Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices 18 Curriculum Development Division Ministry of Education Malaysia 2. (b) Simplify the following. (i) 15 5 15 1 5 3 5 5 3 32 2 2 (2 )( ) x x x x (ii) 6 4 7 x y (iii) 3 2 12 w w (iv) 7 9 7 4y 8y (v) 2 8 6 9 5 9 36 p q p q (vi) 3. Simplify the following expressions: (a) 32 1 2 1 2 5 5 (b) 1 4 3 (c) 4 2 3y x (d) 1 5 4 6 2 s t st (e) 3 3 2 2 1 2m k m n (f) 2 3 6 2 3 2 8 a b ab c 4 3 2 4 2m n 3mn
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices 19 Curriculum Development Division Ministry of Education Malaysia 4. Find the value of each of the following. (a) 4 64 64 3 3 1 (b) 2 5 100 (c) 4 3 81 (d) 2 1 2 1 3 27 (e) m 1 3 2 m 5 1 1 0 a (a ) (a ) (f) 3 4 27 1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices 20 Curriculum Development Division Ministry of Education Malaysia 1. 2 5 10 4 4 4 P 0 4 O 3 4 R 17 4 T 13 4 2. 7 2 3 2 10 10 5 5 T 14 5 10 5 O 5 6 10 5 N 5 5 10 5 B 14 6 10 5 3. 2 2 2 4 2 3 D 4 2 2 E 2 2 2 3 N 2 2 4 3 O 3 4 2 4. y x y x 9 3 2 2 8 M 4 7 2 y x A 4 11 4 x y L 4 1 2 y x K 2 7 4 x y 5. 4 5 2 2 3 A 20 8 2 3 N 9 6 2 3 T 20 6 2 3 S 9 8 2 3 6. 5 2 2 4 m m n n T 7 8 m n U 10 8 m n L 7 6 m n E 10 6 m n Solve the questions to discover the WONDERWORD! You are given 11 multiple choice questions. Choose the correct answer for each of the question. Use the alphabets for each of the answer to form the WONDERWORD! ACTIVITY
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices 21 Curriculum Development Division Ministry of Education Malaysia 7. 3 4 2 3 5 2 5 2 5 2 5 2 F 12 5 2 A 2 5 2 V 6 5 2 E 5 5 2 8. 5 3 2 4 7 Y 15 10 4 7 R 8 7 4 7 M 8 10 4 7 A 15 7 4 7 9. 6 3 9 5 5 25 a b a b L 15 8 15a b I 3 8 5a b S 3 2 5a b T 6 5 15a b 10. 2 3 2 5 5 2 5 2 3 1 3 1 P 5 10 5 2 3 1 E 6 7 5 2 3 1 I 5 7 5 2 3 1 R 6 10 5 2 3 1 11. 3 2 6 7 3 12 p q p q Y 3 3 5 p q A 3 5 4p q R 9 9 3 1 p q D 9 9 3p q Congratulations! You have completed this activity. 1 2 3 4 5 6 7 8 9 10 11 The WONDERWORD IS: ........................................................
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices 22 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF A: 1. (a) 243 (b) 216 (c) 256 (d) 3125 1 (e) 64 27 (f) 25 21 4 (g) 2401 (h) 243 32 2. (a) 5 12m (b) 7 15b (c) 9 18x (d) 8 14p 3. (a) 576 (b) 288 (c) 823543 (d) 6075 16 (e) 250000 (f) 83349 256 ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices 23 Curriculum Development Division Ministry of Education Malaysia 4. (a) 4 2 12 f g (b) 5 2 54r s (c) 7 3 64827w v (d) 2 5 153125 144 h k TEST YOURSELF B: 1. (a) 144 (b) 531 441 (c) 262 144 (d) 729 64 (e) 25 (f) 81 2. (a) 7 q (b) 2 2 1 y (c) 2 3 7 m (d) 3 64b 3. (a) 5 4 2 9 m n (b) 10 6 3 16 c d (c) 3 6 2 f g (d) 7 3 14u v
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices 24 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF C: 1. (a) (b) 1 (c) 2401 64 (d) 15625 729 5 3 6 (e) 125 729 5 3 3 6 (f) 2. (a) (i) 8 2 3 24 (ii) 24 6 2 5 (iii) (iv) 2(5 ) 3 3 2 (v) 3 2 4 7(3 ) (vi) 2 6 14 5 3 (4 ) 2. (b) (i) 15 32x (ii) 24 42 x y (iii) 30 1 w (iv) 7 14 2 y (v) 2 16 q p (vi) 7 18 162m n 32768 2 16777216 24 11 4
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices 25 Curriculum Development Division Ministry of Education Malaysia 3. (a) 32 1 2 1 5 (b) 3 4 (c) 4 8 81 x y (d) 9 2 3 1 t s (e) 6 3 3 8k m n (f) 16 4 6 16 1 b a c 4. (a) 4 (b) 100000 (c) 27 1 (d) 9 (e) 5 a (f) 81 1 ACTIVITY: The WONDERWORD is ONEMALAYSIA
Unit 1: Negative Numbers UNIT 6 COORDINATES AND GRAPHS OF FUNCTIONS B a s i c E s s e n t i a l A d d i t i o n a l M a t h e m a t i c s S k i l l s Curriculum Development Division Ministry of Education Malaysia
TABLE OF CONTENTS Module Overview 1 Part A: Coordinates 2 Part A1: State the Coordinates of the Given Points 4 Activity A1 8 Part A2: Plot the Point on the Cartesian Plane Given Its Coordinates 9 Activity A2 13 Part B: Graphs of Functions 14 Part B1: Mark Numbers on the x-Axis and y-Axis Based on the Scales Given 16 Part B2: Draw Graph of a Function Given a Table for Values of x and y 20 Activity B1 23 Part B3: State the Values of x and y on the Axes 24 Part B4: State the Value of y Given the Value x from the Graph and Vice Versa 28 Activity B2 34 Answers 35
