Graphs of functions — Paper 3 · IGCSE Mathematics 0580

6 (a) Complete the table below for y = x2 − 2x. For Examiner's x −2 −1 0 1 2 3 4 Use y 8 −1 3 8 [3] (b) On the grid below, draw the graph of y = x2 − 2x for −2 x 4. y 8 7 6 5 4 3 2 y = 2 1 _4 _3 _2 _1 0 1 2 3 4 x _1 _2 [4] (c) The line y = 2 is drawn on the diagram. Use your graph to find the values of x that solve the equation x2 − 2x = 2. Answer(c) x = or x = [2] (d) Complete the table below for y = 4 − x. x −4 0 4 y 8 [2] (e) On the grid above, draw the line y = 4 − x for −4 x 4. [1] (f) Write down the x coordinates of the points of intersection of the graphs of y = x2 − 2x and y = 4 − x. Answer(f) x = or x = [2]

14 marks

Mark scheme: 6 a) 3 0 0 1,1,1 b) 7 correct points plotted P3√ P2√ for 5 or 6 points ± ½ sm. sq. P1√ for 4 points. not strict f.t. smooth curve through all correct points C1 incorrectly plotted points should be ignored for C1. Minimum curved, not pointed c) -0.8 to -0.7 c.a.o. 1 ignore any y values 2.7 to 2.8 c.a.o. 1 IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 d) 4 0 1,1 e) correct line drawn through 1 complete line (-4,8) and (4,0) f) -1.7 to -1.4 c.a.o. 1 ignore any y values 2.4 to 2.7 c.a.o. 1 14

This question in 0580/31 Oct/Nov 2004

2 (a) Complete the table of values for y = 1 + 2x – x2. For Examiner's Use x − 3 − 2 − 1 0 1 2 3 4 5 y − 14 − 7 1 − 2 − 14 [3] (b) Draw the graph of y = 1 + 2x – x2 on the grid below. y 4 2 x –3 –2 –1 0 1 2 3 4 5 –2 –4 –6 –8 –10 –12 –14 [4] (c) Use your graph to find the solutions to the equation 1 + 2x – x2 = 0. Answer (c) x = or x = [2] (d) (i) On the grid, draw the line of symmetry of the graph. [1] (ii) Write down the equation of this line of symmetry. Answer(d)(ii) [1]

11 marks

Mark scheme: 2 (a) –2 1 2 –7 3 B2 for 3 correct, B1 for 1 or 2 correct (b) 9 correct points P3 f.t. P2 f.t. for 7 or 8 correct, P1 f.t. for 5 or 6 plotted correct limit for acurracy is ½ small square smooth curve drawn C1 must go through the 9 correct points not dependent on P3 (c) –0.4 ( ± /0.1) 1 please note no f.t. on this part 2.4 ( ± 0.1) 1 (d) (i) correct line drawn 1 accept dotted/dashed line must be full length from (1, –14) to (1,2) (ii) x = 1 1 f.t. f.t. from (d)(i) if x = k any reference to y is X 11

This question in 0580/31 May/June 2005

3 (a) (i) Complete the table of values for y = x 2 − 2 x − 3 . For Examiner's Use x −3 −2 −1 0 1 2 3 4 5 y 12 0 −4 −3 0 5 [3] (ii) Draw the graph of y = x 2 − 2 x − 3 on the grid below. y 12 10 8 6 4 2 x –3 –2 –1 0 1 2 3 4 5 –2 –4 [4] (iii) Use your graph to find the solutions to x 2 −x2 − 3 = −1 . Give your answers to 1 decimal place. Answer(a)(iii) x = or x = [2] 2 (b) (i) Complete the table of values for the equation y = . x x 0.25 0.5 1 2 3 4 5 y 4 1 0.7 0.5 0.4 [1] 2 (ii) On the same grid draw the graph of y = for 0.25 x 5. [3] x (iii) Write down the x co-ordinate of the point of intersection of your two graphs. Answer(b)(iii) x = [1]

14 marks

Mark scheme: 3 (a) (i) 5 1 –3 1 12 1 (ii) 9 correct points plotted P3√ P2 for 7 or 8 or P1 for 5 or 6 correct, smooth curve drawn C1 (iii) –0.8 to –0.7 1 2.6 to 2.8 1 (b) (i) 8 and 2 1 (ii) points P2 P1 for 5 or 6 correct curve C1 (iii) 3.1 to 3.3 1√ ft dep on only 1 point of intersection [14] IGCSE – NOVEMBER 2005 0580/0581 3

This question in 0580/31 Oct/Nov 2005

2 (a) Complete the table for the equation y = − x2 + x + 2. For Examiner's Use x −3 −2 −1 0 1 2 3 4 y −10 0 2 2 0 [3] (b) On the grid below draw the graph of y = − x2 + x + 2. y 3 2 1 x 3 2 1 0 1 2 3 4 1 2 3 4 5 6 7 8 9 10 [4] (c) On the grid, draw the line of symmetry of your graph. [1] (d) Use your graph to find the maximum value of y. Answer(d) y = [1] (e) Draw the line y = 1 on the grid. [1] (f) Write down the two values of x for which − x2 + x + 2 = 1. Answer(f) x = or x = [2]

12 marks

Mark scheme: 2 (a) –4 –4 –10 3 1 for each correct entry (b) 1 P3ft P2 for 6 or 7 correct. ft 8 correctly plotted points, within square. P1 for 4 or 5 correct. ft 2 Allow small errors in the points Smooth curve through 8 points C1 provided shape is maintained. (c) x = 0.5 drawn. 1 must be from (0.5, –9) to curve at least (d) 2.2 to 2.4 1ft (e) y = 1 drawn. 1 must touch curve as min. length (f) (x =) –0.7 to –0.5 1 (x =) 1.5 to 1.7 1 12

This question in 0580/31 Oct/Nov 2006

4 (a) The table shows corresponding values of x and y for the function For Examiner's 60 Use y = (x ≠ 0). x x −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 y −12 −15 −30 60 12 10 [2] (i) Fill in the missing values of y in the table above. (ii) Plot the points on the grid below and draw the graph for −6 x −1 and 1 x 6. y 60 50 40 30 20 10 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –10 –20 –30 –40 –50 –60 [4] (b) Write down the order of rotational symmetry of the graph. Answer(b) [1] (c) Draw the lines of symmetry of the graph on the grid. [2] (d) One line of symmetry intersects the graph at two points. (i) Write down the co-ordinates of these two points. Answer(d)(i) ( , ) and ( , ) [2] (ii) Write down the equation of this line of symmetry. Answer(d)(ii) [1] (e) Find the gradient of the other line of symmetry. Answer(e) [1]

