learn Mathematics Relations Pre-Activit How can relations be used to represent baseball statistics? Read the introduction to Lesson - at the top of page 5 in our tetbook. In 997, Ken Griffe, Jr. had 56 home runs and strikeouts. This can be represented with the ordered pair ( 56, ). Reading the Lesson. Look at page 5 in our tetbook. There ou see the same relation represented b a set of ordered pairs, a table, a graph, and a mapping. a. In the list of ordered pairs, where do ou see the numbers for the domain? the numbers for the range? before the commas; after the commas b. What parts of the table show the domain and the range? The column of numbers under the letter shows the domain, and the column of numbers under the letter shows the range. c. How do the table, the graph, and the mapping show that there are three ordered pairs in the relation? The table has three rows of numbers, the graph shows three points marked with dots, and the mapping uses three arrows.. Which tells ou more about a relation, a list of the ordered pairs in the relation or the domain and range of the relation? Eplain. Sample answer: A list of the ordered pairs tells ou more. You can use it to find the domain and the range, and ou know eactl how the numbers are paired. If ou onl know the domain and the range, ou cannot be sure how the numbers are paired.. Describe how ou would find the inverse of the relation {(, ), (, ), (, 6), (, 8)}. Switch the coordinates in each ordered pair to get {(, ), (, ), (6, ), (8, )}. Helping You Remember. The first letters in two words and their order in the alphabet can sometimes help ou remember their mathematical meaning. Two ke terms in this lesson are domain and range. Describe how the alphabet method could help ou remember their meaning. Sample answer: d comes first for the first coordinate, r comes second for the second coordinate. Glencoe/McGraw-Hill 9 Glencoe Algebra - Enrichment Inverse Relations n each grid below, plot the points in Sets A and B. Then connect the points in Set A with the corresponding points in Set B. Then find the inverses of Set A and Set B, plot the two sets, and connect those points. Set A Set B (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Inverse Set A Set B. (, ) (, ). (, ) (, ). (, ) (, ). (, ) (, ) Set A Set B Inverse Set A Set B (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) 5. (, ) (, ) 6. (, ) (, ) 7. (, ) (, ) 8. (, ) (, ) Set A Set B (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Inverse Set A Set B 9. (, ) (, ). (, ) (, ). (, ) (, ). (, ) (, ). What is the graphical relationship between the line segments ou drew connecting points in Sets A and B and the line segments connecting points in the inverses of those two sets? Answers will var. A possible answer is that the graphs are reflected across the line b their inverses. Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

10 Answers (Lesson -) Lesson - - Stud Guide and Intervention Equations as Relations Solve Equations The equation is an eample of an equation in two variables because it contains two variables, and. The solution of an equation in two variables is an ordered pair of replacements for the variables that results in a true statement when substituted into the equation. Eample Eample Find the solution set for, given the replacement set {(, ), (, ), (, ), (, )}. Make a table. Substitute the and -values of each ordered pair into the equation. True or False ( ) () () () 7 True True False False The ordered pairs (, ), and (, ) result in true statements. The solution set is {(, ), (, )}. Solve b a if the domain is {,,,, }. Make a table. The values of a come from the domain. Substitute each value of a into the equation to determine the corresponding values of b in the range. a a b (a, b) ( ) 5 (, 5) ( ) (, ) () (, ) () (, ) () 7 (, 7) The solution set is {(, 5), (, ), (, ), (, ), (, 7)}. Eercises Find the solution set of each equation, given the replacement set.. ; {(, ),,,,, (, )} {(, ),,, (, )}. 6; {(, ), (, ), (, ), (, )} {(, ), (, )}. 5 ; {(, ), (, ), (, ), (, )} {(, ), (, )} Solve each equation if the domain is (,,,, }.. {(, 8), (, 6), (, ), (, ) (, )} 5. 6 {(, ), (, ), (, 6), (, ), (, )} 6. 