Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS eBook plus Weblink Battleship Game e A bishop can move diagonally any 8 number of (unoccupied) squares at a time. Complete the following sentence: ‘With its next move, the bishop at b3 could capture the . . . . . . . . (name the chess piece) at . . . . . . . . (name the position).’ 7 The game of Battleship uses alphanumeric grids. Navigate to the Battleship weblink in your eBookPLUS. Play the game to test your skills with alphanumeric references. 7 6 5 4 3 2 REFLECTION When we specify an alphanumeric grid reference, does order matter? (That is, does it matter whether the letter is stated first and the number second, or the other way around?) 15B eBook plus eLesson Coordinates and the Cartesian plane eles-0008 1 a b c d e f g h The Cartesian plane ■ ■ ■ ■ ■ The Cartesian plane is named after its inventor, mathematician René Descartes. It is a visual means of describing locations on a plane by using two numbers as coordinates (rather than a letter and a number). The Cartesian plane is formed by two perpendicular lines. The horizontal line is called the x-axis, while the vertical line is referred to as the y-axis. The point where the two axes intersect is called the origin. Both axes must be marked (with marks being evenly spaced) y and numbered. The distance between each mark is one unit. 4 To locate any point on the Cartesian plane we use a pair of numbers called Cartesian coordinates. The numbers are written 3 in a set of brackets and are separated by a comma. The first number in brackets is called the x-coordinate of the point; this 2 shows how far across from the origin the point is located. The second number is called the y-coordinate; this shows how far up 1 (or down) from the origin the point is. 0 If the Cartesian coordinates of the point are known, it can be 1 2 3 4 5 6 x easily located by moving across and up (or down) from the origin the specified number of units. For example, to find the point with coordinates (2, 3), start from the origin and move 2 units to the right and 3 units up. Hint: To help remember the order in which Cartesian coordinates are measured, think about using a ladder. Remember we must always walk across with our ladder and then climb up it. y y 4 5 (2, 3) 3 4 2 3 1 2 0 1 2 3 4 x 1 0 1 2 Chapter 15 3 4 5 6 7 8 9 10 11 12 x Coordinates and the Cartesian plane 507 NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS WORKED EXAMPLE 3 Draw a Cartesian plane with axes extending from 0 to 6 units. Mark the following points with a dot, and label them. 1 a (2, 4) b (5, 0) c (0, 2) d (3 2 , 1) THINK DRAW 1 First rule up and label the axes. 2 Mark each point. a (2, 4) means starting at the origin, go across 2 units, and then up 4 units. b (5, 0) means go across 5 units and up 0 units. It lies on the x-axis. c d (0, 2) means go across 0 units and up 2 units. It lies on the y-axis. 1 y 5 (2, 4) 4 3 (3 1–2 , 1) 2 (0, 2) 1 0 (5, 0) 1 2 3 4 6 x 5 1 (3 2 , 1) means go across 3 2 units and up 1 unit. Label each point. WORKED EXAMPLE 4 Find the Cartesian coordinates for each of the points A, B, C and D. y 5 4 C B 3 THINK WRITE Point A is 3 units across and 1 unit up Point B is 1 unit across and 3 units up. Point C is 0 units across and 4 units up. 1 Point D is 1 unit across and 1 2 units up. A is at (3, 1) B is at (1, 3) C is at (0, 4) 1 D is at (1, 1 2) 2 D A 1 0 1 2 3 4 5 6 x Extending the axes ■ ■ ■ ■ 508 The Cartesian axes can extend infinitely in both directions. y On the x-axis, the values to the left of the origin are 6 negative and decreasing. Likewise, on the y-axis the 5 values below the origin are negative and decreasing. 