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 1 Curriculum Development Division Ministry of Education Malaysia MODULE OVERVIEW 1. The aim of this module is to reinforce pupils’ understanding of the concept of coordinates and graphs. 2. It is hoped that this module will provide a solid foundation for the studies of Additional Mathematics topics such as: Coordinate Geometry Linear Law Linear Programming Trigonometric Functions Statistics Vectors 3. Basically, this module is designed to enhance the pupils’ skills in: stating coordinates of points plotted on a Cartesian plane; plotting points on a Cartesian plane given the coordinates of the points; drawing graphs of functions on a Cartesian plane; and stating the y-coordinate given the x-coordinate of a point on a graph and vice versa. 4. This module consists of two parts. Part A deals with coordinates in two sections whereas Part B covers graphs of functions in four sections. Each section deals with one particular skill. This format provides the teacher with the freedom of choosing any section that is relevant to the skills to be reinforced. 5. Activities are also included to make the reinforcement of basic essential skills more enjoyable and meaningful.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 2 Curriculum Development Division Ministry of Education Malaysia LEARNING OBJECTIVES Upon completion of Part A, pupils will be able to: 1. state the coordinates of points plotted on a Cartesian plane; and 2. plot points on the Cartesian plane, given the coordinates of the points. PART A: COORDINATES TEACHING AND LEARNING STRATEGIES Some pupils may find difficulty in stating the coordinates of a point. The concept of negative coordinates is even more difficult for them to grasp. The reverse process of plotting a point given its coordinates is yet another problem area for some pupils. Strategy: Pupils at Form 4 level know what translation is. Capitalizing on this, the teacher can use the translation = , where O is the origin and P is a point on the Cartesian plane, to state the coordinates of P as (h, k). Likewise, given the coordinates of P as ( h , k ), the pupils can carry out the translation = to determine the position of P on the Cartesian plane. This common approach will definitely make the reinforcement of both the basic skills mentioned above much easier for the pupils. This approach of integrating coordinates with vectors will also give the pupils a head start in the topic of Vectors.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 3 Curriculum Development Division Ministry of Education Malaysia PART A: COORDINATES 1. 2. The translation must start from the origin O horizontally [left or right] and then vertically [up or down] to reach the point P. 3. The appropriate sign must be given to the components of the translation, h and k, as shown in the following table. Component Movement Sign h left – right + k up + down – 4. If there is no horizontal movement, the x-coordinate is 0. If there is no vertical movement, the y-coordinate is 0. 5. With this system, the coordinates of the Origin O are (0, 0). Coordinates of P = (h, k) Start from the origin. x y O ● P h units k units LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 4 Curriculum Development Division Ministry of Education Malaysia PART A1: State the coordinates of the given points. 1. Coordinates of A = (2, 3) 1. Coordinates of A = 2. Coordinates of B = (–3, 1) 2. Coordinates of B = 3. Coordinates of C = (–2, –2) 3. Coordinates of C = EXAMPLES TEST YOURSELF • A 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x Start from the origin, move 2 units to the right. Next, move 3 units up. • A 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x 4 3 2 1 -1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x Next, move 1 unit up. • B Start from the origin, move 3 units to the left. 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x • B 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x C• Start from the origin, move 2 units to the left. Next, move 2 units down. 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x C• EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 5 Curriculum Development Division Ministry of Education Malaysia PART A1: State the coordinates of the given points. 