91028 910280 1SUPERVISOR S USE ONLY Level 1 Mathematics and Statistics, 2014 91028 Investigate relationships between tables, equations and graphs 9.30 am Tuesday 18 November 2014 Credits: Four Achievement Achievement with Merit Achievement with Excellence Investigate relationships between tables, equations and graphs. Investigate relationships between tables, equations and graphs, using relational thinking. Investigate relationships between tables, equations and graphs, using extended abstract thinking. Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page. You should attempt ALL the questions in this booklet. Show ALL working. If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question. Check that this booklet has pages 2 10 in the correct order and that none of these pages is blank. YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION. TOTAL ASSESSOR S USE ONLY New Zealand Qualifications Authority, 2014. All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

2 QUESTION ONE ASSESSOR S USE ONLY (a) Josie is investigating number patterns in supermarket displays. Layer 1 2 3 4 Number of layers of cans, n Number of cans in layer, A 1 4 (i) Complete the table on the right. 2 3 (ii) On the grid below plot the graph that shows the number of cans in each layer. 15 A 4 7 10 5 5 10 n If you need to redraw this graph, use the grid on page 8 (iii) Josie says arranging cans in a single-row layer like this is too unstable and decides to add another 2 rows behind, so there are 3 rows in each layer, as shown below. Give the equation for the number of cans, A, in any layer, n, of a display, where each layer is made of 3 rows. Layer 1 2 3 4 A = Mathematics and Statistics 91028, 2014

3 (iv) A display is 15 layers high. It also has 4 cans in each row of the top layer. ASSESSOR S USE ONLY How many cans would she need for the bottom layer of this display if each layer is made of 3 rows? You must show use of your equation. (b) (i) Sketch the graph of y = x y 2 + 7x 2 10 5 5 5 x 5 If you need to redraw this graph, use the grid on page 8 (ii) 2 n + 7n Josie knows that T = 2 with n single-row layers. is the equation for the total number of cans, T, in a display How would the graph of the relationship between T and n differ from the graph you drew in (b)(i)? Give reasons for your answer. Mathematics and Statistics 91028, 2014

4 QUESTION TWO ASSESSOR S USE ONLY (a) (i) Give the equation of the graph shown below. y 60 50 40 30 20 10 5 5 x 10 (ii) For what values of x is y negative? (b) Raja is investigating number sequences of the form y = x 2 x + q where q is a constant number. If he drew a graph of this, what would q represent on the graph? Mathematics and Statistics 91028, 2014

(c) 5 Raja used his calculator to draw the graph of y = x 2 x on the axes below. y 120 110 100 90 80 70 60 50 40 30 20 ASSESSOR S USE ONLY 10 0 10 8 6 4 2 0 2 4 6 8 10 x (i) What are the co-ordinates of the vertex (lowest point) of the graph above? Remember to show your working. (ii) Describe, in words, how the graph of y = 2x 2 + 2x compares to the graph above. (iii) If the graph from part (c)(i) above was moved 5 units to the right and 15 units up, what would its new equation be? Simplify your answer. Mathematics and Statistics 91028, 2014

6 QUESTION THREE ASSESSOR S USE ONLY (a) Arne owes $450 on his student loan. He is paying it off at the rate of $15 each week. (i) For the first 5 weeks, sketch the graph of the total amount Arne still has to pay, A, against the number of weeks he has been paying off the loan, w. Total still to pay, A 480 450 420 390 If you need to redraw this graph, use the grid on page 9 360 330 300 270 240 210 180 150 120 90 60 30 (ii) 0 5 10 15 20 25 30 35 40 Number of weeks, w Give the equation for the amount A that Arne owes after he makes each weekly payment of $15. A = Mathematics and Statistics 91028, 2014