13 marks

Mark scheme: 4 (a) (i) −10, −20, −60, 30, 20, 15 B2 B1 for –20 (x = –3) or 20 (x = 3) (ii) Their 12 points plotted correctly. P3ft P2ft for 10 or 11 points correct. P1ft for 8 or 9 points or 1 quadrant correct. Smooth curves through all points. C1 Two distinct curves; no part of curves between x = –1 and x = 1 (b) 2 B1 (c) Correct lines ruled B1,B1 Minimum length from x = –3 to x = 3. (d) (i) (2.4 to 2.5, 24 to 25) B1ft ft their points of intersection (−2.4 to −2.5, −24 to −25) B1ft ft their points of intersection (ii) y = 10x oe B1 cao (e) −10 B1 cao [13] IGCSE – May/June 2007 0580/0581 03

This question in 0580/31 May/June 2007

36 , (x ≠ 0). For3 (a) Complete the table for the function y = x Examiner's Use x −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 y −7.2 −9 −18 18 9 7.2 [3] 36 (b) On the grid below, draw the graph of y = for −6 x −1 and 1 x 6. x y 40 30 20 10 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –10 –20 –30 –40 [4] (c) Use your graph to find x when y = 21. Answer(c) x = [1] (d) Complete the table for the function y = x2. For Examiner's Use x −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 y 25 16 4 1 1 4 16 25 [2] (e) On the same grid, draw the graph of y = x2 for −6 x 6. [4] 36(f) Write down the co-ordinates of the point of intersection of the graphs of y = and y = x2. x Answer(f)( , ) [1]

15 marks

Mark scheme: 3 (a) –6, –12, –36, 36, 12, 6 B3 B1 for ± 36, B1 for ± 12, B1 for ± 6 SC1 for any 3 correct (b) 12 points plotted P3 correct points ft within 1 mm P2 for 10 or 11, P1 for 8 or 9, P1 for 1 correct branch 2 curves drawn C1 must be smooth branches of rectangular hyperbola (c) 1.6 to 1.8 B1 ft (d) 36, 9, 0, 9, 36 B2 B1 for 4 correct (e) 13 points plotted P3 correct points ft within 1 mm P2 for 11 or 12 P1 for 9 or 10 curve drawn C1 must be smooth parabola (f) 3.3, 10.9 B1ft x from 3.2 to 3.4, y from 10.0 to 12.0 [15]

This question in 0580/31 Oct/Nov 2007

8 (a) The width of a rectangle is x centimetres. For Examiner's The length of the rectangle is 3 centimetres more than the width. Use Write down an expression, in terms of x, for (i) the length of the rectangle, Answer(a)(i) cm [1] (ii) the area of the rectangle. Answer(a)(ii) cm2 [1] (iii) The area of the rectangle is 7 square centimetres. Show that x2 + 3x − 7 = 0. Answer (a)(iii) [1] (b) (i) Complete the tables of values for the equation y = x2 + 3x − 7. x −5 −4 −3 −2 −1 0 1 2 y 3 −7 −9 −7 3 [3] (ii) On the grid below, draw the graph of y = x2 + 3x − 7 for −5 Y x Y 2. For y Examiner's Use 4 2 x –5 –4 –3 –2 –1 0 1 2 A –2 –4 –6 –8 –10 [4] (c) (i) Use your graph to find the solutions to the equation x2 + 3x − 7 = 0. Answer(c)(i) x = or x = [2] (ii) Find the length of the rectangle in part (a). Answer(c)(ii) cm [1] (d) The point A(1, −1) is marked on the grid. (i) Draw a straight line through A with a gradient of 2. [1] (ii) Write down the equation of this line in the form y = mx + c. Answer(d)(ii) y = [2]

16 marks

Mark scheme: 8 (a) (i) x + 3 B1 (ii) x (x + 3) or x² +3x B1 ft from their (a)(i) (iii) x² +3x = 7 x² +3x - 7 = 0 E1 both lines seen (b) (i) -3, -9, -3 B3 B1, B1, B1 (ii) 8 points correctly plotted P3 ft P2ft or 6 or 7, P1ft for 4 or 5 (+/- 1/2 small square) smooth curve C1 (must go below y = -9) IGCSE – May/June 2008 0580/0581 03 (c) (i) 1.5 to 1.6 B1 ft -4.5 to -4.6 B1 ft ft is their intersections with the x-axis (ii) 4.5 to 4.6 B1 ft ft is their positive (c)(i) + 3 (d) (i) correct line L1 long enough to cross y axis (+/- 1/2 small square) (ii) (y =) 2x - 3 B1,B1ft B1 for 2 (as coefficient of x) B1 ft for their intersection with the y-axis [16]

This question in 0580/31 May/June 2008

7 (a) Complete the table of values for the equation y = x2 + x − 3. For Examiner's Use x −4 −3 −2 −1 0 1 2 3 y 9 −1 −3 −1 9 [3] (b) On the grid, draw the graph of y = x2 + x − 3. y 10 9 8 7 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 –1 –2 –3 –4 [4] (c) Write down the coordinates of the lowest point of the curve. Answer(c) ( , ) [2] (d) (i) Draw the line of symmetry of the graph. [1] (ii) Write down the equation of the line of symmetry. Answer(d)(ii) [1]

11 marks

Mark scheme: y 7 (a) 3, −3, 3 W3 W1 for each correct value (b) 8 correctly plotted points W3ft W2 for 6 or 7 points, W1 for 4 or 5 points Smooth curve W1 Half square accuracy must go below line y = −3 (c) ( −0.5, −3.25) W2ft W1 for one coordinate correct Ft their graph but −1 < x < 0 and y < −3 Allow calculated if exact values (W2 or W1) (d) (i) Line x = −0.5 drawn W1cao Half square accuracy (ii) x = −0.5 oe W1ft Ft any vertical line only