4. Coordinates of D = (4, –3) 4. Coordinates of D = 5. Coordinates of E = (3, 0) 5. Coordinates of E = 6. Coordinates of F = (0, 3) 6. Coordinates of F = EXAMPLES TEST YOURSELF 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x Start from the origin, move 4 units to the right. Next, move 3 units down. • D 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 •3 4 x E Start from the origin, move 3 units to the right. 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x Start from the origin, move 3 units up. • F 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x • D 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 •2 3 4 x E 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x • F EXAMPLES TEST YOURSELF Do not move along the y-axis since y = 0. Do not move along the x-axis since x = 0.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 6 Curriculum Development Division Ministry of Education Malaysia PART A1: State the coordinates of the given points. 7. Coordinates of G = (–2, 0) 7. Coordinates of G = 8. Coordinates of H = (0, –2) 8. Coordinates of H = 9. Coordinates of J = (6, 8) 9. Coordinates of J = EXAMPLES TEST YOURSELF 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x Start from the origin, move 2 units to the left. • G 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –• 1 0 1 2 3 4 x G 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x •H Start from the origin, move 2 units down. 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x •H 8 6 4 2 –2 –4 –6 –8 y –8 –6 –4 –2 0 2 4 6 8 x • J Start from the origin, move 6 units to the right. Next, move 8units up. 8 6 4 2 –2 –4 –6 –8 y –8 –6 –4 –2 0 2 4 6 8 x • J EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 7 Curriculum Development Division Ministry of Education Malaysia PART A1: State the coordinates of the given points. 10. Coordinates of K = (– 6 , 6) 10. Coordinates of K = 11. Coordinates of L = (–15, –20) 11. Coordinates of L = 12. Coordinates of M = (3, – 4) 12. Coordinates of M = 8 6 4 2 –2 –4 –6 –8 y –8 –6 –4 –2 0 2 4 6 8 x Start from the origin, move 6 units to the left. • K Next, move 6 units up. 8 6 4 2 –2 –4 –6 –8 y –8 –6 –4 –2 0 2 4 6 8 x K• 20 15 10 5 –5 –10 –15 –20 y –20 –15 –10 –5 0 5 10 15 20 x • L Next, move 20 units down. Start from the origin, move 15 units to the left. 20 15 10 5 –5 –10 –15 –20 y –20 –15 –10 –5 0 5 10 15 20 x •L M 4 2 –2 –4 y –4 –2 0 2 4 x • Start from the origin, move 3 units to the right. Next, move 4 units down. 4 2 –2 –4 y –4 –2 0 2 4 x • M EXAMPLES TEST YOURSELF EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 8 Curriculum Development Division Ministry of Education Malaysia Write the step by step directions involving integer coordinates that will get the mouse through the maze to the cheese. –6 –5 –4 –3 –2 –1 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 0 y 1 2 3 4 5 6 7 x ACTIVITY A1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 9 Curriculum Development Division Ministry of Education Malaysia PART A2: Plot the point on the Cartesian plane given its coordinates. . 1. Plot point A (3, 4) 1. Plot point A (2, 3) 2. Plot point B (–2, 3) 2. Plot point B (–3, 4) 3. Plot point C (–1, –3) 3. Plot point C (–1, –2) 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 -1 0 1 2 3 4 x 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x EXAMPLES TEST YOURSELF 4 3 2 1 –1 –2 –3 –4 • y A –4 –3 –2 –1 0 1 2 3 4 x 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x • B 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x C• EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 10 Curriculum Development Division Ministry of Education Malaysia PART A2: Plot the point on the Cartesian plane given the coordinates. . 4. Plot point D (2, – 4) 4. Plot point D (1, –3) 5. Plot point E (1, 0) 5. Plot point E (2, 0) 6. Plot point F (0, 4) 6. Plot point F (0, 3) 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x EXAMPLES TEST YOURSELF 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x • D 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 • 1 2 3 4 x E 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x • F EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 11 Curriculum Development Division Ministry of Education Malaysia PART A2: Plot the point on the Cartesian plane given the coordinates. . 