MAL SHIELD. Student Textbook

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1 MAL SHIELD Student Textbook

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3 Queensland GO Maths Student Textbook Level /6 (Year 9) Copyright 008 ORIGO Education Author: Mal Shield Consultant: Kathy Blum Shield, M. J. (Malcolm John). Go maths : level /6. Student textbook. Includes index. For secondary school students. ISBN (pbk.). 1. Mathematics - Textbooks. I. Title. 10 For more information, info@origo.com.au or visit for other contact details. All rights reserved. Unless specifically stated, no part of this publication may be reproduced, copied into, or stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of ORIGO Education. ISBN: Acknowledgement The author and project team would like to acknowledge the Queensland Studies Authority, Brisbane, for its permission to reproduce extracts from the Mathematics: Years 1 to 10 Syllabus. Sections of the topographic map Pialba 9447 and their map keys are Copyright Commonwealth of Australia, Geoscience Australia. All rights reserved. Reproduced by permission of the Chief Executive Officer, Geoscience Australia, Canberra, ACT. Apart from any use as permitted under the Copyright Act 1968, no part of the map sections or the map keys may be reproduced by any process without prior written permission from Geoscience Australia. Requests and inquiries concerning reproduction and rights should be addressed to the Manager Copyright, Geoscience Australia, GPO Box 378, Canberra ACT 601, or by to copyright@ga.gov.au Graphics calculator instructions in this textbook are based on the models CASIO CFX-980GC PLUS and TI-83 Plus, and may therefore vary for different models. When completing the exercises and explorations, you may also need to refer to the instruction manual for the particular graphics calculator you are using. Every effort was made to correctly publish any information regarding computer software, calculators and websites. Any instructions for computer software and calculators are based on the specific version or model that was used to compile the instructions, and may differ for other versions and models. Any instructions for websites are based on the websites at the time of publication. ORIGO Education does not endorse any brand of computer software or calculator.

4 Contents Unit Number Unit Topic Major Outcomes Page 1 Working with Data CD. / CD 6. 1 Numbers and Operations N.1 / N. / N.3 / N 6.1 / N 6. / N Modelling with Functions PA.1 / PA Proportions N.1 / N.3 / N Working with D and 3D Shapes S.1 / S. / S Perimeter, Area, Volume and Capacity M.1 / M Time and Global Position M. / M 6. / S Working with Chance CD.1 / CD Living with Money N Lines, Angles and Shapes S.1 / S Working with Maps S. / S Chance, Data and Relationships CD 6.1 / CD Working with Triangles S.1 / S 6.1 / M Using and Interpreting Mathematical Models PA 6.1 / PA Surface Area, Volume and Capacity M 6.1 / S Working with Algebraic Symbols PA 6.1 / PA Answers 36 Index 30

5 Using and Interpreting Mathematical Models 14 PA 6.1 / PA 6. Linear Equations 1 Find the gradient of the line in this graph. Distance (m) Time (min) Solution Find the gradient using rise run. Choose points on the line that make finding the rise and run easy. The line starts at the point 0 min and 10 m (0, 10) and goes through the point 8 min and 600 m (8, 600). Run = 8 0 = 8 min Rise = = 40 m Gradient = 40 8 = 6. m/min Distance (m) run = 8 0 rise = Time (min) The gradient shows a speed (rate) of 6. metres per minute. 66 GO Maths Unit 14

6 Use the gradient and y-intercept to draw the graph of the function t = 0.p + 3. Solution This is a linear function. The gradient is 0., which means a rise of 0. for a run of 1, or a rise of 1 for a run of. The y-intercept is 3, which means that the line cuts the vertical axis at +3. t rise = 1 run = p 1 1. Find the gradient of the line in each of these graphs. a. b. c. Distance (km) Volume (L) Spending money ($) Time (h) Time (min) Time (days). Use the gradient and y-intercept to draw the graphs of these functions. a. H = k k + 1 b. y = 3x + 1 c. r = -d + 7 d. L = 0.6g Solve each of these equations. a. 3m = 13 b. 1.x = 6 PA 6.1 / PA 6. Using and Interpreting Mathematical Models 67