4 1st quadrant 2nd quadrant The axes divide the Cartesian plane into four sections 3 called quadrants. The quadrants are numbered in 2 an anti-clockwise direction, starting with the top right 1 Origin corner. x 0 -6 -5 -4 -3 -2 -1 If both x- and y-coordinates of the point are positive, it -1 1 2 3 4 5 6 -2 will be located in the first quadrant; if the x-coordinate -3 is negative but the y-coordinate is positive, the point 3rd quadrant 4th quadrant -4 will be in the second quadrant. If the point is in the -5 third quadrant, both the x- and y-coordinates of the -6 point will be negative. Finally, if the point is in the fourth quadrant, its x-coordinate is positive, while its y-coordinate is negative. Maths Quest 7 for the Australian Curriculum NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS ■ If the point is located on the x-axis, its y-coordinate is always 0. Likewise, if the point is on the y-axis, its x-coordinate is always 0. WORKED EXAMPLE 5 Plot the following points on the Cartesian plane. A(-1, 2), B(2, -4), C(0, -3), D(4, 0), E(-5, -2) State the location of each point on the plane (that is, the quadrant, or the axis on which it sits). THINK WRITE 1 Draw a set of axes, ensuring that they are long enough to fit all the values. 2 Plot the points. The first point is one unit to the left and two units up from the origin. The second point is two units to the right and four units down from the origin (and so on). y 5 4 3 A 2 1 D x 0 −5 −4 −3 −2 −1 −1 1 2 3 4 5 −2 E −3 C −4 B −5 3 Look at the plane and state the location of each point. Remember that the quadrants are numbered in an anticlockwise direction, starting at the top right. If the point is on the axis, specify which axis it is. Point A is in the second quadrant. Point B is in the fourth quadrant. Point C is on the y-axis. Point D is on the x-axis. Point E is in the third quadrant. REMEMBER 1. Cartesian coordinates can be used to locate any point on a plane. 2. The Cartesian plane is formed by two perpendicular lines called axes. The horizontal axis is called the x-axis and the vertical axis is called the y-axis. The axes intersect at the point called the origin. y 3. Both axes must be marked (with marks being 6 evenly spaced) and numbered. The distance 5 between each mark is one unit. The axes can 4 1st quadrant 2nd quadrant 3 extend infinitely in both directions. 2 4. The location of any point on the Cartesian 1 Origin plane is given by its Cartesian coordinates. x The Cartesian coordinates are a pair of 0 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 numbers that are separated by a comma and -2 are shown within brackets. The first number is -3 3rd quadrant 4th quadrant called the x-coordinate of the point; this shows -4 how far across (that is, to the left or to the -5 right) from the origin the point is located. The -6 second number is called the y-coordinate; this shows how far up (or down) from the origin the point is. For example, the point (2, 3) is located 2 units to the right and 3 units up from the origin. Chapter 15 Coordinates and the Cartesian plane 509 NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS EXERCISE 15B INDIVIDUAL PATHWAYS eBook plus Activity 15-B-1 Introducing the Cartesian plane doc-2009 The Cartesian plane FLUENCY 1 WE3 Draw a Cartesian plane with axes extending from 0 to 5 units. Mark the following points with a dot, and label them. a (4, 3) b (1, 4) c (3, 3) d (2, 0) e (0, 4) f (0, 0) 2 WE4 Find the Cartesian coordinates for each of the points A–L. y Activity 15-B-2 D 10 More of the Cartesian plane doc-2010 L C 9 Activity 15-B-3 3D planes doc-2011 8 J 7 6 eBook plus Weblink The coordinate plane G B 5 I 4 3 K 2 A 1 0 H F 1 2 3 4 E 5 6 7 8 9 10 11 x 12 3 WE5 Plot the following points on the Cartesian plane. A(–1, –3), B(–2, 5), C(3, –3), D(0, –4), E(–2, –2), F(–5, 0), G(3, 1), H(3, 0), I(–4, –2), J(4, –5) State the location of each point on the plane (that is, the quadrant, or the axis it sits on). UNDERSTANDING 4 Each of these sets of Cartesian axes (except one) has something wrong with it. From the list below, match the mistake in each diagram with one of the sentences. A The units are not marked evenly. B The y-axis is not vertical. C The axes are labelled incorrectly. D The units are not marked on the axes. E There is nothing wrong. a b y 3 3 2 2 1 1 0 510 x 1 2 3 4 Maths Quest 7 for the Australian Curriculum x 0 1 2 3 4 y NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS c y 3 3 2 2 1 1 0 e d y 1 2 3 4 0 x f y 3 8 eBook plus Digital docs 9 4 x y 1 1 2 3 4 x 0 5 From the diagram at right, write down the coordinates 7 3 2 1 6 2 3 2 0 1 1 2 3 4 x y of 2 points which: 5 a have the same x-coordinate C 4 b have the same y-coordinate. 