7. Plot point G (–2, 0) 7. Plot point G (– 4,0) 8. Plot point H (0, – 4) 8. Plot point H (0, –2) 9. Plot point J (6, 4) 9. Plot point J (8, 6) EXAMPLES TEST YOURSELF 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x 4 3 2 1 –1 –2 –3 –4 y –4 –3 –• 2 –1 0 1 2 3 4 x G 8 6 4 2 –2 –4 –6 –8 y –8 –6 –4 –2 0 2 4 6 8 x • J 8 6 4 2 –2 –4 –6 –8 y –8 –6 –4 –2 0 2 4 6 8 x 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x •H EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 12 Curriculum Development Division Ministry of Education Malaysia PART A2: Plot the point on the Cartesian plane given the coordinates. . 10. Plot point K (– 4, 6) 10. Plot point K (– 6, 2) 11. Plot point L (–15, –10) 11. Plot point L (–20, –5) 12. Plot point M (30, –15) 12. Plot point M (10, –25) 29 10 –10 –20 y –20 –10 0 10 20 x •L EXAMPLES TEST YOURSELF 8 4 –4 –8 y –8 –4 0 4 8 x • K 8 4 –4 –8 y -8 -4 0 4 8 x –20 –10 0 10 20 20 10 –10 –20 y x 20 10 –10 –20 y –40 –20 0 20 40 x 20 10 –10 –20 y –40 –20 0 20 40 x •M EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 13 Curriculum Development Division Ministry of Education Malaysia 1. Plot the following points on the Cartesian plane. P(3, 3) , Q(6, 3) , R(3, 1) , S(6, 1) , T(6, –2) , U(3, –2) , A(–3, 3) , B(–5, –1) , C(–2, –1) , D(–3, – 2) , E(1, 1) , F(2, 1). 2. Draw the following line segments: AB, AD, BC, EF, PQ, PR, RS, UT, ST YAKOMI ISLANDS –4 –2 2 4 x 2 4 –2 y 0 –4 , Exclusive News: A group of robbers stole RM 1 million from a bank. They hid the money somewhere near the Yakomi Islands. As an expert in treasure hunting, you are required to locate the money! Carry out the following tasks to get the clue to the location of the money. Mark the location with the symbol. Enjoy yourself ! ACTIVITY A2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 14 Curriculum Development Division Ministry of Education Malaysia LEARNING OBJECTIVES Upon completion of Part B, pupils will be able to: 1. understand and use the concept of scales for the coordinate axes; 2. draw graphs of functions; and 3. state the y-coordinate given the x-coordinate of a point on a graph and vice versa. PART B: GRAPHS OF FUNCTIONS TEACHING AND LEARNING STRATEGIES Drawing a graph on the graph paper is a challenge to some pupils. The concept of scales used on both the x-axis and y-axis is equally difficult. Stating the coordinates of points lying on a particular graph drawn is yet another problematic area. Strategy: Before a proper graph can be drawn, pupils need to know how to mark numbers on the number line, specifically both the axes, given the scales to be used. Practice makes perfect. Thus, basic skill practices in this area are given in Part B1. Combining this basic skills with the knowledge of plotting points on the Cartesian plane, the skill of drawing graphs of functions, given the values of x and y, is then further enhanced in Part B2. Using a similar strategy, Stating the values of numbers on the axes is done in Part B3 followed by Stating coordinates of points on a graph in Part B4. For both the skills mentioned above, only the common scales used in the drawing of graphs are considered.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 15 Curriculum Development Division Ministry of Education Malaysia PART B: GRAPHS OF FUNCTIONS 1. For a standard graph paper, 2 cm is represented by 10 small squares. 2. Some common scales used are as follows: Scale Note 2 cm to 10 units 10 small squares represent 10 units 1 small square represents 1 unit 2 cm to 5 units 10 small squares represent 5 units 1 small square represents 0.5 unit 2 cm to 2 units 10 small squares represent 2 units 1 small square represents 0.2 unit 2 cm to 1 unit 10 small squares represent 1 unit 1 small square represents 0.1 unit 2 cm to 0.1 unit 10 small squares represent 0.1 unit 1 small square represents 0.01 unit 2 cm 2 cm LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 16 Curriculum Development Division Ministry of Education Malaysia PART B1: Mark numbers on the x-axis and y-axis based on the scales given. 1. Mark – 4. 