This question in 0580/31 Oct/Nov 2008

7 y = 9x – x2. For Examiner's (a) Complete the table of values for this equation. Use x 0 1 2 3 4 5 6 7 8 9 y 8 20 20 8 0 [3] (b) On the grid below, draw the graph of y = 9x – x2 for 0 Y x Y 9. y 22 20 18 16 14 12 10 8 6 4 2 x 0 1 2 3 4 5 6 7 8 9 [4] (c) Write down the values of x and y at the highest point of the curve. For Examiner's Use Answer(c) x = y = [2] (d) (i) On the grid, draw the line y = 6 for 0 Y x Y 9. [1] (ii) Use this line to find the solutions of the equation 9x – x2 = 6. Give your answers correct to one decimal place. Answer(d)(ii) x = or x = [2]

12 marks

Mark scheme: 7 (a) x 0 1 2 3 4 5 6 7 8 9 3 W2 for 4 correct y 0 8 14 18 20 20 18 14 8 0 W1 for 3 correct (b) Their 10 points correctly plotted, within P3ft P2ft for 8 or 9 correct half a square. P1ft for 6 or 7 correct Smooth curve through the 10 correct C1 Shape must be correct and the curve goes above points y = 20. (c) (x =) 4.4 to 4.6 1cao (y =) 20.1 to 20.5 1cao (d) (i) Ruled line y = 6 1 (ii) 8.1 to 8.5 Must be to 1 decimal place 1cao SC1 for both correct but not to 1dp e.g. 8.27 and 0.5 to 0.9 Must be to 1 decimal place 1cao 0.73 IGCSE – May/June 2009 0580, 0581 03

This question in 0580/31 May/June 2009

3 For 6 (a) Complete the table of values for the function y = , x ≠ 0. Examiner's x Use x −3 −2.5 −2 −1.5 −1 −0.5 −0.3 0.3 0.5 1 1.5 2 2.5 3 y −1 −1.2 −2 −3 −6 3 2 1.5 1 [3] 3 (b) On the grid below, draw the graph of y = for −3 Y x Y −0.3 and 0.3 Y x Y 3. x y 10 9 8 7 6 5 4 3 2 1 x –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 –6 –7 –8 –9 [5] –10 3 For (c) Use your graph to solve the equation = 7. Examiner's x Use Answer(c) x = [1] 2 x (d) Complete the table of values for y = − 1 . 3 x −3 0 3 y [2] 2 x (e) On the grid, draw the straight line y = − 1 for −3 Y x Y 3. [2] 3 2 x (f) Write down the co-ordinates of the points where the line y = − 1 intersects 3 3 the graph of y = . x Answer(f) ( , ) and ( , ) [2]

15 marks

Mark scheme: 6 (a) –1.5 –10 10 6 1.2 3 B2 for 3 or 4 correct, B1 for 2 correct (b) 14 points plotted accurately P3ft P2ft for 11, 12 or 13 points, P1ft for 8, 9 or 10 2 smooth correct curves C1 No part across y-axis B1 Indep (c) 0.4 to 0.5 1 (d) −3 −1 1 2 B1 for 2 correct (e) Ruled line from (−3, −3) to (3, 1) 2 SC1 for freehand or short ruled line – must meet curve twice or P1 for their 3 points plotted (f) (−1.5, −2) and (3, 1) 1, 1

This question in 0580/31 May/June 2010

4 For 6 (a) Complete the table of values for y = , x ≠ 0 . Examiner's x Use x −4 −3 −2 −1 − 0.5 0.5 1 2 3 4 y −1.3 −2 −8 8 4 2 [2] 4 (b) On the grid below, draw the graph of y = , for – 4 Y x Y – 0.5 and 0.5 Y x Y 4. x y 8 7 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 4 –1 –2 –3 –4 –5 –6 –7 –8 [4] (c) Complete the following statement. For Examiner's Use 4 The point (−2.5, ) lies on the graph of y = . [1] x (d) (i) On the grid, draw the line y = 5. [1] 4 (ii) Use your graphs to solve the equation = 5 . x Answer(d)(ii) x = [1] (e) (i) On the grid, draw the straight line joining the points (− 0.5 , − 8 ) and ( 2 , 2 ). [2] (ii) Find the gradient of this line. Answer(e)(ii) [1] (iii) Write down the equation of this line in the form y = mx + c. Answer(e)(iii) y = [2]

14 marks

Mark scheme: 6 (a) –1, –4, 1.3, 1 2 B1 for –1 and 1 and B1 for –4 and 1.3 (b) 10 points plotted ½ small square P3ft P2 for 8 or 9 points, P1 for 5 or 6 or 7 points accuracy smooth correct curves not across y-axis C1 (c) –1.6 correct or ft 1ft ft from their graph (d) (i) y = 5 drawn 1 (ii) (x =) 0.8 correct or ft 1ft ft from their graph (e) (i) Ruled line drawn from (–0.5, –8) 2 B1 for ruled line drawn from either point not to (2, 2) horizontal or vertical (ii) 4 cao 1 (iii) y = 4x – 6 or 2ft B1 ft y = 4x + k or y = their (e)(ii) x + k or y = their (e)(ii) x + their intercept y = jx – 6 or y = jx + their intercept or y = 4x + their intercept

This question in 0580/31 Oct/Nov 2010

5 For y Examiner's Use 12 10 8 6 A 4 2 x –12 –10 –8 –6 –4 –2 0 2 4 6 8 10 12 –2 –4 B –6 –8 –10 –12 A graph is drawn on the grid. Points A and B are marked on the curves. (a) (i) Write down the co-ordinates of the points A and B. Answer(a)(i) A( , ) and B( , ) [2] (ii) The equation of the graph is xy = n. Write down the value of n. Answer(a)(ii) n = [1] (b) (i) Write down the order of rotational symmetry of the graph. For Examiner's Use Answer(b)(i) [1] (ii) On the grid, draw the lines of symmetry of the graph. [2] (iii) Write down the equation of each line of symmetry. Answer(b)(iii) and [2] (c) (i) One line of symmetry crosses both curves. Write down the x co-ordinates of the points where this line meets each curve. Give your answers to 1 decimal place. Answer(c)(i) x = and x = [2] (ii) On the grid, draw the line which passes through the point (0, 4) and is parallel to the line of symmetry in part (c)(i). [1] (iii) Write down the equation of this line in the form y = mx + c. Answer(c)(iii) y = [2]