5a b {(, 5), (, ), (, 5), (, ), (, 5)} 7. {(, ), (, 9), (, 6), (, ) (, )} {(, ), (, ), (, 6), (, ), (, )} 9. 8 {(, ), (, ), (, ), (, 8), (, )}. 8 {(, ), (, ), (, 8), (, 6) (, )}. {(, 8), (, ), (, ), (, 6), (, )} Glencoe/McGraw-Hill Glencoe Algebra Answers - Stud Guide and Intervention (continued) Equations as Relations Graph Solution Sets You can graph the ordered pairs in the solution set of an equation in two variables. The domain contains values represented b the independent variable. The range contains the corresponding values represented b the dependent variable, which are determined b the given equation. Eample Solve if the domain is (,,, }. Graph the solution set. First solve the equation for in terms of. riginal equation Subtract from each side. Simplif. Divide each side b. 6 Simplif. Substitute each value of from the domain to determine the corresponding value of in the range. 6 (, ) Graph the solution set. 6 ( ) 8 (, 8) 6 () 6 (, 6) 6 () (, ) 6 () (, ) Eercises Solve each equation for the given domain. Graph the solution set.. for {,,, }. for {,,, } {(, ), (, ), (, ), (, )} {(, ), (, ), (, ), (, 5)}. 6 for {,,, 6}. 8 for {,,, } {(, ), (, ), (, ), (6, )} {(, ), (, ), (, ), (, )} Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

11 Answers (Lesson -) Lesson - - Skills Practice Equations as Relations Find the solution set for each equation, given the replacement set.. ; {(, 5), (, 7), (, ), (, )} {(, 5), (, )}. ; {(, ), (, ), (, 6), (, 8)} {(, 6)}. 7 ; {(, ), (, ), (5, ), (, 5)} {(, ), (, ), (5, )}. ; {(, 7), (, ), (, ), (, )} {(, ), (, ), (, )} Solve each equation if the domain is {,,,, 5}. 5. {(, ), (, ), (, ), (, 6), (5, 9)} 6. {(, 8), (, 5), (, ), (, ), (5, )} 7. {(, ), (, ), (, ), (, 5), (5, )} 8. {(, ), (, ), (, ), (, ), (5, )} 9. {(, 5), (, ), (, ), (, ), (5, )}. {(, 8), (, 6), (, ), (, ), (5, 6)}. 7 {(, ), (, 9), (, 7), (, ), (5, )}. 6 {(, 7), (, 5), (, ), (, ), (5, 7)} Solve each equation for the given domain. Graph the solution set.. 5 for { 5,,,, }. for {,,,, } {( 5, 5), (, ), (, ), (, ), (, 5)} {(, 5), (, ), (, ), (, ), (, 5)} 5. for {,,,, } 6. 6 for {,,,, 6} {(, 5), (, ), (, ), (, ), (, 5)} {(, 6), (, ), (, ), (, ), (6, )} Glencoe/McGraw-Hill Glencoe Algebra - Practice (Average) Equations as Relations Find the solution set for each equation, given the replacement set.. 5; {(, ), (, 7), (, 8), (, 7)} {(, 8), (, 7)}. ; {(, ), (,.5), (,.5), (,.5)} {(,.5), (,.5)} Solve each equation if the domain is {,,,, 5}.. {(, 8), (, 6), (, ), (, ), (5, 6)}. 8 {(, ), (, 9), (, 6), (, 5), (5, )} 5. {(, 9), (, 7), (, ), (, ), (5, 5)} 6. 6 {(, ), (,.5), (, ), (,.5), (5,.5)} 7. 6 {(, 5), (,.5), (, ), (,.5), (5,.5)} 8. 6 {(, 6), (, ), (, 8), (, 6), (5, )} Solve each equation for the given domain. Graph the solution set for {,,,, 5}. for {,,,, } {(, ), (,.5), (, ), (, ), (5,.5)} {(, ),,, (, ), (, ), (, )} EARTH SCIENCE For Eercises and, use the following information. Earth moves at a rate of kilometers per second around the Sun. The equation d t relates the distance d in kilometers Earth moves to time t in seconds.. Find the set of ordered pairs when t {,,, 5, 7}. {(, ), (, 6), (, 9), (5, 5), (7, )} Distance Earth Travels Distance (km) 6. Graph the set of ordered pairs Time (s) GEMETRY For Eercises 5, use the following information. The equation for the area of a triangle is A bh. Suppose the area of triangle DEF is square inches.. Solve the equation for h. h. State the independent and dependent variables. b is independent; h is dependent. 5. Choose 5 values for b and find the corresponding values for h. Sample answer: {(5, ), (6, ), (, 6), (, 5), (5, )} 6 b Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