7 (iii) Arne decides that he wants to go overseas for a holiday. He wants to finish repaying all of his loan at the end of 25 weeks. He looks at what he has left to pay and decides he can repay the loan exactly at the end of 25 weeks if he increases the amount he repays to $20 a week. ASSESSOR S USE ONLY Add this section to your graph. (iv) After how many weeks did he start paying $20 per week? (v) Give the equation for the graph representing the increased payments. (b) Give the equation of the graph below. 10 y 5 10 5 5 10 x 5 10 Mathematics and Statistics 91028, 2014

8 If you need to redraw your graph from Question One (a)(ii), draw it on the grid below. Make sure it is clear which graph you want marked. ASSESSOR S USE ONLY 15 A 10 5 5 10 n If you need to redraw your graph from Question One (b)(i), draw it on the grid below. Make sure it is clear which graph you want marked. y 10 5 5 5 x 5 Mathematics and Statistics 91028, 2014

9 If you need to redraw your graph from Question Three (a)(i), draw it on the grid below. Make sure it is clear which graph you want marked. ASSESSOR S USE ONLY Total still to pay, A 480 450 420 390 360 330 300 270 240 210 180 150 120 90 60 30 0 5 10 15 20 25 30 35 40 Number of weeks, w Mathematics and Statistics 91028, 2014

NCEA Level 1 Mathematics and Statistics (91028) 2014 page 1 of 3 Assessment Schedule 2014 Mathematics and Statistics: Investigate relationships between tables, equations and graphs (91028) Evidence Statement Q Evidence Achievement Achievement with Merit Achievement with Excellence ONE (a)(i) Table completed 1 4 2 5 3 6 4 7 Table correct. 7 10 (ii) Graph of A = n + 3 Correct graph with points joined. Correct graph with discrete points. (iii) A = 3n + 9 Correct. (iv) 54 Correct. (b)(i) Graph y = (x 2 + 7x) / 2 Graph of x 2 + 7x Some errors. Correct graph. (ii) Discrete points you cannot have fractional number of cans. First point is (1,4) or No points for n < 1. There would be no display if you didn t have any cans, and the number of cans cannot be negative. One reason. Either ONE reason with explanation. OR TWO reasons without explanation. Full explanation.

NCEA Level 1 Mathematics and Statistics (91028) 2014 page 2 of 3 NØ N1 N2 A3 A4 M5 M6 E7 E8 No response; no relevant evidence. Some evidence. 1 of u 2 of u 3 of u 1 of r 2 of r 1 of t 2 of t Q Evidence Achievement Achievement with Merit Achievement with Excellence TWO (a)(i) y = (x 2)(x + 4) Some errors in equation. Equation correct. (ii) 4 < x < 2 ONE section correct OR correct with minor errors. BOTH sections identified. (b) y intercept. The value of the function when x = 0 Correct. (c)(i) (0.5, 0.25) ONE coordinate correct. Correct coordinates. (ii) All points would be twice as high as they are now, then reflected in the x-axis. Some aspect correctly identified. Incomplete description. Full description. The parabola would be twice as steep and it would be upsidedown. (iii) y = (x 5) 2 (x 5) + 15 = x 2 11x + 45 15 added. 5 and 15 in equation. Simplified equation. NØ N1 N2 A3 A4 M5 M6 E7 E8 No response; no relevant evidence. Some evidence. 1 of u 2 of u 3 of u 1 of r 2 of r 1 of t 2 of t

NCEA Level 1 Mathematics and Statistics (91028) 2014 page 3 of 3 Questio n Evidence Achievement Achievement with Merit Achievement with Excellence THREE (a)(i) Graph of A = 450 15w As a step function (end points do not need to be open / closed correctly). Drawn as continuous line. Discrete graph with dots. Step function. (ii) A = 450 15w Gradient correct. Equation correct. (iii) Graph A = 500 20w drawn from w = 10 Drawn as continuous line. Discrete graph with dots. (iv) 10 Correct solution. (v) y = 20w + 500 Gradient correct. Correct equation. (b) y = 2 (x + 1) 3 ONE error in equation. Correct equation. NØ N1 N2 A3 A4 M5 M6 E7 E8 No response; no relevant evidence. Some evidence. 1 of u 2 of u 3 of u 1 of r 2 of r 1 of t 2 of t Cut Scores Not Achieved Achievement Achievement with Merit Achievement with Excellence Score range 0 6 7 13 14 18 19 24