7 Solution Apply the same operations to both sides of the equation to keep it in balance. The aim is to finish with the unknown by itself on one side of the equation. a. 3m = 13 3m + = m = 18 3m 3 = 18 3 m = 6 b. 1.x = 6 1.x = x =.4 1.x 1. =.4 1. x = Solve each of these equations. 1. 4F + = 17. 3g = = b m = 1. z = 6. y 1.8 = = r 8. 4M + = x + 40 = 400 SECTION 14.1 Modelling with Linear Functions Mathematicians use functions to model situations, for example, scientific and economic situations. Such mathematical models represent situations as relationships between variables. The models can be used to predict future values of variables so that planning can take place An international phone call is charged by a connection fee of 7c and then at the rate of 1c per minute. a. Name the independent ndent variable in this situation. b. Name the dependent nt variable in this situation. c. Write a general rule using symbols to find the cost of a call of any length. d. Use your rule to calculate the cost of a call of 16 min. e. Use your rule to find the length of a call that can be made for $ GO Maths Unit 14

8 . A student has $7 in her moneybox. She decides to place $10 in the moneybox each Monday from now on. a. Name the independent variable in this situation. b. Name the dependent variable in this situation. c. Write a general rule using symbols to find how much she has in the moneybox after any number of Mondays. d. Are the variables in this relationship discrete or continuous? e. Draw a graph to represent the relationship between the variables for the next 6 Mondays. f. Use your rule to find the amount in the moneybox after 1 Mondays. g. Use your rule to find how long it will take to have a total of at least $00 in the moneybox. 3. A bushwalker is on a track km away from his car. He starts walking back on the track towards his car at a constant speed of 4 km/h. a. Name the independent variable in this situation. b. Name the dependent variable in this situation. c. Write a general rule using symbols to find how far away ay from his car he is after any number of hours of walking. d. Are the variables in this relationship discrete or continuous? ous? e. Draw a graph to represent the relationship between the variables for the first 7 h. f. Use your graph to find how far he is from the car after 3 _ 1 h of walking. g Use your rule to find how long it will take him to reach his car. 1 In this exploration, you will model a situation involving an investment earning simple interest. You will need a sheet of grid paper. You have calculated the interest on an investment earning simple interest in Unit 9. Imagine an investment of $100 at 8% p.a. simple interest. 1. How much interest does the investment earn each year?. Calculate the value of the investment after years and after years. 3. Write a rule to find the value of the investment after any number of years. 4. Write a short paragraph explaining why the rule is a linear function.. What is the gradient of the linear function? Think carefully about the units of the gradient. 6. In this situation, are the variables discrete or continuous? PA 6.1 / PA 6. Using and Interpreting Mathematical Models 69

9 7. Draw a graph to represent the relationship. 8. Write a new rule for an investment of $100 at 1% p.a. simple interest. 9. Plot this relationship on the same graph as the 8% p.a. investment. 10. How do the patterns of the points for the two investments compare? Write a short explanation. SECTION 14. Working with the Equations of Lines Previously you have found the equations of linear functions by calculating and inserting ing the gradient (m) and y-intercept (c) into the general form of a linear equation: y = mx + c. The other symbols in this equation, x and y, represent the co-ordinates of any point on the straight t line representing the function. That is, they represent the independent and dependent dent variables in the function. Mathematicians often name the axes of the graph after them: x-axis s (independent variable, horizontal) and y-axis (dependent variable, vertical). 4 a. Write the equation of a linear function that has a gradient of - and a y-intercept of. b. Draw the graph of the function using a four-quadrant uad grid. c. Find the y co-ordinates of the points on the line with x co-ordinates of 4 and -1. d. Find the x co-ordinates of the points on the line with y co-ordinates ordinate of 3 and -4. Solution a. Use the general eral form of a linear function, y = mx + c. Therefore, m = - (gradient) and c = (y-intercept). The equation is y = -x x + b. Use the y-intercept - t and gradient to fix the position of the straight line. A run of 3 and a rise of -6 have been en used here. c. Reading from the graph, when x = 4, y = -3. When x = -1, y = 7. d. Reading from the graph, when y = 3, x = 1. When y = -4, x = 4.. y 7 6 run = rise = x 70 GO Maths Unit 14