13 marks

Mark scheme: 5 (a) (i) (2, 6) and (–3, –4) 2 B1 for one pair correct (ii) (n =) 12 cao 1 (b) (i) 2 cao 1 (ii) Lines of symmetry drawn 1, 1 (iii) y = x oe and y = –x oe cao 1, 1 (c) (i) (x =) 3.3 to 3.7 and 1ft ft their graph (x =) –3.3 to –3.7 1ft (ii) Line parallel to line in (c)(i) 1ft (c)(i) line must be linear through (0, 4) (iii) y = x + 4 oe 2ft B1 for y = mx + 4 (m ≠ 0) or for y = x + k (k ≠ 0) B1ft for y = mx + ‘4’ (m ≠ 0) or for y = ‘m’x + k (k ≠ 0) IGCSE – October/November 2010 0580 32

This question in 0580/32 Oct/Nov 2010

9 (a) (i) Complete the table for y = 12 – x2. For Examiner's Use x 0 1 2 3 4 y 12 11 – 4 [2] (ii) On the grid, draw the graph of y = 12 – x2 for 0 Y x Y 4. y 12 11

2 marks

Mark scheme: 9 (a) (i) 8, 3 1, 1 (ii) 5 points correctly plotted 2ft P1 for 4 correct points ft Smooth curve through their 5 1 points (iii) 3.4 Y x Y 3.6 1ft ft their intersection with x-axis (b) (i) 3, 2, 1.5 1, 1, 1 B1 each (ii) 8 points correctly plotted 2ft P1 for 6 or 7 points Smooth branch of rectangular 1 hyperbola through 12 points (c) (1 < x Y 1.2, 10.6 Y y < 11) 1ft ft to same accuracy intersections of their two (2.6 Y x < 3, 4.2 Y y Y 4.5) 1ft graphs

This question in 0580/33 Oct/Nov 2010

18 For 7 (a) The table shows some values for y = . Examiner's x Use x O9 O6 O4 O3 O2 2 3 4 6 9 y O2 O4.5 O9 4.5 3 (i) Complete the table. [2] 18 (ii) On the grid, draw the graph of y = for O9 Y x Y O2 and 2 Y x Y 9 . x y 9 8 7 6 5 4 3 2 1 x –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 –1 –2 –3 –4 –5 –6 –7 –8 –9 [4] 18 (iii) Use your graph to solve the equation = O5 . x Answer(a)(iii) x = [1] (b) (i) Complete the table of values for y = 2x + 3 . For Examiner's Use x O4 O3 2 3 y O5 7 [2] (ii) On the grid, draw the graph of y = 2x + 3 for O4 Y x Y 3 . [1] (iii) Find the co-ordinates of the points of intersection of the graphs of 18 y = and y = 2x + 3 . x Answer(b)(iii) ( , ) and ( , ) [2]

12 marks

Mark scheme: 7 (a) (i) −3, −6, 9, 6, 2 2 B1 for 4 correct (ii) Graph P3ft P2ft for 8 or 9 points correct P1ft for 6 or 7 points correct C1 Correct curve and not crossing axis (iii) −3.7 to −3.5 1ft ft their curve (b) (i) −3, 9 1, 1 (ii) Ruled continuous line y = 2x + 3 1 Line long enough to intersect both parts (iii) (2.2 to 2.5, 7.5 to 7.8) 1ft ft their line intersection with the curves (−4.0 to −3.7, −4.8 to −4.5) 1ft

This question in 0580/32 Oct/Nov 2011

6 (a) Complete the table for y = 4 + 2x O x2. For Examiner's Use x –2 –1 0 1 2 3 4 y 1 5 1 [2] (b) On the grid, draw the graph of y = 4 + 2x O x2 for –2 Y x Y 4 . y 6 5 4 3 2 1 x –2 –1 0 1 2 3 4 –1 –2 –3 –4 [4] (c) (i) Draw the line of symmetry of the graph. [1] (ii) Write down the equation of this line of symmetry. Answer(c)(ii) [1] (d) Use your graph to solve the equation 4 + 2x O x2 = 0. Answer(d) x = or x = [2]

10 marks

Mark scheme: 6 (a) –4, …, 4, …, 4, …, –4 2 B1 for both –4s B1 for both 4s (b) 7 points plotted ft 3ft P2 for 5 or 6 points plotted ft P1 for 3 or 4 Reasonable curve through at least 6 1ft Only ft if shape parabola points (c) (i) The line x = 1 drawn 1ft (ii) x = 1 1ft (d) –1.4 to –1.1, 3.1 to 3.4 2ft B1 B1ft if not in these ranges

This question in 0580/33 Oct/Nov 2011

10 For 4 (a) The table shows some values of y = . Examiner's x Use x –8 –5 –4 –2 –1 1 2 4 5 8 y –1.25 –5 10 2 (i) Complete the table. [2] 10 (ii) On the grid opposite, draw the graph of y = for −8 Y x Y −1 and 1 Y x Y=8 . [4] x (b) (i) On the same grid, draw the straight line through the points (−3, −5) and (1, 3). Extend the line to the edges of the grid. [2] 10 (ii) Find the co-ordinates of the points of intersection of this line with the graph of y = . x Answer(b)(ii) ( , ) and ( , ) [2] (c) For the line in part (b)(i) (i) work out the gradient, Answer(c)(i) [2] (ii) write down the equation in the form y = mx + c . Answer(c)(ii) y = [1] y For Examiner's 10 Use 9 8 7 6 5 4 3 2 1 x –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10

13 marks

Mark scheme: 4 (a) (i) −2, −2.5, −10 2 B1 for 4 or 5 correct 5, 2.5, 1.25 (ii) 10 points correctly plotted 3ft B2ft for 8 or 9 points correctly plotted. B1ft for 6 or 7 points correctly plotted Smooth curve 1 (b) (i) Ruled line through both given points 2 B1 for not ruled but otherwise correct or through just 1 of the points (ii) (−2.5, −4),(2, 5) 2ft B1 for 1 correct. ft their line and their curve. (c) (i) 2 cao 2 M1 for change in y / change in x for 2 correct points (ii) (y =) 2x + 1 1ft Ft (y=) their (c)(i) x + intercept of their line in (b)(i)