7, 16 and 27on the x-axis. Scale: 2 cm to 10 units. [ 1 small square represents 1 unit ] 1. Mark – 6 4, 15 and 26 on the x-axis. Scale: 2 cm to 10 units. [ 1 small square represents 1 unit ] 2. Mark –7, –2, 3 and 8on the x-axis. Scale: 2 cm to 5 units. [ 1 small square represents 0.5 unit ] 2. Mark –8, –3, 2 and 6, on the x-axis. Scale: 2 cm to 5 units. [ 1 small square represents 0.5 unit ] 3. Mark –3.4, – 0.8, 1 and 2.6, on the x-axis. Scale: 2 cm to 2 units. [ 1 small square represents 0.2 unit ] 3. Mark –3.2, –1, 1.2 and 2.8 on the x-axis. Scale: 2 cm to 2 units. [ 1 small square represents 0.2 unit ] 4. Mark –1.3, – 0.6, 0.5 and 1.6 on the x-axis. Scale: 2 cm to 1 unit. [ 1 small square represents 0.1 unit ] 4. Mark –1.7, – 0.7, 0.7 and 1.5 on the x-axis. Scale: 2 cm to 1 unit. [ 1 small square represents 0.1 unit ] –10 0 10 20 30 x –4 7 16 27 x x –10 –5 0 5 10 x –7 –2 3 8 x –4 –2 2 4 x –3.4 –0.8 0 1 2.6 x –2 – 1 1 2 x –1.3 –0.6 0 0.5 1.6 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 17 Curriculum Development Division Ministry of Education Malaysia PART B1: Mark numbers on the x-axis and y-axis based on the scales given. 5. Mark – 0.15, – 0.04, 0.03 and 0.17 on the x-axis. Scale: 2 cm to 0.1 unit [ 1 small square represents 0.01 unit ] 5. Mark – 0.17, – 0.06, 0.04 and 0.13 on the x-axis. Scale: 2 cm to 0.1 unit [ 1 small square represents 0.01 unit ] 6. Mark –13, –8, 2 and 14 on the y-axis. Scale: 2 cm to 10 units [ 1 small square represents 1 unit ] 6. Mark –16, – 4, 5 and 15 on the y-axis. Scale: 2 cm to 10 units [ 1 small square represents 1 unit ] x –0.2 –0.15 –0.1 –0.04 0 0.03 0.1 0.17 0.2 x y y 0 10 –20 20 –10 –13 –8 2 14 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 18 Curriculum Development Division Ministry of Education Malaysia PART B1: Mark numbers on the x-axis and y-axis based on the scales given. 7. Mark –9, –3, 1 and 7 on the y-axis. Scale: 2 cm to 5 units. [ 1 small square represents 0.5 unit ] 7. Mark –7, – 4, 2 and 6 on the y-axis. Scale: 2 cm to 5 units. [ 1 small square represents 0.5 unit ] 8. Mark –3.2, – 0.6, 1.4 and 2.4 on the y-axis. Scale: 2 cm to 2 units. [ 1 small square represents 0.2 unit ] 8. Mark –3.4, –1.4, 0.8 and 2.8 on the y-axis. Scale: 2 cm to 2 units. [ 1 small square represents 0.2 unit ] y y y 0 5 –10 10 –9 –3 1 7 –5 y 0 –4 4 –2 –3.2 –0.6 2 1.4 2.4 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 19 Curriculum Development Division Ministry of Education Malaysia PART B1: Mark numbers on the x-axis and y-axis based on the scales given. 9. Mark –1.6, – 0.4, 0.4 and 1.5 on the y-axis. Scale: 2 cm to 1 unit. [ 1 small square represents 0.1 unit ] 9. Mark –1.5, – 0.8, 0.3 and 1.7 on the y-axis. Scale: 2 cm to 1 unit. [ 1 small square represents 0.1 unit ] 10. Mark – 0.17, – 0.06, 0.08 and 0.16 on the y-axis. Scale: 2 cm to 0.1 unit. [ 1 small square represents 0.01 unit ] 10. Mark – 0.18, – 0.03, 0.05 and 0.14 on the y-axis. Scale: 2 cm to 0.1 units. [ 1 small square represents 0.01 unit ] y y y 0 1 –2 2 –1 0.4 1.5 – 0.4 –1.6 y 0 0.2 – 0.17 –0.1 – 0.06 0.1 0.08 0.16 –0.2 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 20 Curriculum Development Division Ministry of Education Malaysia PART B2: Draw graph of a function given a table for values of x and y. 1. The table shows some values of two variables, x and y, of a function. x –2 –1 0 1 2 y –2 0 2 4 6 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of the function. 1. The table shows some values of two variables, x and y, of a function. x –3 –2 –1 0 1 y –2 0 2 4 6 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of the function. 2. The table shows some values of two variables, x and y, of a function. x –2 –1 0 1 2 y 5 3 1 –1 –3 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of the function. 2. The table shows some values of two variables, x and y, of a function. x –2 –1 0 1 2 y 7 5 3 1 –1 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of the function. –1 1 x –2 2 –2 6 4 2 y 0 –1 1 x –2 2 –2 6 4 2 y 0 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 21 Curriculum Development Division Ministry of Education Malaysia PART B2: Draw graph of a function given a table for values of x and y. 3. The table shows some values of two variables, x and y, of a function. x – 4 –3 –2 –1 0 1 2 y 15 5 –1 –3 –1 5 15 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of the function. 3. The table shows some values of two variables, x and y, of a function. x –1 0 1 2 3 4 5 y 19 4 –5 –8 –5 4 19 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of the function. 