10 4 1. a. Write the equation of a linear function that has a gradient of and a y-intercept of 1. b. Draw the graph of the function using a four-quadrant grid. c. Find the y co-ordinates of the points on the line with x co-ordinates of and -. d. Find the x co-ordinates of the points on the line with y co-ordinates of and -4.. a. Write the equation of a linear function that has a gradient of 1 and a y-intercept of -3. b. Draw the graph of the function using a four-quadrant grid. c. Find the y co-ordinates of the points on the line with x co-ordinates of 3 and -1. d. Find the x co-ordinates of the points on the line with y co-ordinates of and a. Write the equation of a linear function that has a gradient of -1. and a y-intercept of 4. b. Draw the graph of the function using a four-quadrant grid. c. Find the y co-ordinates of the points on the line with x co-ordinates ordinates of 4 and -1. d. Find the x co-ordinates of the points on the line with y co-ordinates of 6 and -3. When working with lines and points, it is often necessary essary to perform operations with positive and negative integers. You practised these in Unit. Here is a summary of the rules. For addition and subtraction, think of the direction of movement ment along the number line For example 3 4 = -1 Move four steps s to the left from = Move four steps to the right from -. For multiplication and division, think of the signs of the two numbers. + + gives + + gives + gives gives + 3 = = -6-3 = = gives + + gives + gives gives + 6 = = -3-6 = = 3 PA 6.1 / PA 6. Using and Interpreting Mathematical Models 71

11 Calculate each of these In Exercise 4, you found unknown co-ordinates of points on lines representing linear functions by reading from the graph of the function. When we know the equation representing a linear function, we can find unknown co-ordinates of points on the line by solving equations. For the linear function y = x +, find a. the value of y when x = 6 b. the value of x when y = -3 Solution a. Substitute the value x = 6 into the equation. b. Substitute the value y = -3 into the equation. Remember that x means x. y = x + y = x + -3 = x + y = 6 + Solve this equation using the balance method. y = = x + y = 17-8 = x -8 = x -4 = x x = -4 6 Find the x or y value in each of these linear functions. 1. y = x + 1. Find y when x =.. y = 4x 3. Find y when x = y = x 6. Find x when y = y = -x + 3. Find y when x =.. y = x 7. Find x when y = y = -3x +. Find x when y = y = + x. Find x when y = y = 4 x. Find y when x = y = -0.x 6. Find x when y = y = 8 x. Find x when y = 3. 7 GO Maths Unit 14

12 SECTION 14.3 Lines of Best Fit In Unit 1, you worked with two-variable data by plotting points on a scatterplot. The possible relationship between the two variables was then represented by a line of best fit. When the line of best fit is linear, it is possible to use the gradient and y-intercept to represent the relationship with a linear equation. 6 This table shows the number of years of experience and total sales for a week for 10 salespeople. ple. Experience (years) Sales ($) Experience (years) Sales ($) a. Draw a scatterplot to represent the data. b. Draw a line of best fit to represent the possible relationship between the two variables. c. Find the equation of the line of best fit. d. Use the equation to predict the sales of someone with 3. years of experience. Solution a. Sales ($ 000) Experience (years) PA 6.1 / PA 6. Using and Interpreting Mathematical Models 73

13 b. Calculate the mean of each variable to fix the position of the line of best fit. Then position the line of best fit through that point so that it best fits the plotted points. Mean experience = = 3.1 years Mean sales = $ = $ Sales ($ 000) Experience (years) c. It is a linear line of best fit, so its equation will be in the form of y = mx + c. The y-intercept is the point where the line of best fit meets the vertical axis. So c = $6700. Use two convenient points on the line to find the gradient (m). Rise = $1 00 $9000 = $300 Run = Sales ($ 000) = 3 years Gradient = rise run = = Experience (years) Remember that the line has been fitted to the data by eye and is not necessarily accurate. So round the gradient to $100 per year. The equation representing the relationship is S = 100t , where S stands for the value of sales in dollars and t stands for length of experience in years. d. For 3. years of experience, t = 3.. Substitute 3. into the equation to find the predicted sales. S = = = $ GO Maths Unit 14