91028 910280 1SUPERVISOR S USE ONLY Level 1 Mathematics and Statistics, 2015 91028 Investigate relationships between tables, equations and graphs 9.30 a.m. Monday 9 November 2015 Credits: Four Achievement Achievement with Merit Achievement with Excellence Investigate relationships between tables, equations and graphs. Investigate relationships between tables, equations and graphs, using relational thinking. Investigate relationships between tables, equations and graphs, using extended abstract thinking. Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page. You should attempt ALL the questions in this booklet. Show ALL working. If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question. Check that this booklet has pages 2 16 in the correct order and that none of these pages is blank. YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION. TOTAL ASSESSOR S USE ONLY New Zealand Qualifications Authority, 2015. All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

2 QUESTION ONE ASSESSOR S USE ONLY A plant is growing on the surface of a pond. Hank noticed the plant on Day 1. Two days later Hank was worried about the plant and started measuring the area that the plant covered. (a) Each day (at 5 pm) Hank measures the area of water (in square metres) covered by the plant. He records his measurements in the table below. Day, d Area covered by plant, A 1 2 3 4 4 8 5 16 6 32 7 64 8 128 d is the number of days since Hank first noticed the plant. (i) Show how the area of the pond covered by the plant changes with time. Area covered by plant 140 If you need to redraw this graph, use the grid on page 14. 130 120 110 100 Area in square metres, A 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 Day, d Mathematics and Statistics 91028, 2015

3 (ii) The plant followed the same pattern of growth from the time when it was first noticed. ASSESSOR S USE ONLY What area of the pond was covered by the plant when it was first noticed? Explain your answer. (iii) Give the equation that describes the area of the plant covering the pond after d days. (iv) If no intervention takes place, on which day will Hank first measure the area of the plant to be more than 500 square metres? Mathematics and Statistics 91028, 2015

4 (b) Hank and some friends start removing the plant on Day 9. ASSESSOR S USE ONLY The graph of the area covered by the plant from Day 8 (when it covers 128 square metres), below, shows what Hank hopes will happen to the area of pond covered by the plant. 140 Area covered by plant 120 Area in square metres, A 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Day, d (i) What is the equation for the area covered by the plant as shown in this graph? Mathematics and Statistics 91028, 2015

5 (ii) What is unrealistic about this graph? Write at least TWO comments with justification. ASSESSOR S USE ONLY Mathematics and Statistics 91028, 2015

6 QUESTION TWO ASSESSOR S USE ONLY The next year, when the plant begins to grow back, Hank tries to stop it from spreading across the pond so quickly. As soon as he notices the plant, he begins removing it. The graph of the area of pond covered by the plant in this year is shown below: 100 Area covered by plant Area in square metres, A 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Day, d (a) How much more area is the plant covering each day? (b) What day will it be when the plant covers 200 square metres if the conditions remain the same? Show your working. Mathematics and Statistics 91028, 2015

7 After 7 days removing some of the plant by himself, Hank decides to get help. ASSESSOR S USE ONLY (c) One friend helps on Day 8 and Day 9. The area covered by the plant stays the same for Day 8 and Day 9. 100 Area covered by plant Area in square metres, A 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Day, d If you need to redraw this graph, use the grid on page 14. (i) What is the equation of this new section of the graph on Day 8 and Day 9? (ii) What does this section of the graph mean? (d) Two more friends come to help. Now the area covered by the plant decreases by 15 square metres each day until the plant is completely removed. (i) (ii) Draw a graph on the grid above to show the area of pond covered by the plant from Day 10. On what day will there be no plant left? Mathematics and Statistics 91028, 2015

8 (e) The equation of the line for Day 9 onwards is A = 225 15d. ASSESSOR S USE ONLY If Hank s 2 friends had come on Day 8, what would the equation of this line have been? Explain your reasoning. Mathematics and Statistics 91028, 2015