12 Lesson - - Reading to Learn Mathematics Equations as Relations Pre-Activit Wh are equations of relations important in traveling? Read the introduction to Lesson - at the top of page in our tetbook. In the equation p.69d, p represents pounds and d represents dollars. How man variables are in the equation p.69d? two Reading the Lesson. Suppose ou make the following table to solve an equation that uses the domain {,,,, }. (, ) 7 (, 7) 6 (, 6) 5 (, 5) (, ) (, ) a. What is the equation? b. Which column shows the domain? the first column c. Which column shows the range? the third column d. Which column shows the solution set? the fourth column. The solution set of the equation for a given domain is {(, ), (, ), (, ), (7, )}. Tell whether each sentence is true or false. If false, replace the underlined word(s) to make a true sentence. a. The domain contains the values represented b the independent variable. true b. The domain contains the numbers,,, and. false; range c. For each number in the domain, the range contains a corresponding number that is a value of the dependent variable. true. What is meant b solving an equation for in terms of? isolating on one side of the equation Helping You Remember. Remember, when ou solve an equation for a given variable, that variable becomes the dependent variable. Write an equation and describe how ou would identif the dependent variable. Sample answer: shows that ou have solved for, so is the dependent variable. Glencoe/McGraw-Hill 5 Glencoe Algebra Answers Coordinate Geometr and Area How would ou find the area of a triangle whose vertices have the coordinates A(-, ), B(, ), and C(, )? When a figure has no sides parallel to either ais, the height and base are difficult to find. ne method of finding the area is to enclose the figure in a rectangle and subtract the area of the surrounding triangles from the area of the rectangle. Area of rectangle DEFC 6 square units A Area of triangle I ()() II D C Area of triangle II ()() E III B F Area of triangle III ()() Total square units Area of triangle ABC 6, or 6 square units Find the areas of the figures with the following vertices.. A(-, -6), B(, ),. A(6, -), B(8, -),. A(, ), B(, 7), C(, ) C(, -6) C(6, ), D(9, -) square units square units 55 square units C B B A C A D A C B Glencoe/McGraw-Hill 6 Glencoe Algebra I Answers (Lesson -) - Enrichment A B C Glencoe/McGraw-Hill A Glencoe Algebra

13 Answers (Lesson -5) Lesson -5-5 Stud Guide and Intervention Graphing Linear Equations Identif Linear Equations A linear equation is an equation that can be written in the form A B C. This is called the standard form of a linear equation. Standard Form of a Linear Equation A B C, where A, A and B are not both zero, and A, B, and C are integers whose GCF is. Eample Eample Determine whether 6 is a linear equation. If so, write the equation in standard form. First rewrite the equation so both variables are on the same side of the equation. 6 riginal equation 6 Add to each side. 6 Simplif. The equation is now in standard form, with A, B and C 6. This is a linear equation. Determine whether is a linear equation. If so, write the equation in standard form. Since the term has two variables, the equation cannot be written in the form A B C. Therefore, this is not a linear equation. Eercises Determine whether each equation is a linear equation. If so, write the equation in standard form es; es; es; no es; 6 no es; 9 es; 8 es;... 8 es; es; 6 8 no a b 8 b es; 6 no es; 6a b es; 8 no no Glencoe/McGraw-Hill 7 Glencoe Algebra -5 Stud Guide and Intervention (continued) Graphing Linear Equations Graph Linear Equations The graph of a linear equation is a line. The line represents all solutions to the linear equation. Also, ever ordered pair on this line satisfies the equation. Eample Graph the equation. Solve the equation for. riginal equation Add to each side. Simplif. Select five values for the domain and make a table. Then graph the ordered pairs and draw a line through the points. (, ) ( ) (, ) ( ) (, ) () (, ) () (, ) () 5 (, 5) Eercises Graph each equation Glencoe/McGraw-Hill 8 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

14 Answers (Lesson -5) Lesson -5-5 Skills Practice Graphing Linear Equations Determine whether each equation is a linear equation. If so, write the equation in standard form no es; es; es; 5 es; 6 7 no es; es; es; Graph each equation Glencoe/McGraw-Hill 9 Glencoe Algebra Answers -5 Practice (Average) Graphing Linear Equations Determine whether each equation is a linear equation. If so, write the equation in standard form no es; es; a 5 b es; 5 es; 8 es; 5a b es; 6 es; no Graph each equation CMMUNICATINS For Eercises 5, use the following information. A telephone compan charges $.95 per month for long distance calls plus $.5 per minute. The monthl cost c of long distance calls can be described b the equation c.5m.95, where m is the number of minutes.. Find the -intercept of the graph of the equation. (,.95). Graph the equation. 