14 7 1. A household water tank with a capacity of 000 L is fitted with a gauge that shows the amount of water in the tank. Before the rain started, the gauge showed 600 L. During the rain, the owner looked at the gauge every hour and recorded the reading. Time (h) Volume (L) a. Draw a scatterplot to represent the data. b. Draw a line of best fit to represent the possible relationship between the two variables. ables. c. Find the equation of the line of best fit. d. Use the equation to predict the volume of water in the tank after 8 h if it keeps raining in a similar way.. A student started training for the 400-m track race. For the first 6 weeks of training, she was timed in a trial run at the end of each week. Week Time (s) a. Draw a scatterplot to represent the data. b. Draw a line of best fit to represent the possible relationship between een the two variables. c. Find the equation of the line of best fit. d. Use the equation to predict her time for 400 m at the end of the 10th week of training. e. Do you think that the prediction in Part d is realistic? Explain in a short paragraph. In this exploration, you will l further investigate data that you collected in Unit In Unit 1, Exploration 4 you gathered data from the class on foot length and hand span and represented the data in a scatterplot. In Unit 1, Exploration you drew a line of best fit on the scatterplot. Find that scatterplot. If you have not yet performed Explorations 4 and, gather data on foot length and hand span for each member of your class, represent the data in a scatterplot and draw a line of best fit.. Find the equation of the line of best fit. 3. Use your equation to predict the hand spans of some people with really big feet and really small feet. PA 6.1 / PA 6. Using and Interpreting Mathematical Models 7

15 4. Do you think your predictions would be realistic? Explain and justify your predictions in a paragraph.. If you have data for at least eight girls and eight boys, divide the data into two sets based on gender. 6. Draw a scatterplot and line of best fit for each set of data. 7. Find the equations of the two lines of best fit. 8. Compare the equation for girls and the equation for boys. Are they very similar or are they different? How do their gradients and intercepts compare? Write a paragraph explaining the similarities and differences. SECTION 14.4 Points and Gradients You have been finding the gradient of a straight line using the rule gradient = rise run or rise run. The rise is the change in y-position and the run is the change in x-position. You have been finding each change by subtracting the appropriate co-ordinates (7, ) (, ) Run = 7 = Rise = = 3 Gradient = run rise = 3 The same rules apply when working on the general x y plane (four-quadrant grid). We can think of the rise as the change in y-position - and the run as the change in x-position. The changes can be found from the co-ordinates of two points on the line by subtraction. y (x 1, y 1 ) y y 1 x 1 x (x, y ) x Gradient = run rise change in y = change in x y = y 1 x x 1 76 GO Maths Unit 14

16 7 Find the gradient of the straight line that passes through the points (3, 1) and (-, ). Solution y x It does not matter which of the given points is called (x 1, y 1 ) and which is called (x, y ). However, once the choice has been made, it must be consistently followed. Let (3, 1) be (x 1, y 1 ) and (-, ) be (x, y ). Gradient = (y y ) 1 (x x 1 ) ( 1) = (- 3) = 4 - = - 4 (Positive negative gives negative.) 8 Draw axes for an x y plane on a sheet of grid paper. Label each axis from -7 to 7. Follow parts a, b, and c for each pair of points below. a. Plot the points on the x y plane. b. Draw the straight line that passes through the two points. c. Use the co-ordinates of the two points to find the gradient of the straight line. 1. (1, ) and (, 6). (-3, -) and (3, ) 3. (-, 1) and (, ) 4. (-4, 6) and (, -). (4, 0) and (6, -) 6. (-4, 0) and (0, ) 3 In this exploration, you will use a graphics calculator to find the gradients and equations of lines. You will need a graphics calculator. Part A Find the equation of the straight line through (0, 4) and (6, -). PA 6.1 / PA 6. Using and Interpreting Mathematical Models 77