(f) Hank had a dream that he and his friends made the area of the pond covered by the plant follow the parabola given below: 9 ASSESSOR S USE ONLY Day Hank s dream 0 20.00 1 37.33 2 52.00 3 64.00 4 73.33 5 80.00 6 84.00 7 85.33 8 84.00 9 80.00 10 73.33 11 64.00 12 52.00 13 37.33 14 20.00 15 0.00 Area in square metres, A 100 90 80 70 60 50 40 30 20 10 0 0 Area covered by plant 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Day, d What is the equation of this graph? Mathematics and Statistics 91028, 2015

10 QUESTION THREE ASSESSOR S USE ONLY (a) Jodie sets her friends a mathematical problem. She says: I think of an integer When I add 1 to my number, I get A But if I take 4 off my number, I get B When A is multiplied by B, I get an answer of 6. What s my number? Her friends start by writing a table: x: Jodie s number A = x + 1 B = x 4 y = AB 0 1 4 4 1 2 3 6 2 3 2 3 4 4 (i) Draw the graph of y against x. Use the set of axes below. 10 8 6 y If you need to redraw this graph, use the grid on page 15 4 2 6 5 4 3 2 1 1 2 3 4 5 6 2 x 4 6 8 10. Mathematics and Statistics 91028, 2015

11 (ii) What is the equation of the graph that matches the table above, in terms of x? ASSESSOR S USE ONLY (iii) Explain how Jodie s number can be found from the graph if the answer is 6. y = (iv) Suppose Jodie had said A multiplied by B gives me 10. What does your graph tell you about the solutions to this new problem? Mathematics and Statistics 91028, 2015

12 (b) Tom thinks of a puzzle to challenge Jodie. He starts by saying: I think of a two-digit number. I multiply it by 4 and take away 100 ASSESSOR S USE ONLY (i) What equation would you use to describe this relationship? (ii) Draw the graph of this relationship on the axes below. 350 300 250 200 y If you need to redraw this graph, use the grid on page 15 150 100 50 50 20 40 60 80 100 x 100 150 Mathematics and Statistics 91028, 2015

13 (iii) Tom s whole puzzle is: Guess my 2-digit number: If I multiply it by 4 and take away 100 I get the same as when I add 47 to it and then multiply the result by 1.12 ASSESSOR S USE ONLY Explain how the solution to Tom s question can be found, and give the solution as accurately as possible. Mathematics and Statistics 91028, 2015

14 If you need to redraw your graph from Question One (a)(i), draw it on the grid below. Make sure it is clear which graph you want marked. ASSESSOR S USE ONLY 140 Area covered by plant 130 120 110 100 Area in square metres, A 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 Day, d If you need to redraw your graph from Question Two (c), draw it on the grid below. Make sure it is clear which graph you want marked. 100 Area covered by plant Area in square metres, A 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Day, d Mathematics and Statistics 91028, 2015

15 If you need to redraw your graph from Question Three (a)(i), draw it on the grid below. Make sure it is clear which graph you want marked. ASSESSOR S USE ONLY 10 y 8 6 4 2 6 5 4 3 2 1 1 2 3 4 5 6 2 x 4 6 8 10 If you need to redraw your graph from Question Three (b)(ii), draw it on the grid below. Make sure it is clear which graph you want marked. 350 y 300 250 200 150 100 50 50 20 40 60 80 100 x 100 150 Mathematics and Statistics 91028, 2015