8 6 Long Distance 8 6 Time (minutes) 5. If ou talk minutes, what is the monthl cost for long distance? $.95 MARINE BILGY For Eercises 6 and 7, use the following information. Killer whales usuall swim at a rate of. 9.7 kilometers per hour, though the can travel up to 8. kilometers per hour. Suppose a migrating killer whale is swimming at an average rate of.5 kilometers per hour. The distance d the whale has traveled in t hours can be predicted b the equation d.5t. 6. Graph the equation. 7. Use the graph to predict the time it takes the killer whale to travel kilometers. between 6 h and 7 h Killer Whale Travels Distance (km) Cost ($) Time (hours) Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A5 Glencoe Algebra

15 Answers (Lesson -5) Lesson -5-5 Reading to Learn Mathematics Graphing Linear Equations Pre-Activit How can linear equations be used in nutrition? Read the introduction to Lesson -5 at the top of page 8 in our tetbook. C In the equation f. 9, what are the independent and dependent variables? C is independent, f is dependent. Reading the Lesson. Describe the graph of a linear equation. The graph is a straight line.. Determine whether each equation is a linear equation. Eplain. Equation Linear or non-linear? Eplanation a. linear The equation can be written as. b. 7 non-linear has two variables. c. non-linear The variable has an eponent of. d. 5 linear The equation can be written as.. What do the terms -intercept and -intercept mean? The -intercept is the -coordinate of the point where the graph of an equation crosses the -ais, and the -intercept is the -coordinate of the point where the graph crosses the -ais. Helping You Remember. Describe the method ou would use to graph 8. Sample answer: Find the - and -intercepts, which are and. Plot the points for the ordered pairs (, ) and (, ), and draw a line that connects the points. Glencoe/McGraw-Hill Glencoe Algebra -5 Enrichment Taicab Graphs You have used a rectangular coordinate sstem to graph equations such as on a coordinate plane. In a coordinate plane, the numbers in an ordered pair (, ) can be an two real numbers. A taicab plane is different from the usual coordinate plane. The onl points allowed are those that eist along the horizontal and vertical grid lines. You ma think of the points as taicabs that must sta on the streets. The taicab graph shows the equations and. Notice that one of the graphs is no longer a straight line. It is now a collection of separate points.. Graph these equations on the taicab plane at the right Use our graphs for these problems. 5. Which of the equations has the same graph in both the usual coordinate plane and the taicab plane? Describe the form of equations that have the same graph in both the usual coordinate plane and the taicab plane. A and B, where A and B are integers In the taicab plane, distances are not measured diagonall, but along the streets. Write the tai-distance between each pair of points. 7. (, ) and (5, ) 8. (, ) and (-, ) 9. (, ) and (,.5) 7 units 5 units.5 units. (, ) and (, ). (, ) and (-, ). (, ) and (-, ) units units 6 units Draw these graphs on the taicab grid at the right.. The set of points whose tai-distance from (, ) is units. indicated b crosses. The set of points whose tai-distance from (, ) is units. indicated b dots Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A6 Glencoe Algebra

16 Answers (Lesson -6) Lesson -6-6 Stud Guide and Intervention Functions Identif Functions Relations in which each element of the domain is paired with eactl one element of the range are called functions. Eample Eample Determine whether the relation {(6, ), (, ), (7, ), (, )} is a function. Eplain. Since each element of the domain is paired with eactl one element of the range, this relation is a function. is a function. Since the equation is in the form A B C, the graph of the equation will be a line, as shown at the right. If ou draw a vertical line through each value of, the vertical line passes through just one point of the graph. Thus, the line represents a function. Determine whether 6 Eercises Determine whether