17 CASIO 1. Clear the memory before you start.. Return to the main menu and use the key to select STAT. Press EXE. This should bring you to a screen with four lists. 3. Enter the x values for the two points in List 1: 0 EXE 6 EXE. 4. Use the key to move to List and enter the y-values there: 4 EXE ( ) EXE.. Select F1 (GRPH) then F1 again for GPH1. You will see the two points plotted on the graph. 6. Press F1 (X). On the screen you will see the equation of a linear function as y = ax + b, where a is the gradient and b is the y-intercept. You will also see a = -1 and b = 4, so the equation is y = -1x + 4. Ignore the other information on the screen for now. 7. Press F6 (DRAW) and you will see the straight line drawn through the two points. TI 1. Clear the memory before you start.. Press the STAT key. The screen should now show EDIT and 1: Edit highlighted. Press ENTER ER and a page with lists should appear. 3. Enter the x-values for the two points in List 1 (L1): 0 ENTER 6 ENTER. 4. Use the key to move to List (L) and enter the y-values: 4 ENTER ( ) ENTER.. Press nd Y= (STAT PLOT), then press ENTER when 1: Plot1 is highlighted. Make sure Plot1 is selected and then select On. Use the and keys to select the first Type and then + as the Mark. (To select each of these, press ENTER when the cursor is on top of them.) 6. Press the GRAPH key and you will l see the two points displayed on a graph. It is a little difficult to see the point (0, 4). 7. Press the STAT key and use the key to highlight CALC. Scroll down to select 4: LinReg(ax+b). LinReg stands for linear regression and you can see e that it shows ax + b. The linear function it uses is y = ax + b, where a is the gradient and b is the y-intercept. - Press ENTER twice and you will see on the screen a = -1 and b = 4, so the equation is y = -1x x You can draw the line on the graph by pressing Y= and then ( ) 1 X,T,θ,n + 4 ENTER and then GRAPH. Part B 1. Select a set of data from Exercise 7 (on page 7).. Enter the data into lists in your calculator (clear the memory first) and follow the same procedure as in Part A to find the equation of the line of best fit. 3. Compare the gradient and y-intercept found with the calculator with the ones you found in Exercise GO Maths Unit 14

18 SECTION 14. Developing Quadratic Functions In real-world situations, there are many relationships between two variables that are not linear. You have worked with some of these non-linear relationships, such as temperature changing over time and tide height changing over time. These are complex relationships that cannot be easily represented with an equation. Some real-world situations with non-linear relationships can be modelled using a type of function called a quadratic function. One such situation is the motion of an object falling under the influence of gravity. The simplest quadratic function is y = x. The general form of a quadratic function is y = ax + bx + c. You can see that a quadratic function includes a term with the square of the independent variable. Some examples of quadratic functions are y = x x +, y = x + 4x and y = -3x. 8 For the quadratic function y = x + 4x 6, find the value of y for these x-values. a. x = -3 b. x = 0 Solution a. When x = -3, y = (-3) = = = -3 6 b. When x = 0, y = = = -6 = -9 9 For each of these quadratic functions, find the value of y for the given value of x. 1. y = x + x + 6, when x =. y = x 3x + 4, when x = 3 3. y = x + x + 6, when x = 4. y = x x + 9, when x = -. y = x + x + 6, when x = - 6. y = 3x + x, when x = y = x + 3x 7, when x = 0 8. y = -x x, when x = 9. y = 6 x, when x = y = 1 x + 3x, when x = -4 PA 6.1 / PA 6. Using and Interpreting Mathematical Models 79

19 4 In this exploration, you will investigate some of the properties of quadratic functions. You will need a sheet of grid paper and a graphics calculator. Part A 1. Work with the function y = x. Make a table and calculate the values of y for the integer values of x from -4 to 4.. Use the values in your table to draw the graph of y = x. Assume that the variables are continuous. 3. Write a short paragraph describing the shape of the line that represents y = x. Think about the symmetry of the shape and the way the gradient changes. 4. Work with the function y = x +. Make a new table and calculate the values of y for the integer values of x from -4 to 4. Use the values in your table to draw the graph of y = x + on the same set of axes you used for the first graph.. How do the two lines compare? Write a short paragraph describing how they are similar and how they are different. 6. What do you think the graph of the function y = x + 3 will look like? Without making a table, draw the shape you think it will have on the same set of axes as the other two graphs. 7. Choose three values for x between -4 and 4. Calculate the value of y when x equals each of these numbers for the function y = x + 3. Check that these co-ordinates are on the line you drew in Step Without calculating the co-ordinates of points, draw the graph of the function y = x 1. Part B In Part A you calculated and plotted the co-ordinates of points to draw graphs of quadratic functions on grid paper. A graphics calculator can be used to produce the graphs of functions very quickly. You may have previously used a graphics calculator for drawing graphs of linear functions (Unit 3). The steps for drawing graphs of quadratic functions are very similar. CASIO 1. First clear the memory and then return to the main menu. Use the key to select GRAPH and press EXE.. You will see Graph Func :Y= at the top of the display. The calculator is now ready for you to enter a function. The X,θ,T key is used to enter the independent variable symbol (x). Enter X,θ,T x and press EXE. 3. Press the F6 key (DRAW) and the graph of the function will appear. 80 GO Maths Unit 14

20 4. To better display the graph, you will need to set the ranges of values for both axes. To do this, select SHIFT F3 (V-Window). You can now enter the greatest and least values you need on the axes. Use the ( ) key to enter negative values. Set the x-values from - to and the y-values from -4 to 10. Leave the scale values at 1. Press EXE after entering each x-value and y-value, and use the key to move over the x scale value. When you have completed these steps, press EXIT.. Press F6 to draw the graph again. You should now see more of the line. 6. To go back to the screen to enter more functions, press EXIT. Experiment with different quadratic functions to see how their shapes change. TI 1. First clear the memory and return to a blank screen.. Press the Y= key. The calculator is now ready for you to enter a function. The X,T,θ,n key is used to enter the independent variable symbol (x). Enter X,T,θ,n x and press ENTER. 3. Press GRAPH and the graph of the function will appear. 4. To better display the graph, you will need to set the ranges of values for both axes. To do this, press nd ZOOM (FORMAT). The features that are on are highlighted. You need to have RectGC and AxesOn highlighted. Press WINDOW. You can now enter the greatest and least values you need on the axes. Use the ( ) key to enter negative values and press ENTER after entering each one. Set the x-values from - to and the y-values from -4 to 10. Leave the scale values at 1 and leave Xres= at 1. When you have completed these steps, press GRAPH.. Experiment with different quadratic atic functions to see how their shapes change. To enter the next function, press Y= and move the cursor down to the next empty Y=. SECTION 14.6 Working with Inequalities There are situations that can be modelled mathematically by statements that look like equations but do not have an equals (=) symbol. Instead they have an inequality symbol: < (less than), > (greater than), (less than or equal to), (greater than or equal to). These statements tements are called inequalities. The number of people sitting around a table with six chairs can be modelled with the inequality n 6. This means that there can be any number of people up to six sitting at the table. The solutions to n 6 are 0, 1,, 3, 4,, 6. In this situation negative values are not possible, but in some situations they are. Similarly, in this situation only whole number values are possible (the variable is discrete), but in some situations decimal values are possible (the variable is continuous). Inequalities usually have several possible solutions and sometimes have many possible solutions. For example, in the city of Moscow the maximum daily temperatures for the month of January were all below ºC. The inequality is T < and some of the possible solutions are 4 ºC, 0 ºC, -3 ºC, -3. C, -11 ºC. There are also many other solutions. PA 6.1 / PA 6. Using and Interpreting Mathematical Models 81