This question in 0580/32 May/June 2012

8 (a) Complete the table of values for y = x2 – 2x + 5 . For Examiner's Use x –3 –2 –1 0 1 2 3 4 5 y 20 8 8 20 [3] (b) On the grid, draw the graph of y = x2 – 2x + 5 for −3 Y x Y 5 . y 22 20 18 16 14 12 10 8 6 4 2 x –4 –3 –2 –1 0 1 2 3 4 5 6 [4] (c) (i) On the grid, draw the line of symmetry of the graph. [1] (ii) Write down the equation of the line of symmetry. Answer(c)(ii) [1] (d) (i) On the grid, draw the line y = 12 . [1] For Examiner's (ii) Use your graph to solve the equation x2 – 2x + 5 = 12 . Use Answer(d)(ii) x = or x = [2] (e) The equation of a straight line is y = 6 – 3x . (i) Write down the gradient of this line. Answer(e)(i) [1] (ii) Write down the co-ordinates of the point where this line crosses the y-axis. Answer(e)(ii) ( , ) [1] (iii) Write down the equation of a line parallel to y = 6 – 3x . Answer(e)(iii) [1] (f) Simplify 3(2x + 1) O=2(6 – 3x) . Answer(f) [2]

17 marks

Mark scheme: 8 (a) (20) 13 (8) 5 4 5 (8) 13 (20) 3 B2 for 4 correct B1 for 2 or 3 correct or a correct substitution seen (b) correctly plotting 9 points and 4 P3 for correctly plotting 9 points, P2 for correctly connecting with a smooth curved line plotting 7 or 8 points and P1 for 5 or 6 points C1 for a smooth curve (c) (i) correct line of symmetry cao 1 (ii) x = 1 1ft ft their line (d) (i) correct line 1 (ii) –1.9 to –1.7 and 3.7 to 3.9 1ft,1ft SC1 for correct co-ordinates (e) (i) –3 cao 1 (ii) (0,6) cao 1 (iii) y = c – 3x 1 c can be any number except 6 (f) 12x – 9 or 3(4x – 3) 2 B1 for 6x + 3, –12 + 6x, 12x or –9 IGCSE – May/June 2012 0580 33

This question in 0580/33 May/June 2012

9 (a) Complete the table of values for y = 8 + 3x – x2. For Examiner's Use x –3 –2 –1 0 1 2 3 4 5 6 y –10 8 10 10 –10 [3] (b) On the grid, draw the graph of y = 8 + 3x – x2 for –3 Y x Y 6 . y 12 10 8 6 4 2 x –3 –2 –1 0 1 2 3 4 5 6 –2 –4 –6 –8 –10 [4] (c) Write down the equation of the line of symmetry of the graph. Answer(c) [1] (d) (i) On the grid, draw the graph of y = 6 . [1] (ii) Use your graphs to solve the equation 8 + 3x – x2 = 6 . Answer(d)(ii) x = or x = [2]

11 marks

Mark scheme: 9 (a) y-values –2, 4, 8, 4, –2 3 B2 for 3 or 4 correct B1 for 2 correct (b) 10 correctly plotted points 3ft B2ft for 8 or 9 points B1ft for 6 or 7 points Smooth curve through 10 correct 1 Curve must pass above y = 10 points and correct shape. (c) x = 1.5 oe 1 (d) (i) Line y = 6 drawn 1 (ii) x = 3.5 to 3.7 1ft Ft their curve and their line drawn x = – 0.7 to – 0.5 1ft IGCSE – October/November 2012 0580 31

This question in 0580/31 Oct/Nov 2012

8 For 2 (a) The table shows some values of the function y = x O . Examiner's x Use x O8 O6 O5 O4 O2 O1 1 2 4 5 6 8 y O7 O4.7 O3.4 O2 7 O2 3.4 4.7 7 (i) Complete the table. [3] 8 (ii) On the grid on the opposite page, draw the graph of y = x O for x O8 Y x Y O1, 1 Y x Y 8 . [5] (iii) Write down the order of rotational symmetry of the graph. Answer(a)(iii) [1] 8 (iv) Use your graph to solve the equation x O = 0 . x Answer(a)(iv) x = or x = [2] 1 (b) (i) Write down the gradient of the line y = x + 1 . 2 Answer(b)(i) [1] 1 (ii) Complete the table below for the line y = x + 1 . 2 x O8 O4 0 4 8 y O3 3 [2] 1 (iii) On the grid, draw the line y = x + 1 for O8 Y x Y 8 . [1] 2 8 1 (c) Write down the co-ordinates of the points of intersection of y = x O and y = x + 1 . x 2 Answer(c) ( , ) and ( , ) [2] y For Examiner's 8 Use 7 6 5 4 3 2 1 x–8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 –3 –4 –5 –6 –7 –8

17 marks

Mark scheme: 2 (a) (i) 2 −7 2 1,1,1 (ii) 12 correctly plotted points 3ft P2ft for 10 or 11 correct. P1ft for 8 or 9 correct 2 smooth curves through 12 C1 correct points and correct shape Two separate branches not B1 crossing the y-axis (iii) 2 1 (iv) 2.7 to 3.0, 1 −3.0 to −2.7 1 IGCSE – October/November 2012 0580 32 (b) (i) 1 or 0.5 2 1 (ii) −1 1 5 2 B1 for 2 correct (iii) Correct ruled continuous line 1 drawn (c) (5.0 to 5.2, 3.5 to 3.7) 1ft Ft ± 0.1 from their intersections (−3.2 to –3.0, −0.7 to –0.5) 1ft

This question in 0580/32 Oct/Nov 2012

9 (a) Complete the table of values for y = x2 + 2x O=4 . For Examiner's Use x O4 O3 O2 O1 0 1 2 3 y 4 O4 O4 11 [3] (b) On the grid, draw the graph of y = x2 + 2x O=4 for O4 Y x Y 3 . y 12 11 10 9 8 7 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 –6 [4] (c) (i) Draw the line of symmetry on the graph. [1] For Examiner's Use (ii) Write down the equation of this line of symmetry. Answer(c)(ii) [1] (d) Use your graph to solve the equation x2 + 2x O=4 = 3 Answer(d) x = or x = [2] Question 10 is printed on the next page.