each relation is a function.... X Y es es no no no es 7. {(, ), (, ), (6, )} 8. {(, ), (, ), (, )} 9. {(, ), (, )} es no es.. 8. es no no Glencoe/McGraw-Hill Glencoe Algebra Answers -6 Stud Guide and Intervention (continued) Functions Function Values Equations that are functions can be written in a form called function notation. For eample, can be written as f(). In the function, represents the elements of the domain, and f() represents the elements of the range. Suppose ou want to find the value in the range that corresponds to the element in the domain. This is written f() and is read f of. The value of f() is found b substituting for in the equation. Eample If f(), find each value. a. f() f() () Replace with. 9 Multipl. 5 Simplif. b. f( ) f( ) ( ) Replace with. 6 Multipl. Simplif. Eercises If f() and g(), find each value.. f(). g(). f( 5). g( ) 5. f() 6. g() 7. f() 8. f 9. g 5 6. f(a ). f(k ). g(c) a k c 8c. f(). f() 5. g( ) 6 Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A7 Glencoe Algebra

17 Answers (Lesson -6) Lesson -6-6 Skills Practice Functions Determine whether each relation is a function.. X Y es. X Y es. X Y no es 5. no 6. no {(, 5), (, ), (, ), (5, ), (, 5)} es 8. {(6, ), (, ), (5, ), (, 6), (6, 5)} no 9. 5 es. es. es. no. no If f() and g(), find each value.. f() 5. f(8) 6 6. f( ) 7. g() 8. g( ) 9. g( 6). f() 9. f(). g(). g( ) 6. f( ) 5 5. g(b) 9b b Glencoe/McGraw-Hill 5 Glencoe Algebra -6 Practice (Average) Functions Determine whether each relation is a function.. X Y es. no. es {(, ), (, ), (, 6), ( 6, ), (, 6)} 5. {(6, ), (, ), (, ), (, 6), (, 6)} es no 6. no 7. es If f() 6 and g(), find each value. 8. f() 9. f 7. g( ) 5. g 9. f(7) 9. g( ) 8. f(h 9) 5. g() 6. [g(b) ] h 8 b b WAGES For Eercises 7 and 8, use the following information. Martin earns $7.5 per hour proofreading ads at a local newspaper. His weekl wage w can be described b the equation w 7.5h, where h is the number of hours worked. 7. Write the equation in functional notation. f(h) 7.5h 8. Find f(5), f(), and f(5)..5, 5, 87.5 ELECTRICITY For Eercises 9, use the following information. The table shows the relationship between resistance R and current I in a circuit. Resistance (ohms) Current (amperes) Is the relationship a function? Eplain. Yes; for each value in the domain, there is onl one value in the range.. If the relation can be represented b the equation IR, rewrite the equation in functional notation so that the resistance R is a function of the current I. f(i ). What is the resistance in a circuit when the current is.5 ampere? ohms I Glencoe/McGraw-Hill 6 Glencoe Algebra Glencoe/McGraw-Hill A8 Glencoe Algebra

18 Answers (Lesson -6) Lesson -6-6 Reading to Learn Mathematics Functions Pre-Activit How are functions used in meteorolog? Read the introduction to Lesson -6 at the top of page 6 in our tetbook. If pressure is the independent variable and temperature is the dependent variable, what are the ordered pairs for this set of data? {(, ), (6, ), (997, ), (995, ), (995, 8), (, ), (6, ), (, ), (6, 6), (9, 9)} Reading the Lesson. The statement, Relations in which each element of the range is paired with eactl one element of the domain are called functions, is false. How can ou change the underlined words to make the statement true? Change range to domain and domain to range.. Describe how each method shows that the relation represented is a function. a. mapping The elements of the domain are X Y 7 paired with corresponding elements in the range. Each element of the domain has onl one arrow going from it. b. vertical line test No vertical line passes through more than one point of the graph. Helping You Remember. A student who was tring to help a friend remember how functions are different from relations that are not functions gave the following advice: Just remember that functions are ver strict and never give ou a choice. Eplain how this might help ou remember what a function is. Sample answer: A function alwas pairs each element in the domain with eactl one element in the range. If two people start with the same element of the domain, the are forced to pair it with the same element in the range. The second person cannot pick a different number from the first person. Glencoe/McGraw-Hill 