NCEA Level 1 Mathematics and Statistics (91028) 2015 page 1 of 5 Assessment Schedule 2015 Mathematics and Statistics: Investigate relationships between tables, equations and graphs (91028) Evidence Statement One Expected coverage Achievement Merit Excellence (a)(i) Exponential curve, starting at (3,4) and curving smoothly to (8,128). (See Appendix A.) Correct curve drawn, using discrete points or continuous curve. (ii) 1 m 2, which means that 2 days before he started measuring, there was already a patch of plant 1 m 2 in area. Correct value identified. Valid interpretation of the answer given. (iii) A = 0.5 2 d A = 2 d Correct equation. OR A = 2 d 1 (iv) 10 days Correct answer. (b)(i) A = 25.6d + 332.8 Correct gradient OR Correct intercept. Correct equation. (ii) Grade levels are independent relating to their understanding of the nature of the points. So the evidence provided is not related to an earlier criterion. They might not be able to remove exactly the same amount of plant each day. It would not be a straight line since the amount it grew back would be bigger when there was more plant. Comment relates people s ability to remove plant to the graph s gradient. Comment develops how the slope might vary over the 5 days (with justification). NØ N1 N2 A3 A4 M5 M6 E7 E8 No response or no relevant evidence Some relevant evidence. 1u 2u 3u 1r 2r 1t 2t

NCEA Level 1 Mathematics and Statistics (91028) 2015 page 2 of 5 Two Expected coverage Achievement Merit Excellence (a) 10 m 2 Correct response. (b) 18 days Correct response. Correct response with correct working. (c)(i) (ii) Equation given as A = 90 It means 1. The amount of plant is staying the same. 2. The rate of removal is keeping pace with the growth. Correct equation. Correct equation. AND Makes one point. (d)(i) Line drawn has slope of 15 and starts at (9,90). Correct line drawn. (ii) Day 15 Correct response given. (e) If original equation is A = 225 15d, the new one will be A = 225 15(d + 2) = 15d + 195 Since it needs to be shifted 2 days to the left. y = 210 15d acceptable Correct equation. Correct equation and complete description of rationale behind the new equation in terms of translation of the graph. (f) TWO correct of: Correct equation. A = 4 3 (d +1)(d 15) ^ Factor of 4/3 (d + 1) (d 15) OR A = 4 3 (d 7)2 + 256 3 OR A = 4 3 d 2 + 56 3 d + 20 Factor of 4/3 (d 7) 256/3 or 85.333 Two terms correct. OR 1.33 instead of 4 3 OR GC answer A = 1.3x² + 18.66x + 19.99 NØ N1 N2 A3 A4 M5 M6 E7 E8 No response or no relevant evidence Some relevant evidence. 1u 2u 3u 1r 2r 1t 2t

NCEA Level 1 Mathematics and Statistics (91028) 2015 page 3 of 5 Three Expected coverage Achievement Merit Excellence (a) x A = x + 1 B = x 4 y = AB 0 1 4 4 1 2 3 6 2 3 2 6 3 4 1 4 4 5 0 0 (i) Correct parabola drawn but curve used. Discrete graph of correct parabola drawn. (ii) y = (x + 1)(x 4) OR y = x² 3x 4 OR y = (x 1.5)² 6.25 Correct equation. (iii) The solutions will be where the curve reaches a height of 6 (or hits the line y = 6). This will happen when x = 5. Since a parabola is symmetrical, this will happen twice, once at x = 5, and the other time when x = 2. [Graph shown completed for 2 < x < 0] One solution obtained and explained. OR Two solutions. Clear explanation for why there are two solutions from graph. Does not have to have solutions. (iv) Since the graph never drops to a height of 10, we can see that there will be no (real) solutions. (b)(i) y = 4x 100 OR other variables. Acknowledges impossibility. Correct equation. Complete, clear explanation. (ii) Graph of y = 4x 100 Drawn correctly (iii) You could find the solution by plotting y = 1.12(x + 47) on the same axes and looking for the point of intersection. This may be difficult to do accurately. Solution is x = 53. Correct graph of y = 1.12(x + 47) or use of that equation in response. OR correct equation. OR answer only. Estimate of the solution read off the graph to give 50 < x < 55. Exact solution obtained by refining the graph estimate or by algebra. NØ N1 N2 A3 A4 M5 M6 E7 E8 No response or no relevant evidence Some relevant evidence. 1u 2u 3u 1r 2r 1t 2t