3 Messages can be sent in code using a grid like the one B D 2 drawn below, where the letter B is represented by the coordinates (2, 1). 1 A Use the diagram to decode the answer to the following E 0 riddle. 1 2 3 4 5 x Q Where did they put the man who was run over by a steamroller? y U V W X Y A (4, 2)(4, 3)(3, 2)(5, 3)(4, 4)(1, 4)(4, 2)(5, 4)(1, 1) 5 (2, 3)(4, 2)(4, 3)(3, 5)(1, 1)(3, 4)(4, 1)(4, 4)(4, 4)(4, 2) P Q R S T 4 (4, 5) (4, 4)(5, 1)(2, 5)(5, 1)(4, 3)(5, 1)(4, 2)(2, 2)(3, 2) K L M N O 3 (5, 4)(1, 1)(4, 3)(4, 1)(4, 3)(4, 2)(4, 3)(5, 1) F G H I J Rule up a Cartesian plane with both axes extending from 0 to 2 A B C D E 10 units. Plot the following points and join them in the order 1 given to make a geometric figure. Name each shape. 0 a (2, 2)–(5, 2)–(2, 6)–(2, 2) 1 2 3 4 5 x b (4, 4)–(8, 4)–(6, 8)–(4, 4) c (1, 1)–(10, 1)–(8, 9)–(2, 9)–(1, 1) d (0, 0)–(8, 0)–(10, 10)–(2, 10)–(0, 0) Here is an exercise which may require care and concentration. On graph paper or in your exercise book rule up a pair of Cartesian axes. The x-axis must go from 0 to 26 and the y-axis from 0 to 24. Plot the following points and join them in the order given. (0, 15)–(4, 17)–(9, 22)–(10, 21)–(12, 24)–(16, 22)–(15, 21)– (18, 19)–(20, 24)– (22, 18)–(26, 12)–(26, 10)–(23, 4)–(20, 3)–(18, 4)–(14, 7)–(11, 7)–(4, 6)–(2, 7)– (212 , 8)–(0, 15) N School Complete the picture by joining (19, 2)–(21, 2)–(20, 0)–(19, 2). What is the area of a rectangle formed by connecting the points (2, 1), (7, 1), (7, 4) and (2, 4) on a Cartesian plane? Spreadsheet Plotting points doc-0002 Home Chapter 15 Coordinates and the Cartesian plane 511 NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS 10 Consider the following set of points: A(2, 5) B(–4, –12) C(3, –7) D(0, –2) E(–10, 0) F(0, 0) G(–8, 15) H(–9, –24) I(18, –18) J(24, 0). Which of the following statements is true? a Points A and J are in the first quadrant. c Only point I is in the fourth quadrant. e Point F is at the origin. g Point D is two units to the left of point F. b Points B and H are in the third quadrant. d Only one point is in the second quadrant. f Point J is not on the same axis as point E. y 3 2 1 11 Consider the triangle ABC at right. a State the coordinates of the vertices of the triangle ABC. b Find the area of the triangle. c Reflect the triangle in the x-axis. (You need to copy it into your workbook first.) What are the new coordinates of the vertices? d Now reflect the triangle you have obtained in part c into the y-axis, and state the new coordinates of the vertices. 0 −5 −4 −3 −2 −1 −1 A C 1 2 3 4 5 x −2 −3 −4 REFLECTION eBook plus Why must the x-coordinate always be written first and the y-coordinate second? Digital docs WorkSHEET 15.1 doc-2004 15C B Plotting simple linear relationships ■ ■ ■ ■ When a set of points is plotted on the Cartesian plane, a pattern may be formed. If a pattern forms a straight line, we call it a linear pattern. The coordinates of the points that form a pattern can be presented as a set, or in a table. If shown in a table (similar to the one shown below), the coordinates of each point should be read ‘in columns’; that is, the top number in each column gives the x-coordinate and the bottom number gives the corresponding y-coordinate of the point. Consider, for example, the table of values and the set of points below. Both show the same information. x 0 1 2 3 y 8 6 4 2 (0, 8) (1, 6) (2, 4) (3, 2) The Cartesian coordinates of the points are ordered pairs. That is, the first number always represents the x-coordinate, and the second always represents the y-coordinate of a point. A set of ordered pairs forms a relation between x and y. If the points form a linear pattern when plotted, we say that the relation between x and y is linear. WORKED EXAMPLE 6 Plot the following set of points on the Cartesian plane, and comment on any pattern formed. (1, 3) (2, 4) (3, 5) (4, 6) (5, 7) THINK 1 512 Look at the coordinates of the points in the set: the x-values range between 1 and 5, while the y-values range between 3 and 7. Draw a set of axes, ensuring they are long enough to fit all the values. Maths Quest 7 for the Australian Curriculum WRITE