7 Glencoe Algebra Answers -6 Enrichment Composite Functions Three things are needed to have a function a set called the domain, a set called the range, and a rule that matches each element in the domain with onl one element in the range. Here is an eample. Rule: f() f() 5 5 f() f() () f() () 5 f(-) ( ) 6 5 Suppose we have three sets A, B, and C and two functions described as shown below. Rule: f() Rule: g( ) A B C g( ) g() () 5 f() g[f()] 5 Let s find a rule that will match elements of set A with elements of set C without finding an elements in set B. In other words, let s find a rule for the composite function g[f()]. Since f(), g[ f()] g( ). Since g( ), g( ) ( ), or 6. Therefore, g[ f()] 6. Find a rule for the composite function g[f()].. f() and g( ). f() and g( ) g[f()] 6 g[f()]. f() and g( ). f() and g( ) g[f()] 6 g[f()] 5. Is it alwas the case that g[ f()] f[ g()]? Justif our answer. No. For eample, in Eercise, f [g()] f( ) ( ) 6, not 6. Glencoe/McGraw-Hill 8 Glencoe Algebra Glencoe/McGraw-Hill A9 Glencoe Algebra

19 Answers (Lesson -7) Lesson -7-7 Stud Guide and Intervention Arithmetic Sequences Recognize Arithmetic Sequences A sequence is a set of numbers in a specific order. If the difference between successive terms is constant, then the sequence is called an arithmetic sequence. Arithmetic Sequence a numerical pattern that increases or decreases at a constant rate or value called the common difference Eample Eample Determine whether the sequence,, 5, 7, 9,, is an arithmetic sequence. Justif our answer. If possible, find the common difference between the terms. Since, 5, and so on, the common difference is. Since the difference between the terms of,, 5, 7, 9,, is constant, this is an arithmetic sequence. Determine whether the sequence,,, 8, 6,, is an arithmetic sequence. Justif our answer. If possible, find the common difference between the terms. Since and, there is no common difference. Since the difference between the terms of,,, 8, 6,, is not constant, this is not an arithmetic sequence. Eercises Determine whether each sequence is an arithmetic sequence. If it is, state the common difference.., 5, 9,, 7,. 8,,,, 8,.,, 9, 7, 8, es; es; no., 5, 5,, 6, 5., 5,, 5,, 6. 8, 6,,,,, no es; 5 es; 7., 8,, 6, 8. 5,,, 9, 9..,.,.,., 5., es; no es;. 8, 7, 6, 5,,..5,.5,.5,.5,.5,.,, 6, 6, es; es; no.,, 8,,., 6, 9,, 5. 7,, 7,, es; es; es; 7 Glencoe/McGraw-Hill 9 Glencoe Algebra -7 Stud Guide and Intervention (continued) Arithmetic Sequences Write Arithmetic Sequences You can use the common difference of an arithmetic sequence to find the net term of the sequence. Each term after the first term is found b adding the preceding term and the common difference. Terms of an Arithmetic Sequence If a is the first term of an arithmetic sequence with common difference d, then the sequence is a, a d, a d, a d,. nth Term of an Arithmetic Sequence a n a (n )d Find the net three terms of the arithmetic sequence 8,, 6,,. Find the common difference b subtracting successive terms. 8 The common difference is. Add to the last given term,, to get the net term. Continue adding until the net three terms are found. Eample Eample The net three terms are, 8, 5. Write an equation for the nth term of the sequence, 5, 8,,. In this sequence, a is. Find the common difference. 5 8 The common difference is. Use the formula for the nth term to write an equation. a n a (n )d Formula for the nth term a n (n ) a, d a n n Distributive Propert a n n 9 Simplif. The equation for the nth term is a n n 9. Eercises Find the net three terms of each arithmetic sequence.. 9,, 7,, 5,.,,, 8,,. 9, 5,, 7, 9,, 7 6,, 5, 59, 65., 5,, 5, 5..5, 5, 7.5,, 6..,., 5., 6.,, 5,.5, 5, , 8., 9. Find the nth term of each arithmetic sequence described. 7. a 6, d, n 8. a, d, n 8 9. a, d 5, n 9. a, d, n 5. a, d, n. a, d, n 6 6 Write an equation for the nth term of the arithmetic sequence..,, 5, 7,.,, 7,, 5., 9,, 9, a n n a n n a n 5n Glencoe/McGraw-Hill 5 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