NCEA Level 1 Mathematics and Statistics (91028) 2015 page 4 of 5 Cut Scores Not Achieved Achievement Achievement with Merit Achievement with Excellence 0 8 9 14 15 19 20 24

NCEA Level 1 Mathematics and Statistics (91028) 2015 page 5 of 5 Appendix A Question One (a) Appendix B Question Three (b)(ii) and (iii)

91028 910280 1SUPERVISOR S USE ONLY Level 1 Mathematics and Statistics, 2016 91028 Investigate relationships between tables, equations and graphs 9.30 a.m. Thursday 17 November 2016 Credits: Four Achievement Achievement with Merit Achievement with Excellence Investigate relationships between tables, equations and graphs. Investigate relationships between tables, equations and graphs, using relational thinking. Investigate relationships between tables, equations and graphs, using extended abstract thinking. Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page. You should attempt ALL the questions in this booklet. Show ALL working. If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question. Check that this booklet has pages 2 16 in the correct order and that none of these pages is blank. YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION. TOTAL ASSESSOR S USE ONLY New Zealand Qualifications Authority, 2016. All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

2 QUESTION ONE ASSESSOR S USE ONLY Tama and Pita have three different babysitters to choose from: Matt, Talia, and Sasha. (a) The graph of the amounts that Matt and Talia charge is shown below. $P 80 Amount charged for babysitting 70 60 50 40 30 20 Matt Talia If you need to redraw your answer, use the graph on page 13. 10 0 1 2 3 4 5 6 7 t Number of hours babysitting (i) How much would Matt be paid if he babysits for 4.5 hours? (ii) Once Matt has babysat for 5 or more hours, he increases his charge for the additional hours to $15 an hour or part of an hour that he babysits. On the grid above, show the amount Matt would charge if he babysits for 5 or more hours. (iii) Find the average amount Matt charges per hour if he babysits for 6 hours. (iv) Talia charges an average of $10 per hour for any amount of time that she works. This is shown on the graph above with the red line. Give the equation of the graph. Mathematics and Statistics 91028, 2016

3 (v) Sasha will babysit for up to 7 hours for $55. ASSESSOR S USE ONLY Make recommendations on who Tama and Pita should have as their babysitter, based on the amount that each babysitter charges. y (b) (i) Give the equation of the graph shown on the right. 6 4 2 6 4 2 2 4 x 2 4 (ii) The graph is then translated a units to the right and up b units. Give: the equation of the translated graph the x-value at the vertex. 6 Mathematics and Statistics 91028, 2016

5 QUESTION TWO ASSESSOR S USE ONLY (a) (i) Maria is investigating a set of rectangles that have an area modelled by A = (x 2 40x). Sketch the graph of the possible range of areas of the rectangles as the value of x changes. A (cm 2 ) 500 400 300 200 100 If you need to redraw this graph, use the grid on page 13. 50 40 30 20 10 0 10 20 30 40 50 x (cm) 100 (ii) What is the maximum possible area of the rectangles? (iii) For what values of x are the areas less than 300 cm 2? (iv) What is the maximum area of another set of rectangles that have an area, A = (x 2 mx)? Mathematics and Statistics 91028, 2016

(b) The points listed in the table below lie on a parabola. x y 2 6 1 0 3 4 5 6 6 ASSESSOR S USE ONLY (i) Sketch the parabola represented by these points, and give the coordinates of the intercepts and the vertex. y 10 9 8 7 6 5 4 If you need to redraw this graph, use the grid on page 14. 5 4 3 2 1 3 2 1 1 1 2 3 4 5 6 x 2 3 4 5 6 7 8 9 10 Mathematics and Statistics 91028, 2016

7 (ii) Give the equation of the graph. ASSESSOR S USE ONLY Mathematics and Statistics 91028, 2016

9 QUESTION THREE ASSESSOR S USE ONLY (a) (i) Give the equation of the graph below. y 10 9 8 7 6 5 4 3 2 1 4 3 2 1 0 1 2 3 4 1 x (ii) Give the equation of the resulting graph if the graph above is reflected in the y axis. y = Mathematics and Statistics 91028, 2016