11 marks

Mark scheme: 9 (a) −1, −5, −1, 4 3 B2 3 correct B1 2 correct (b) 8 correct points plotted 3ft B2ft 6 or 7 points plotted ft B1ft 4 or 5 points plotted ft Smooth curve through 8 correct points 1 and correct shape (c) (i) x = − 1 drawn 1 (ii) x = − 1 oe cao 1 (d) 1.8 to 1.9 and –3.8 to – 3.9 2 ft B1 B1 IGCSE – October/November 2012 0580 33

This question in 0580/33 Oct/Nov 2012

9 (a) (i) Complete the table of values for y = x2 + x . For Examiner′s Use x –3 –2 –1 0 1 2 3 y 6 0 0 6 [2] (ii) On the grid, draw the graph of y = x2 + x for –3 Ğ x Ğ 3 . y 14 13 12 11 10 9 8 7 6 5 4 3 2 1 x –3 –2 –1 0 1 2 3 –1 –2 [4] (iii) On the grid, draw the line y = 10. [1] (iv) Use both your graphs to solve x2 + x = 10 for –3 Ğ x Ğ 3 . Answer(a)(iv) x = … [1] 2 For (b) Another line, L, has the equation y = 3 x – 5 . Examiner′s Use (i) Write down the gradient of L. Answer(b)(i) … [1] (ii) Write down the equation of a straight line that is parallel to L. Answer(b)(ii) … [1] (c) y 5 K 4 3 2 1 x –2 –1 0 1 2 3 4 –1 –2 –3 –4 –5 –6 –7 Write the equation of the line, K, in the form y = mx + c . Answer(c) y = … [3] _____________________________________________________________________________________

13 marks

Mark scheme: 9 (a) (i) 2 and 2 1 all in the correct places 12 1 (ii) 7 points correctly plotted 3ft P2ft for 5 or 6 points correctly plotted P1ft for 3 or 4 points correctly plotted correct curve through the 7points 1 (iii) correct line 1 Must be ruled and continuous (iv) 2.6 – 2.8 1ft ft their curve and their line (b) (i) 2 1 3 2 (ii) y = x + c 1 c not –5 3 (c) [y =] 2x – 3 3 M2 for y = 2x + p rise or M1 for attempt at gradient i.e. run B1 for y = qx – 3 q≠0

This question in 0580/31 May/June 2013

7 (a) Complete the table of values for the function y = x2 – 5x + 2 . For Examiner′s Use x –1 0 1 2 3 4 5 y –2 –4 –4 2 [2] (b) On the grid, draw the graph of y = x2 – 5x + 2 for –1 Ğ x Ğ=5 . y 9 8 7 6 5 4 3 2 1 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6 –7 [4] (c) (i) Write down the co-ordinates of the lowest point of the graph of y = x2 – 5x + 2 . For Examiner′s Use Answer(c)(i) ( … , … ) [1] (ii) On the grid, draw the line y = –1 . [1] (iii) Write down the x co-ordinates of the two points where y = –1 crosses the graph of y = x2 – 5x + 2 . Answer(c)(iii) x = … and x = … [2] (d) The point (5, 2) is refl ected in the y-axis. Write down the co-ordinates of the image of the point. Answer(d) ( … , … ) [1] (e) Write down the equation of the line, l, drawn on the grid below. Give your answer in the form y = mx + c . y 7 l 6 5 4 3 2 1 x –3 –2 –1 0 1 2 3 4 5 –1 –2 –3 –4 –5 Answer(e) y = … [3] _____________________________________________________________________________________

14 marks

Mark scheme: 7 (a) 8, 2, –2, 2 B1 for 2 correct y values (b) 7 correctly plotted points 3ft P2ft for 5 or 6 correctly plotted points P1ft for 3 or 4 correctly plotted points Correct smooth curve going below 1 y = –4 at lowest point (c) (i) ( 2.5cao , –4.25) 1 (ii) y = – 1 drawn 1 must be ruled and continuous (iii) 0.5 to 0.9, 4.1 to 4.5 1ft,1ft ft is the x coordinates of the intersection of their line and their curve (d) (– 5, 2) 1 (e) [y] = – 2 x + 3 3 M2 for y = – 2 x + p or y = 2x + 3 or M1 for y = 2x + q or for attempt at rise/run even if negative not shown B1 for y = kx + 3 k≠0 4 M1 f [ 60]

This question in 0580/32 May/June 2013

4 (a) The table shows some values of y = x2 – 2x – 1. For Examiner′s Use x –3 –2 –1 0 1 2 3 4 y 14 2 –1 –2 7 (i) Complete the table. [2] (ii) On the grid, draw the graph of y = x2 – 2x – 1 for –3 Y x Y 4. y 16 14 12 10 8 6 4 2 x –3 –2 –1 0 1 2 3 4 –2 –4 [4] (b) Write down the equation of the line of symmetry of the graph. For Examiner′s Use Answer(b) … [1] (c) The point with co-ordinates (–3, 7) lies on the line y = –x + 4 . (i) Write down the co-ordinates of two other points on this line. Use x co-ordinates so that –3 < x Y 4 . Answer(c)(i) ( … , … ) and ( … , … ) [2] (ii) On the grid, draw the line y = –x + 4 for –3 Y x Y 4 . [1] (iii) Use both graphs to fi nd the solutions of the equation x2 – 2x – 1 = –x + 4 . Answer(c)(iii) x = … or x = … [2] _____________________________________________________________________________________

12 marks

Mark scheme: 4 (a) (i) 7, –1, 2 2 B1 for any 2 correct (ii) 8 points plotted 3ft P2ft for 6 or 7 correct P1ft for 4 or 5 correct Correct smooth curve 1 (b) x = 1 1 (c) (i) Two correct points 1,1 x –2 –1 –0 –1 –2 –3– 4 y –6 –5 –4– 3– 2 –1 –0 (ii) Correct line drawn 1 Must be ruled and continuous (iii) –1.9 to –1.7, 2.7 to 2.9 2ft 1 for each correct

This question in 0580/33 May/June 2013

7 (a) Complete the table of values for y = x2 – x + 2 . For Examiner′s Use x –3 –2 –1 0 1 2 3 4 y 8 2 4 [3] (b) On the grid, draw the graph of y = x2 – x + 2 for −3 Y x Ğ 4 . y 16 14 12 10 8 6 4 2 x –3 –2 –1 0 1 2 3 4 [4] (c) Write down the equation of the line of symmetry of the graph. For Examiner′s Use Answer(c) … [1] (d) (i) On the grid, draw the line y = 9 . [1] (ii) Solve the equation x2 – x + 2 = 9 . Answer(d)(ii) x = … or x = … [2] _____________________________________________________________________________________