4. The table shows some values of two variables, x and y, of a function. x –2 –1 0 1 2 3 4 y –7 –2 1 2 1 –2 –7 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of the function. 4. The table shows some values of two variables, x and y, of a function. x –2 –1 0 1 2 3 y –8 –4 –2 –2 – 4 –8 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of the function. y 10 5 15 –5 x –4 –3 1 2 0 –2 –1 0 y 2 –6 –2 –4 x –2 –1 1 2 3 4 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 22 Curriculum Development Division Ministry of Education Malaysia PART B2: Draw graph of a function given a table for values of x and y. 5. The table shows some values of two variables, x and y, of a function. x –2 –1 0 1 2 y –7 –1 1 3 11 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of the function. 5. The table shows some values of two variables, x and y, of a function. x –2 –1 0 1 2 y –6 2 4 6 16 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of the function. 6. The table shows some values of two variables, x and y, of a function. x –3 –2 –1 0 1 2 3 y 22 5 0 1 2 –3 –20 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 10 units on the y-axis, draw the graph of the function. 6. The table shows some values of two variables, x and y, of a function. x –3 –2 –1 0 1 2 3 y 21 4 –1 0 1 – 4 –21 By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 10 units on the y-axis, draw the graph of the function. x –3 –2 –1 0 1 2 3 y 20 –20 –10 10 y 10 5 15 –5 x –2 –1 1 2 0 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 23 Curriculum Development Division Ministry of Education Malaysia Each table below shows the values of x and y for a certain function. The graphs of all these functions, when drawn on the same axes, form a beautiful logo. Draw the logo on the graph paper provided by using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis. FUNCTION 1 FUNCTION 2 x – 4 –3 –2 –1 0 x 0 1 2 3 4 y 16 17 18 19 20 y 20 19 18 17 16 FUNCTION 3 x – 4 –3 –2 –1 0 1 2 3 4 y 16 9 4 1 0 1 4 9 16 FUNCTION 4 x –3 –2 –1 0 1 2 3 y 9 14 17 18 17 14 9 FUNCTION 5 x –3 –2 –1.5 –1 – 0.5 0 y 9 8 7.9 7 4.6 0 FUNCTION 6 x 0 0.5 1 1.5 2 3 y 0 4.6 7 7.9 8 9 x y 0 ACTIVITY B1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 24 Curriculum Development Division Ministry of Education Malaysia PART B3: State the values of x and y on the axes. 1. State the values of a, b, c and d on the x-axis below. Scale: 2 cm to 10 units. [ 1 small square represents 1 unit ] a = 7, b = 13, c = – 4, d = –14 1. State the values of a, b, c and d on the x-axis below. 2. State the values of a, b, c and d on the x-axis below. Scale: 2 cm to 5 units. [ 1 small square represents 0.5 unit ] a = 2, b = 7.5, c = –3, d = –8.5 2. State the values of a, b, c and d on the x-axis below. 3. State the values of a, b, c and d on the x-axis below. Scale: 2 cm to 2 units. [ 1 small square represents 0.2 unit ] a = 0.6, b = 3.4, c = –1.2, d = –2.6 3. State the values of a, b, c and d on the x-axis below. –20 10 20 x d –10 c 0 a b –20 10 20 x d –10 c 0 a b –10 –5 0 5 10 x d c a b –10 –5 0 5 10 x d c a b – 4 –2 c 2 4 x d 0 a b – 4 – 2 c 2 4 x d 0 a b EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 25 Curriculum Development Division Ministry of Education Malaysia PART B3: State the values of x and y on the axes. 4. State the values of a, b, c and d on the x-axis below. Scale: 2 cm to 1 unit. [ 1 small square represents 0.1 unit ] a = 0.8, b = 1.4, c = – 0.3, d = –1.6 4. State the values of a, b, c and d on the x-axis below. 5. State the values of a, b, c and d on the x-axis below. Scale: 2 cm to 0.1 unit. [ 1 small square represents 0.01 unit ] a = 0.04, b = 0.14, c = – 0.03, d = – 0.16 5. State the values of a, b, c and d on the x-axis below. 6. State the values of a, b, c and d on the y-axis below. Scale: 2 cm to 10 units. [ 1 small square represents 1 unit ] a = 3, b = 17 c = – 6, d = –15 6. State the values of a, b, c and d on the y-axis below. –2 –1 1 2 x d c 0 a b –2 –1 1 2 x d c 0 a b c x –0.2 d –0.1 0 a 0.1 b 0.2 c x – 0.2 d –0.1 0 a 0.1 b 0.2 y 0 10 –20 20 –10 d c a b y 0 10 –20 20 –10 d c a b EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 26 Curriculum Development Division Ministry of Education Malaysia PART B3: State the values of x and y on the axes. 7. State the values of a, b, c and d on the y-axis below. Scale: 2 cm to 5 units. [ 1 small square represents 0.5 unit ] a = 4, b = 9.5 c = –2, d = –7.5 7. State the values of a, b, c and d on the y-axis below. 