(b) A new fun park was very popular when it opened. In the first three months, an average of 4000 people visited the park each month. After the first three months, the attendance began to drop by approximately 15% each month for the next nine months. After the first three months, the approximate number of visitors to the park can be modelled by: P = 4000 0.85 n 3, where n is the number of months since the park opened. 10 ASSESSOR S USE ONLY (i) Complete the table below showing the approximate number of people who visited the fun park during each month for the first year. Month (n) Approximate number of people visiting park this month (P) 1 2 3 4 5 6 2457 7 2088 8 1775 9 1509 10 1282 11 1090 12 926 Mathematics and Statistics 91028, 2016

11 (ii) Draw the graph showing the approximate number of people visiting the fun park each month. ASSESSOR S USE ONLY P 4500 4000 3500 3000 2500 If you need to redraw this graph, use the grid on page 15. 2000 1500 1000 500 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 n 500 Mathematics and Statistics 91028, 2016

12 (iii) At the end of the month that the number of visitors dropped below 2000 for the first time, the management decided to open only on weekends. ASSESSOR S USE ONLY Find how many months of the year the park was open only on the weekends, and explain by using the features of the graph, how this information can be found. (iv) In the second year, more people visit the park during the first three months. As the year progresses, the number of people visiting the park declines at the same rate as it did for the first year. The managers want to limit to a maximum of 2 months, the period when the park is running just on weekends. What is the average number of people who would need to be visiting the park each month in the first three months if this was to be achieved? Mathematics and Statistics 91028, 2016

13 If you need to redraw your answer from Question One (a), draw it on the graph below. Make sure it is clear which graph you want marked. ASSESSOR S USE ONLY Amount charged for babysitting $P 80 70 60 50 40 30 20 10 Matt Talia 0 1 2 3 4 5 6 7 t Number of hours babysitting If you need to redraw your graph from Question Two (a)(i), draw it on the grid below. Make sure it is clear which graph you want marked. A (cm 2 ) 500 400 300 200 100 50 40 30 20 10 0 10 20 30 40 50 x (cm) 100 Mathematics and Statistics 91028, 2016

14 If you need to redraw your graph from Question Two (b)(i), draw it on the grid below. Make sure it is clear which graph you want marked. ASSESSOR S USE ONLY y 10 9 8 7 6 5 4 3 2 1 5 4 3 2 1 1 1 2 3 4 5 6 x 2 3 4 5 6 7 8 9 10 Mathematics and Statistics 91028, 2016

15 If you need to redraw your graph from Question Three (b)(ii), draw it on the grid below. Make sure it is clear which graph you want marked. ASSESSOR S USE ONLY P 4500 4000 3500 3000 2500 2000 1500 1000 500 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 n 500 Mathematics and Statistics 91028, 2016

NCEA Level 1 Mathematics and Statistics (91028) 2016 page 1 of 4 Assessment Schedule 2016 Mathematics and Statistics: Investigate relationships between tables, equations and graphs (91028) Evidence Statement One Expected coverage Achievement (u) Merit (r) Excellence (t) (a)(i) $50 Correct value. (ii) Lines joining (5,65) to (6,65) open circle and (6,80) to (7,80) open circle. Correct step functions without open circle or one correct step. Correct step functions with open circles. (iii) Gradient (slope) of the graph of line joining (0,0) to (6,80). 80 6 = $13.33 Accept 65 6. Correct solution calculated. (iv) y = 10x OR (P = 10t) Correct equation. (v) Let t be time for baby sitting If t < 5.5 hours, Talia is the cheapest t > 5.5, Sasha is the cheapest t < 3, Talia is cheapest 3 < t < 5.5 Talia is the cheapest Matt is never the cheapest. 1 region correct. 2 regions correct. 3 correct descriptions. (b)(i) y = 2x 2 + 5x 3 OR y = 2(x 0.5)(x + 3) OR y = 2(x + 1.25) 2 6.125 OR y = (2x 1)(x + 3) 1 term incorrect. OR equivalent. Correct equation. (ii) y = 2(x a) 2 + 5(x a) 3 + b y = 2(x 2 2ax + a 2 ) + 5x 5a 3 + b y = 2x 2 4ax + 2a 2 + 5x 5a 3 + b y = 2x 2 + (5 4a)x + 2a 2 5a 3 + b y = 2(x ( 1.25 + a) 2 6.125 + b y = 2(x + 1.25 a)² 6.125 + b x value of turning point is 1.25 + a turning point is ( 1.25 + a, 6.125 + b) Just a or just b. Initial equation. x value of turning point or coordinates of turning point. N1: one question attempted A3: 2 of u or 1 of r M5: 2 of r E7: 1 of t N2: 1u A4: 3 of u or 1 of r and 1 of u M6: 3 of r E8: 2 of t