11 marks

Mark scheme: 7 (a) 14, 4, 2, 8, 14 3 B2 for 4 correct B1 for 2 or 3 correct (b) 8 points correctly plotted P3FT P2FT for 6 or 7 points correctly plotted P1FT for 4 or 5 points correctly plotted Smooth and correct curve through all C1 correct points 1 (c) x = 0.5 or x = 1 2 (d) (i) y = 9 ruled 1 (ii) –2.15 to –2.25 1FT 3.15 to 3.25 1FT

This question in 0580/31 Oct/Nov 2013

5 Examiner′s5 (a) Complete the table of values for y = . x Use x –5 –4 –3 –2 –1 1 2 3 4 5 y –1.67 –2.5 –5 5 1.67 1.25 [2] 5 (b) On the grid, draw the graph of y = for –5 Y x Y –1 and 1 Y x Y 5. x y 6 5 4 3 2 1 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6 [4] 5 (c) Use your graph to solve the equation = 4 . x Answer(c) x = … [1] (d) (i) On the grid, draw the line x = –3.5 . [1] (ii) On the grid, plot the point (5, –3) and label it P. [1] (iii) Draw the line that passes through P and is perpendicular to x = –3.5 . [1] _____________________________________________________________________________________

10 marks

Mark scheme: 5 (a) –1 –1.25 2.5 1 2 B1 for two correct (b) 10 correctly plotted points P3FT P2FT for 8 or 9 correctly plotted P1FT for 6 or 7 correctly plotted Two correct smooth curves through C1 all correct points and not across y-axis (c) 1.15 to 1.35 1FT (d) (i) Line x = –3.5 ruled 1 (ii) (5, –3) plotted 1 (iii) line y = –3 ruled 1FT IGCSE – October/November 2013 0580 32

This question in 0580/32 Oct/Nov 2013

5 (a) (i) Complete the table for y = 5 + 3x – x2. For Examiner′s Use x –2 –1 0 1 2 3 4 5 y –5 5 7 5 –5 [3] (ii) On the grid, draw the graph of y = 5 + 3x – x2 for –2 Y x Y 5. y 8 7 6 5 4 3 2 1 x –2 –1 0 1 2 3 4 5 –1 –2 –3 –4 –5 –6 [4] (b) Use your graph to solve the equation 5 + 3x – x2 = 0 . Answer(b) x = … or x = … [2] (c) (i) On the grid, draw the line of symmetry of y = 5 + 3x – x2. [1] For Examiner′s Use (ii) Write down the equation of this line of symmetry. Answer(c)(ii) … [1] (d) (i) On the grid, draw a straight line from (–1, 1) to (3, 5). [1] (ii) Work out the gradient of this line. Answer(d)(ii) … [2] (iii) Write down the equation of this line in the form y = mx + c. Answer(d)(iii) y = … [1] _____________________________________________________________________________________

15 marks

Mark scheme: 5 (a) (i) 1, 7, 1 1, 1, 1 (ii) 8 points correctly plotted P3FT P2FT for 6 or 7 correct P1FT for 4 or 5 correct Correct smooth curve through all 8 C1 correct points IGCSE – October/November 2013 0580 33 (b) –1.1 to –1.3 and 4.1 to 4.3 1FT, 1FT (c) (i) Line x = 1.5 drawn 1 (ii) x = 1.5 oe 1FT Equation of their line in (c)(i) (d) (i) Ruled continuous line drawn 1 rise (ii) 1 2 M1 for for their line run (iii) [y =] x + 2 1FT their (d)(ii) + their 2

This question in 0580/33 Oct/Nov 2013

86 (a) (i) Complete the table of values for y = , x ≠ 0 . x x –8 –4 –2 –1 1 2 4 8 y –2 2 [3] 8 (ii) On the grid, draw the graph of y = for –8 Ğ x Ğ –1 and 1 Ğ x Ğ 8 . x y 8 6 4 2 x –8 –6 –4 –2 0 2 4 6 8 –2 –4 –6 –8 [4] (iii) Write down the order of rotational symmetry of your graph. Answer(a)(iii) … [1] (b) (i) Complete this table of values for y = 1.5x + 3 . x –6 –4 –2 0 2 y –6 3 [2] (ii) On the grid, draw the graph of y = 1.5x + 3 . [1] 8 (c) Use your graphs to solve the equation = 1.5x + 3 . x Answer(c) x = … or x = … [2] (d) Write down the gradient of the graph of y = 1.5x + 3 . Answer(d) … [1] __________________________________________________________________________________________

14 marks

Mark scheme: 6 (a) (i) −1, −4, −8, 8, 4, 1. 3 1 for each symmetrical pair (ii) 8 points correctly plotted, within ½ square. 3FT B2FT for 6 or 7 correct Or B1 FT for 4 or 5 correct 2 smooth correct curves, not joined 1 (iii) 2 1 IGCSE – May/June 2014 0580 31 (b) (i) −3 0 6 2 B1 for two correct (ii) Correct ruled line 1 (c) 1.4 to 1.6 and −3.6 to −3.4 1FT,1FT FT from their graph ±0.1 (d) 1.5 1

This question in 0580/31 May/June 2014

6 (a) Complete the table of values for y = x2 + 2x – 3 . x –4 –3 –2 –1 0 1 2 3 4 y 0 –3 –4 –3 0 5 21 [2] (b) On the grid, draw the graph of y = x2 + 2x – 3 for –4 Ğ x Ğ 4 . y 25 20 15 10 5 x –4 –3 –2 –1 0 1 2 3 4 –5 [4] (c) On the grid, draw the line y = 10 . [1] (d) Use your graphs to solve the equation x2 + 2x – 3 = 10 for –4 Y x Y 4 . Answer(d) x = … [1] __________________________________________________________________________________________

8 marks

Mark scheme: 6 (a) 5 12 2 B1, B1 (b) 9 points plotted correctly 3FT B2FT for 7 or 8 points correctly plotted B1FT for 5 or 6 points correctly plotted correct smooth curve through all 1 9 correct points (c) correct ruled line 1 minimum length must touch y axis and curve (d) 2.7 to 2.8 1FT FT their curve and ruled line IGCSE – May/June 2014 0580 32