21 10 1. A car has five seats. Write an inequality for the number of people in the car. Write three possible solutions for the inequality.. A climber needs a rope at least 30 m long to abseil down a cliff. Write an inequality for the possible lengths of rope she might take. Write three possible solutions for the inequality. Remember that they do not have to be whole numbers of metres. 3. It is often freezing in the morning in Stanthorpe during July. The highest minimum temperature e one July was 8 ºC. Write an inequality for the possible morning temperatures in Stanthorpe that July. Write four possible solutions, including two below 0 ºC. 4. A bus has a driver s seat and seats for 1 passengers. Write two inequalities that model the total number of people seated on the moving bus. Write three possible solutions that fit both inequalities. ies.. A group trip overseas needs at least 10 people to make it viable and can take up to 4 people. e. Write two inequalities that model the total number of people who could go on the trip. Write three possible solutions that fit both inequalities. Imagine a table with six chairs placed around it. Each chair can be occupied by a boy or a girl or nobody. So there could be four girls, or four boys and two girls, or three girls and one boy, or several other combinations. There are two variables in this situation, the number of girls and the number of boys. The situation specifies that the number of girls plus the number of boys must be less than or equal to 6. We can represent this condition with the inequality g + b 6, where g represents the number of girls and b represents the number of boys. Both variables are discrete as there can only be whole numbers of boys and girls. The possible combinations can be represented as points on a graph. Number of boys Number of girls You can see that there are 8 possible combinations, including no-one sitting at the table. Each of the possible combinations shown on the graph is a solution to the inequality. 8 GO Maths Unit 14

22 In this exploration, you will use a graph to model a situation involving inequalities and two variables. You will need some grid paper. Part A Imagine a five-seater car travelling down the road. The car can carry combinations of adults and children. 1. Name the two variables in this situation.. Write an inequality that models the situation. 3. In this situation, there are some combinations that would not be allowed as solutions. Which combinations are they? 4. Write an inequality to represent each of the conditions you found in Step 3.. Draw a graph that shows all of the allowable combinations. 6. Describe the pattern of the points that represent the possible combinations of adults and children in the car. Part B Imagine a teacher is organising a tennis activity for boys and girls. The students who participate can play singles or doubles. There are two courts available and the teacher wants all participants to be playing tennis. 1. What is the least number of students needed for the activity?. What is the most number of students who can be involved? 3. Write an inequality to represent each of the conditions you found in Steps 1 and. 4. Draw a graph to show all the possible combinations of girls and boys who can participate.. Describe the pattern of the points that represent the possible combinations of girl and boy tennis players. PA 6.1 / PA 6. Using and Interpreting Mathematical Models 83

23 More Functions Previously you used linear functions to model real-world situations, such as driving at a constant speed and earning simple interest on an investment. There are many types of functions that can be used to model complex real-world situations. In the explorations in this section you will investigate the properties of some of these functions. 6 In this exploration, you will further investigate the properties of quadratic functions. You will need a graphics calculator and some grid paper. The general form of a quadratic function is y = ax + bx + c. In Exploration 4 you investigated the properties of quadratic functions of the type y = x + c by comparing the shapes of the graphs for various values of c. These e quadratic functions had a = 1 and b = Write a short statement describing the way the shape of the graph of a quadratic function changes as the value of c changes. Include a comparison with the shape of y = x.. Your task now is to investigate the effect of the value of b on the graph of a quadratic atic function. Investigate functions of the form y = x + bx. These functions have a = 1 and c = 0. Use your graphics calculator to investigate the effect of changing the value of b. 3. Write a short report explaining the effect of different values of b on the shape of the graph of a quadratic function. Draw graphs to illustrate your explanation. 7 In this exploration, you will investigate the properties of simple cubic functions. You will need a graphics calculator and some grid paper. 1. Use your graphics calculator to investigate cubic functions of the general form y = x 3 + c.. Write a short report explaining the effect of different values of c on the shape of the graph of a cubic function. Draw graphs to illustrate your explanation. 3. How does the change in the graph compare with the changes for different values of c in y = x + c? 84 GO Maths Unit 14

24 GO Maths is the first core mathematics program in Queensland to provide a developmentally appropriate teaching sequence for Prep to Year 9. It has been developed by a team of practising mathematics educators who have a proven track record of translating suitable research findings into effective classroom practice. The GO Maths core program honestly reflects the content, intent and methodology of the 004 syllabus is a complete package that offers teachers a structure from which to implement and assess an outcomes-based curriculum is written and published in Queensland for Queensland schools is fresh and inviting for all students The other GO Maths components at this level are the GO Maths teacher sourcebook the GO Maths student portfolio Visit for more information on the program. The mathematical content of this student textbook reflects what is generally taught in Year 9 in Queensland schools. Product Code: GMS 66X Q