8. State the values of a, b, c and d on the y-axis below. Scale: 2 cm to 2 units. [ 1 small square represents 0.2 unit ] a = 0.8, b = 3.2 c = –1.2, d = –2.6 8. State the values of a, b, c and d on the y-axis below. y 0 5 –10 10 d c a b –5 y 0 5 –10 10 d c a b –5 y 0 –4 4 –2 d c 2 a b y 0 –4 4 –2 d c 2 a b EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 27 Curriculum Development Division Ministry of Education Malaysia PART B3: State the values of x and y on the axes. 9. State the values of a, b, c and d on the y-axis below. Scale: 2 cm to 1 unit. [ 1 small square represents 0.1 unit ] a = 0.7, b = 1.2 c = – 0.6, d = –1.4 9. State the values of a, b, c and d on the y-axis below. 10. State the values of a, b, c and d on the y-axis below. Scale: 2 cm to 0.1 unit. [ 1 small square represents 0.01 unit ] a = 0.03, b = 0.07 c = – 0.04, d = – 0.18 10. State the values of a, b, c and d on the y-axis below. y 0 1 –2 2 –1 a b c d y 0 1 –2 2 –1 a b c d y 0 d –0.1 c a –0.2 0.2 b 0.1 y 0 d c a –0.2 0.2 b 0.1 –0.1 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 28 Curriculum Development Division Ministry of Education Malaysia PART B4: State the value of y given the value x from the graph and vice versa. 1. Based on the graph below, find the value of y when (a) x = 1.5 (b) x = –2.8 (a) 7 (b) –1.6 1. Based on the graph below, find the value of y when (a) x = 0.6 (b) x = –1.7 (a) (b) 2. Based on the graph below, find the value of y when ( a ) x = 0.14 ( b ) x = – 0.26 (a) 1.5 (b) 11.5 2. Based on the graph below, find the value of y when ( a ) x = 0.07 ( b ) x = – 0.18 (a) (b) –1 1 x –2 2 –2 6 4 2 y 0 – 2.8 1.5 7 – 1.6 –1 1 x –2 2 –2 6 4 2 y 0 – 0.26 1.5 0.14 11.5 x – 0. 2 –0.1 0.1 0.2 y 10 –10 5 –5 0 x –0. 2 –0.1 0.1 0.2 y 10 –10 5 – 5 0 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 29 Curriculum Development Division Ministry of Education Malaysia PART B4: State the value of y given the value x from the graph and vice versa. 3. Based on the graph below, find the value of y when ( a ) x = 0.6 ( b ) x = –2.7 ( a ) 11 ( b ) –3.5 3. Based on the graph below, find the value of y when ( a ) x = 1.2 ( b ) x = –1.8 ( a ) ( b ) 4. Based on the graph below, find the value of y when (a) x = 1.4 (b) x = –1.5 (a) 3 (b) –5.8 4. Based on the graph below, find the value of y when (a) x = 2.7 (b) x = –2.1 (a) (b) y 10 5 15 –5 x – 4 –3 1 2 0 –2 –1 11 0.6 – 2.7 – 3.5 y 10 5 15 –5 x – 4 –3 1 2 0 –2 –1 x –2 –1 1 2 3 4 0 y 2 – 6 – 2 – 4 1.4 3 – 1.5 – 5.8 x –2 –1 1 2 3 4 0 y 2 – 6 – 2 – 4 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 30 Curriculum Development Division Ministry of Education Malaysia PART B4: State the value of y given the value x from the graph and vice versa. 5. Based on the graph below, find the value of y when (a) x = 1.7 (b) x = –1.3 (a) 5.5 (b) –3.5 5. Based on the graph below, find the value of y when (a) x = 1.2 (b) x = –1.9 (a) (b) 6. Based on the graph below, find the value of y when (a) x = 1.6 (b) x = –2.3 (a) –9 (b) 25 6. Based on the graph below, find the value of y when (a) x = 2.8 (b) x = –2.6 (a) (b) y 10 5 15 –5 –2 x –1 1 2 0 5.5 1.7 – 1.3 – 3.5 y 10 5 15 –5 –2 x –1 1 2 0 x –3 –2 –1 0 1 2 3 y 20 –20 –10 10 1.6 – 9 – 2.3 25 x –3 –2 –1 0 1 2 3 y 20 –20 –10 10 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 31 Curriculum Development Division Ministry of Education Malaysia PART B4: State the value of y given the value x from the graph and vice versa. 7. Based on the graph below, find the value of x when (a) y = 5.4 (b) y = –1.6 (a) 1.4 (b) –2.8 7. Based on the graph below, find the value of x when (a) y = 2.8 (b) y = –2.4 (a) (b) 8. Based on the graph below, find the value of x when ( a ) y = 4 ( b ) y = –7.5 (a) – 0.07 (b) 0.08 8. Based on the graph below, find the value of x when ( a ) y = 6.5 ( b ) y = –7 (a) (b) –1 1 x –2 2 –2 6 4 2 y 0 x –0. 2 –0.1 0.1 0.2 y 10 –10 5 – 5 0 –1 1 x –2 2 –2 6 4 2 y 0 – 2.8 1.4 5.4 – 1.6 – 0.07 4 0.08 – 7.5 x –0. 2 –0.1 0.1 0.2 y 10 –10 5 –5 0 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 32 Curriculum Development Division Ministry of Education Malaysia PART B4: State the value of y given the value x from the graph and vice versa. 