20 Answers (Lesson -7) Lesson -7-7 Skills Practice Arithmetic Sequences Determine whether each sequence is an arithmetic sequence. If it is, state the common difference.., 7, 9,, no. 5,,, 9, es;. 7,,, 6, es;. 6, 5,,, no 5. 5,,,, es; 6. 9,, 5, 8, es; Find the net three terms of each arithmetic sequence. 7., 7,, 5, 9,, 7 8.,, 8, 6,,, 9.,, 9, 7, 5,,., 5, 8,,, 7,. 9,, 9,, 9,, 9. 6, 7,,,, 9, 8 Find the nth term of each arithmetic sequence described.. a 6, d, n 9. a, d 5, n 8 5. a, d, n 5 6. a, d, n a, d 8, n a 8, d 6, n ,, 8,, for n 7 88.,,,, for n 67.,, 8, 6, for n 6 8., 7,,, for n 5 8 Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence.. 7,, 9, 5,., 6,, 8, 5. 7,,,, a n 6n a n n a n n a n a n a n 6 n 6 n 6 n 8 Glencoe/McGraw-Hill 5 Glencoe Algebra Answers -7 Practice (Average) Arithmetic Sequences Determine whether each sequence is an arithmetic sequence. If it is, state the common difference..,, 5,,. 5,, 9, 6,..,.,.,., es; 8 es; 7 no Find the net three terms of each arithmetic sequence.. 8, 76, 7, 6, 5. 9, 5,, 7, 6.,,,, 58, 5, 6 7,, 5,, Find the nth term of each arithmetic sequence described. 7. a 7, d 9, n a, d, n ,, 8,, for n 7..,.8, 5.5, 6., for n. 7. a 8, d, n 5 8. a, d, n 7 Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence.. 9,, 7,,. 5,,,, 5. 9,,, 55, a n n 5 a n n 8 a n n 7 a n a n a n n 6 n 6 n BANKING For Eercises 6 and 7, use the following information. Chem deposited $5. in a savings account. Each week thereafter, he deposits $5. into the account. 6. Write a formula to find the total amount Chem has deposited for an particular number of weeks after his initial deposit. a n 5n 5 7. How much has Chem deposited weeks after his initial deposit? $65 8. STRE DISPLAY Tamika is stacking boes of tissue for a store displa. Each row of tissues has fewer boes than the row below. The first row has boes of tissues. How man boes will there be in the tenth row? 5 Glencoe/McGraw-Hill 5 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

21 Answers (Lesson -7) Lesson -7-7 Reading to Learn Mathematics Arithmetic Sequences Pre-Activit How are arithmetic sequences used to solve problems in science? Read the introduction to Lesson -7 at the top of page in our tetbook. Describe the pattern in the data. The altitude of the probe increases b 8. feet each second. Reading the Lesson. Do the recorded altitudes in the introduction form an arithmetic sequence? Eplain. Yes; the difference between the successive terms is the constant 8... What is meant b successive terms? terms that come one right after the other. Complete the table. Pattern Is the sequence increasing Is there a common difference? or decreasing? If so, what is it? a., 5, 8,,, increasing es; b. 55, 5, 5,, decreasing es; 5 c.,,, 9, 6, increasing no d.,,,, decreasing es; e..6,.9,.,.5, increasing es;. Helping You Remember. Use the pattern, 7,, 5, to eplain how ou would help someone else learn how to find the th term of an arithmetic sequence. Find the common difference of. To find the th term, use the formula a n a (n )d. Substitute for n, for a, and for d. The th term is 9. Glencoe/McGraw-Hill 5 Glencoe Algebra -7 Enrichment Arithmetic Series An arithmetic series is a series in which each term after the first ma be found b adding the same number to the preceding term. Let S stand for the following series in which each term is more than the preceding one. S The series remains the same if we S reverse the order of all the terms. So S let us reverse the order of the terms S and add one series to the other, term S 7() b term. This is shown at the right. Let a represent the first term of the series. Let represent the last term of the series. Let n represent the number of terms in the series. S 7() 77 In the preceding eample, a, l, and n 7. Notice that when ou add the two series, term b term, the sum of each pair of terms is. That sum can be found b adding the first and last terms, or a. Notice also that there are 7, or n, such sums. Therefore, the value of S is 7(), or n(a ) in the general case. Since this is twice the sum of the series, ou can use the following formula to find the sum of an arithmetic series. S n(a ) 7() Eample Find the sum: ( 9) 9 a, 9, n 9, so S 5 Eample Find the sum: 9 ( 5) ( ) 7 5 7( 9 5) 7 6 a 9, 5, n 7, so S Find the sum of each arithmetic series ( 6) ( ) ( 6) ( ) 9 8. even whole numbers from through odd whole numbers between and 5 Glencoe/McGraw-Hill 5 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra