Straight line graphs PDF

9 CHAPTER Straight line graphs 3D printers and the number plane Number plane geometry is the basis for 3-dimensional After one full l...

9 CHAPTER Straight line graphs 3D printers and the number plane Number plane geometry is the basis for 3-dimensional After one full layer, the print platform moves printing. Imagine a virtual number plane with downward slightly and the computer directs the positive x - and y -axes drawn on the print platform nozzle where to plot the next layer. Each layer is of a 3D printer. Computer software tells the printer called a ‘slice’ and can be slightly different in shape head or nozzle how far across and how far up to from the previous slice. Thousands of slices slowly move on this 2D number plane. The nozzle draws build a 3D object. a line or curve forming a layer about 0.2 mm thick. Many different materials can be used in place of the Objects made using 3D printing include: artificial traditional ink, such as plastic, glass, metal, wood, body parts (e.g. hands, legs and ears); scale models rubber or wax. of buildings; vehicle, plane or rocket engine parts ISBN 978-1-108-77281-5 © Greenwood et al. 2019 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. Updated July 2021  Online resources A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more. In this chapter 9A The number plane (CONSOLIDATING) 9B Rules, tables and graphs 9C Finding the rule using tables 9D Using graphs to solve linear equations 9E The x- and y-intercepts 9F Gradient (EXTENDING) 9G Gradient–intercept form (EXTENDING) 9H Applications of straight line graphs 9I Non-linear graphs (EXTENDING) Australian Curriculum NUMBER AND ALGEBRA Linear and non-linear relationships Plot linear relationships on the Cartesian plane with and without the use of digital technologies (ACMNA193) Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution (ACMNA194) © ACARA including labels; jogging shoe samples testing colour and design features; smartphone cases; 3D maps; anatomy structures for a surgeon to study before a delicate operation (e.g. a patient’s skull); replica fossils; models of engineering projects; and movie characters. There are endless possibilities for the practical applications of mathematics. ISBN 978-1-108-77281-5 © Greenwood et al. 2019 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. Updated July 2021  586 Chapter 9 Straight line graphs 9A The number plane CONSOLIDATING Learning intentions To understand that coordinates can be used to describe locations in two-dimensional space on a number plane To know the location of the four quadrants of a number plane To be able to plot points at given coordinates On a number plane, a pair of coordinates gives the exact position of a point. The number plane is also called the Cartesian plane after its inventor, René Descartes, who lived in France in the 17th century. The number plane extends both a horizontal axis (x) and vertical axis (y) to include negative numbers. The point where these axes cross over is called the origin and it provides a reference point for all other points on the plane. CAD (computer-aided design) software uses a number plane with points located by their coordinates and straight lines modelled with linear equations. Architects, surveyors, engineers and industrial designers all use CAD. y LESSON STARTER Make the shape 4 In groups or as a class, see if you can remember how to plot 3 points on a number plane. Then decide what type of shape 2 is formed by each set. 1 A(0, 0), B(3, 1), C(0, 4) x −4 −3 −2 −1−1O 1 2 3 4 A(−2, 3), B(−2, −1), C(−1, −1), D(−1, 3) A(−3, −4), B(2, −4), C(0, −1), D(−1, −1) −2 Discuss the basic rules for plotting points on a number plane. −3 −4 KEY IDEAS A number plane (or Cartesian plane) includes a vertical y y-axis and a horizontal x-axis intersecting at right angles. There are 4 quadrants labelled as shown. 4 Quadrant 2 3 Quadrant 1 A point on a number plane has coordinates (x, y). 2 The x-coordinate is listed first followed by the 1 (0, 0) Origin y-coordinate. x −4 −3 −2 −1−1O 1 2 3 4 The point (0, 0) is called the origin (O). −2 horizontal vertical ( origin origin ) Quadrant 3 −3 Quadrant 4 (x, y) = units from , units from −4 ISBN 978-1-108-77281-5 © Greenwood et al. 2019 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. Updated July 2021  9A The number plane 587 BUILDING UNDERSTANDING 1 State the missing parts to complete these sentences. a The coordinates of the origin are ________. b The vertical axis is called the __-axis. c The quadrant that has positive coordinates for both x and y is the _______ quadrant. d The quadrant that has negative coordinates for both x and y is the _______ quadrant. e The point (−2, 3) has x-coordinate _______. f The point (1, −5) has y-coordinate _______. 2 State the missing number for the coordinates of the points a–h. y g(— , 3) a(3, —) 4 3 f (−3, —) 2 h (— , 2) 1 x −4 −3 −2 −1−1O 1 2 3 4 −2 b (3, —) e(— , −1) −3 c(1, —) −4 d (— , −4) 3 State the coordinates of the points labelled A to M. y 5 C 4 F D 3 2 G E 1 H A B x O −5 −4 −3 −2 −1−1 1 2 3 4 5 −2 I −3 K L −4 −5 J M ISBN 978-1-108-77281-5 © Greenwood et al. 2019 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. Updated July 2021  588 Chapter 9 Straight line graphs Example 1 Plotting points on a number plane Draw a number plane extending from −4 to 4 on both axes, and then plot and label these points. a A(2, 3) b B(0, 4) c C(−1, 2.5) d D(−3.5, 0) e E(−2, −2.5) f F(2, −4) SOLUTION EXPLANATION y The x-coordinate is listed first followed by the y-coordinate. 4 B For each point start at the origin (0, 0) and move 3 A C left or right or up and down to suit both x- and 2 y-coordinates. For point C(−1, 2.5), for example, 1 D move 1 to the left and 2.5 up. x −4 −3 −2 −1−1O 1 2 3 4 −2 E −3 −4 F Now you try Draw a number plane extending from −4 to 4 on both axes, and then plot and label these points. a A(4, 1) b B(0, 2) c C(−2, 3.5) d D(−2.5, 0) e E(−1, −3.5) f F(4, −3) Exercise 9A FLUENCY 1, 2–3(1/2) 2–3(1/2) 2–3(1/2) Example 1 1 Draw a number plane extending from −4 to 4 on both axes, and then plot and label these points. a A(2, 3) b B(0, 1) c C(−2, 1.5) d D(−2.5, 0) e E(−3, −3.5) f F(4, −3) Example 1 2 Draw a number plane extending from − 4 to 4 on both axes, and then the plot and label these points. a A(4, 1) b B(2, 3) c C(0, 1) d D(−1, 3) e E(−3, 3) f F(−2, 0) g G(−3, −1) h H(−1, − 4) i I(0, −2.5) j J(3.5, 3.5) k K(3.5, −1) l L(1.5, − 4) m M(−3.5, −3.5) n N(−3.5, 0.5) o O(0, 0) p P(2.5, −3.5) ISBN 978-1-108-77281-5 © Greenwood et al. 2019 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. Updated July 2021  9A The number plane 589 3 Using a scale extending from −5 to 5 on both axes, plot and then join the points for each part. Describe the basic picture formed. (2 ) (2 2) 1 1 1 a (−2, −2), (2, −2), (2, 2), (1, 3), (1, 4), ,4 , ,3 , (0, 4), (−2, 2), (−2, −2) b (2, 1), (0, 3), (−1, 3), (−3, 1), (−4, 1), (−5, 2), (−5, −2), (−4, −1), (−3, −1), (−1, −3), (0, −3), (2, −1), (1, 0), (2, 1) PROBLEM-SOLVING 4–5(1/2) 4–5(1/2) 5–6(1/2) 4 One point in each set is not ‘in line’ with the other points. Name the point in each case. a A(1, 2), B(2, 4), C(3, 4), D(4, 5), E(5, 6) b A(−5, 3), B(−4, 1), C(−3, 0), D(−2, −3), E(−1, −5) c A(−4, −3), B(−2, −2), C(0, −1), D(2, 0), E(3, 1) d A(6, −4), B(0, −1), C(4, −3), D(3, −2), E(−2, 0) 5 Each set of points forms a basic shape. Describe the shape without drawing a diagram if you can. a A(−2, 4), B(−1, −1), C(3, 0) b A(−3, 1), B(2, 1), C(2, −6), D(−3, −6) c A(−4, 2), B(3, 2), C(4, 0), D(−3, 0) d A(−1, 0), B(1, 3), C(3, 0), D(1, −9) 6 The midpoint of a line segment (or interval) is the point that cuts the segment in half. Find the midpoint of the line segment joining these pairs of points. a (1, 3) and (3, 5) b (−4, 1) and (−6, 3) c (−2, −3) and (0, −2) d (3, −5) and (6, −4) REASONING 7 7, 8 8, 9 7 List all the points, using only integer values of x and y that lie on the line segment joining these pairs of points. a (1, −3) and (1, 2) b (−2, 0) and (3, 0) c (−3, 4) and (2, −1) d (−3, −6) and (3, 12) 8 If (a, b) is a point on a number plane, name the quadrant or quadrants that matches the given description. a a > 0 and b < 0 b a < 0 and b > 0 c a