9. Based on the graph below, find the values of x when (a) y = 8.5 (b) y = 0 (a) –3.1 , 2.1 (b) –2 , 1 9. Based on the graph below, find the values of x when (a) y = 3.5 (b) y = 0 (a) (b) 10. Based on the graph below, find the values of x when (a) y = 2.6 (b) y = – 4.8 (a) 0.6 , 2.1 (b) –1.2 , 3.9 10. Based on the graph below, find the values of x when (a) y = 1.2 (b) y = – 4.4 (a) (b) x –2 –1 1 2 3 4 0 y 2 – 6 – 2 – 4 x – 4 –3 –2 –1 1 2 – 3.1 2.1 8.5 0 y 10 5 15 –5 x –2 –1 1 2 3 4 0 y 2 – 6 – 2 – 4 0.6 2.1 – 1.2 3.9 2.6 – 4.8 x – 4 –3 –2 –1 1 2 0 y 10 5 15 –5 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 33 Curriculum Development Division Ministry of Education Malaysia PART B4: State the value of y given the value x from the graph and vice versa. 11. Based on the graph below, find the value of x when (a) y = 14 (b) y = –17 (a) 2.6 (b) –2.3 11. Based on the graph below, find the value of x when (a) y = 11 (b) y = –23 (a) (b) 12. Based on the graph below, find the value of x when (a) y = 6.5 (b) y = 0 (c) y = –6 (a) – 0.8 (b) 1.3 (c) 2.3 12. Based on the graph below, find the value of x when (a) y = 7.5 (b ) y = 0 (c) y = –9 (a) (b) (c) x –3 –2 –1 0 1 2 3 y 20 –20 –10 10 2.6 – 2.3 – 17 14 x –3 –2 –1 0 1 2 3 y 20 –20 –10 10 y 10 5 15 –5 –2 x –1 1 2 0 y 10 5 15 –5 –2 x –1 1 2 0 6.5 – 6 – 0.8 1.3 2.3 EXAMPLES TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 34 Curriculum Development Division Ministry of Education Malaysia Task 1: Two points on the graph given are (6.5, k) and (h, 45). Find the values of h and k. Task 2: Smuggling takes place at the locations with coordinates (h, k). State each location in terms of coordinates. 0 5 10 15 20 25 30 35 40 45 50 55 60 y 1 2 3 4 5 6 7 8 9 x There is smuggling at sea and you know two possible locations. As a responsible citizen, you need to report to the marine police these two locations. ACTIVITY B2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 35 Curriculum Development Division Ministry of Education Malaysia PART A: PART A1: 1. A (4, 2) 2. B (– 4, 3) 2. 3. C (–3, –3) 4. D (3, – 4) 5. E (2, 0) 6. F (0, 2) 7. G (–1, 0) 8. H (0, –1) 9. J (8, 6) 10. K (– 4, 8) 11. L (–10, –15) 12. M (4, –3) ACTIVITY A1: Start at (5, 3). Then, move in order to (4, 3), (4, –3), (3, –3), (3, 2), (1, 2) , (1, –3) , (–3, –3) , (–3, 3), (– 4, 3), (– 4, 5), (–3, 5) and (–3, 6). ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 36 Curriculum Development Division Ministry of Education Malaysia PART A2: 1. 4. 2. 5. 3. 6. 4 3 2 1 –1 –2 –3 -–4 –4 –3 –2 –1 0 1 2 3 4 y x • B 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x • A 4 3 2 1 –1 –2 –3 –4 –4 –3 –2 –1 0 1 2 3 4 y x • D 4 3 2 1 –1 –2 –3 –4 –4 –3 –2 –1 0 1 2 3 4 y • x E 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x C• 4 3 2 1 –1 –2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x • F
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 37 Curriculum Development Division Ministry of Education Malaysia 7. 10. 8. 11. 9. 12. 4 3 2 1 –1 –2 –3 –4 y –•4 –3 –2 –1 0 1 2 3 4 x G • K 8 4 –4 –8 y –8 –4 0 4 8 x 4 3 2 1 –1 -2 –3 –4 y –4 –3 –2 –1 0 1 2 3 4 x – • H –20 –10 0 10 20 20 10 –10 –20 y x •L 8 6 4 2 –2 –4 –6 –8 y –8 –6 –4 –2 0 2 4 6 8 x • J • M 20 10 –10 –20 y –40 –20 0 20 40 x
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 38 Curriculum Development Division Ministry of Education Malaysia ACTIVITY A2: YAKOMI ISLANDS 2 4 –2 y O –4 RM 1 million U A B C D E F P Q R S T –4 –2 2 4 x ,
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 39 Curriculum Development Division Ministry of Education Malaysia PART B1: 1 2. 3. 4. 5. 6. 7. 8. 9. 10. –10 0 10 20 30 x –6 4 15 26 –10 –5 0 5 10 x –8 –3 2 6 –4 –2 2 4 x –3.2 –1 0 1.2 2.8 –2 –1 1 2 x –1.7 –0.7 0 0.7 1.5 x –0.2 –0.16 –0.1 –0.06 0 0.04 0.1 0.13 0.2 y 0 10 –20 20 –10 –16 –4 5 15 y 0 5 –10 10 –7 –4 2 6 –5 y 0 1 –2 2 –1 0.3 1.7 –0.8 –1.5 y 0 0.2 – 0.18 – 0.1 – 0.03 0.1 0.05 0.14 – 0.2 y 0 –4 4 –2 –3.4 –1.4 2 0.8 2.8
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions 40 Curriculum Development Division Ministry of Education Malaysia PART B2: 1. 2. 3. 4. 5. 6. –2 6 4 2 y 0 x –3 –2 –1 1 –1 1 x –2 2 –2 6 4 2 y 0 x –1 1 4 5 0 y 10 5 15 –5 2 3 y –4 –8 –2 –6 0 x –2 –1 1 2 3 y 10 5 15 –5 –2 x –1 1 2 0 x –3 –2 – 1 0 1 2 3 y 20 –20 –10 10