NCEA Level 1 Mathematics and Statistics (91028) 2016 page 2 of 4 TWO Expected coverage Achievement (u) Merit (r) Excellence (t) (a)(i) Graph of A = x 2 + 40x Correct graph without restriction to the first quadrant. Accurate graph in the first quarter. (ii) Maximum area is 400 (when x = 20). Correct response. (iii) 0 < x < 10 and 30 < x < 40. 30 < x < 10 OR 10 > x > 30 Indicates x < 10 AND x > 30. 0 < x < 10 AND 30 < x < 40 (iv) x value = m 2 x = m 2 Correct simplified solution. m Area = 2 2 m m 2 = m2 4 (b)(i) Any TWO of intercepts or vertex and ONE feature of graph correct. Symmetrical parabola. Any THREE of intercepts and vertex and one error on graph. (Does not have to be coordinates.) Symmetrical parabola. Totally correct, giving coordinates of intercepts and vertex and correct graph. Symmetrical parabola. x-intercepts ( 1,0) and (4,0) y-intercept (0,4) Max coordinates (1.5, 6.25) accept y between 6 < x 7 (on graph). (ii) y = x 2 + 3x + 4 OR y = (x + 1)(x 4) Correct equation. N1: one question attempted N2: 1u A3: 2 of u or 1 of r A4: 3 of u or 1 of r and 1 of u M5: 2 of r M6: 3 of r E7: 1 of t E8: 2 of t

NCEA Level 1 Mathematics and Statistics (91028) 2016 page 3 of 4 THREE Expected coverage Achievement (u) Merit (r) Excellence (t) (a)(i) y = 2 x 1 or equivalent. One minor error Correct equation. (ii) y = 2 x 1 One minor error. Correct equation. (b)(i) Month (n) Approximate number of people visiting park this month (P) 1 4000 2 4000 3 4000 4 3400 5 2890 6 2457 7 2088 8 1775 9 1509 10 1282 11 1090 12 926 At least two months correct. Table correctly completed. (ii) At least 5 points consistent. Consistent graph drawn as discrete or continuous. Allow one point incorrect. 4000 for first 3 months and then from table. (iii) Solution found by finding the x-coordinate of where the graph of P = 2000 intersects the graph of P = 4000 0.85 n OR from table OR by calculation Partial description and 7th month. OR Closed for 5 months. Stated closed for 4 months with correct 2000 = 4000 0.85 x x = 4.26 4 only (CAO) Number of months = 7.26 At the end of the 8th month, there will be fewer than 2000 visitors. (10 3) (iv) 2000 = A 0.85 A = 6238.7 The required number is at least 6239. Demonstrated use of 0.85 and consistent with graph. Required number of people found. N1: table in b correct N2: 1u A3: 2 of u or 1 of r A4: 3 of u or 1 of r and 1 of u M5: 2 of r M6: 3 of r E7: 1 of t E8: 2 of t

NCEA Level 1 Mathematics and Statistics (91028) 2016 page 4 of 4 Cut Scores Not Achieved Achievement Achievement with Merit Achievement with Excellence 0 6 7 13 14 18 19 24