This question in 0580/32 May/June 2014

6 (a) (i) Complete the table of values for y = 8 – x2. x –3 –2 –1 0 1 2 3 y –1 8 7 –1 [2] (ii) On the grid, draw the graph of y = 8 – x2 for –3 Y x Y 3 . y 12 11 10 9 8 7 6 5 4 3 2 1 x –3 –2 –1 0 1 2 3 –1 –2 –3 –4 [4] (iii) Write down the equation of the line of symmetry of the graph. Answer(a)(iii) … [1] (iv) Use your graph to solve the equation 8 – x2 = 0. Answer(a)(iv) x = … or x = … [2] (b) (i) On the grid, plot the points (–2, 8) and (2.5, –1). Draw a straight line through these points. [2] (ii) Find the equation of your line in the form y = mx + c. Answer(b)(ii) y = … [3] (iii) Write down the co-ordinates of the point of intersection of your line with y = 8 – x2. Answer(b)(iii) ( … , … ) [1] __________________________________________________________________________________________

15 marks

Mark scheme: 6 (a) (i) 4, 7, 4 2 B1 for 2 correct (ii) 7 points correctly plotted 3FT B2 for 5 or 6 correct B1 for 3 or 4 correct Correct curve through the points 1 (iii) x = 0 1 (iv) 2.7 to 2.9, –2.7 to –2.9 1, 1 (b) (i) Points correctly plotted and a ruled line 2 B1 for 1 correct plot. (even if line is through points and beyond them. not drawn) (ii) [y =] –2x + 4 3 B2 for –2x + j or B1 for kx + 4 k ≠ 0 or [gradient =] riserun correct values (iii) 1 ( –1.2 to –1.4, 6.4 to 6.6)

This question in 0580/31 Oct/Nov 2014

20 .6 (a) (i) Complete the table of values for y = x x –8 –5 –4 –2.5 2.5 4 5 8 y –2.5 –4 8 4 [2] 20 (ii) On the grid, draw the graph of y = for –8 Y x Y –2.5 and 2.5 Y x Y 8. x y 9 8 7 6 5 4 3 2 1 x –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 –3 –4 –5 –6 –7 –8 –9 [4] 20 (iii) By drawing a suitable line on your graph solve the equation = 6. x Answer(a)(iii) x = … [2] (b) x –8 0 8 y 1 (i) Complete the table for y = x – 1. [2] 2 1 (ii) On the grid, draw the graph of y = x – 1 for –8 Y x Y 8. [1] 2 1 (iii) Write down the gradient of y = x – 1. 2 Answer(b)(iii) … [1] 20 1 (c) Write down the values of x at the points of intersection of the graphs of y = and y = x – 1. x 2 Answer(c) x = … and x = … [2] __________________________________________________________________________________________

14 marks

Mark scheme: 6 (a) (i) –5 –8 5 2.5 2 B1 for 3 correct (ii) 8 points correctly plotted B3FT B2FT for 6 or 7 correct points Correct curve 1 B1FT for 4 or 5 correct points (iii) Ruled line y = 6 drawn 1 Independent marks 3.1 to 3.6 1 (b) (i) –5 –1 3 2 B1 for 2 correct (ii) Ruled correct line 1 (iii) 1 1 oe 2 (c) 7.2 to 7.6 1FT –5.2 to –5.6 1FT

This question in 0580/32 Oct/Nov 2014

5 (a) (i) Complete the table of values for y = x2 + x – 4. x –4 –3 –2 –1 0 1 2 3 y –2 –4 –2 8 [2] (ii) On the grid, draw the graph of y = x2 + x – 4 for - 4 G x G 3 . y 8 7 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 [4] (b) (i) Write down the co-ordinates of the lowest point of the graph. Answer(b)(i) ( … , … ) [1] (ii) Write down the equation of the line of symmetry of the graph. Answer(b)(ii) … [1] (c) Use your graph to solve the equation x2 + x – 4 = –3. Answer(c) x = … or x = … [2] (d) y 3 2 1 x –4 –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 –6 –7 L (i) In the diagram, a line L has been drawn on a 1 cm2 grid. Write down the equation of the line L. Give your answer in the form y = mx + c. Answer(d)(i) y = … [2] (ii) Find the area of the shaded triangle. Answer(d)(ii) … cm2 [1] __________________________________________________________________________________________

13 marks

Mark scheme: 5 (a) (i) 8, 2, –4, 2 2 B1 for 3 correct values (ii) Correctly plotted points and smooth 4 B3FT for 8 correct correct curve B2FT for 6 or 7 correct B1FT for 4 or 5 correct C1 for correct smooth curve passing below y = –4 (b) (i) (–0.5, k) where –4.5 ≤ k < –4 1 (ii) x = –0.5 1 (c) –1.8 ≤ x ≤ –1.4, 0.4 ≤ x ≤ 0.8 2FT B1FT, B1FT for values from their graph rise (d) (i) 2x – 3 2 M1 for or better run If zero scored, SC1 for kx – 3 (ii) 9 1

This question in 0580/32 Feb/March 2015

9 (a) (i) Complete the table of values for y = –x2 + 5x . x –1 0 1 2 3 4 5 6 y –6 4 4 0 [2] (ii) On the grid, draw the graph of y = –x2 + 5x for –1  x  6 . y 7 6 5 4 3 2 1 x –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6 [4] (b) Write down the co-ordinates of the highest point on the graph. Answer(b) ( … , … ) [1] (c) Use your graph to solve the equation –x2 + 5x = –3. Answer(c) x = … or x = … [2] (d) (i) On the grid, draw the line of symmetry for the graph. [1] (ii) Write down the equation of the line of symmetry for the graph. Answer(d)(ii) … [1] (iii) The curve passes through the points (–10, –150) and (k, –150). Use the symmetry of the curve to find the value of k. Answer(d)(iii) k = … [1] __________________________________________________________________________________________

12 marks

Mark scheme: 9 (a) (i) 0, 6, 6, –6 2 B1 for any 3 correct (ii) 8 points correctly plotted 4 B3FT for 7 or 8 correct correct smooth curve B2FT for 5 or 6 correct B1FT for 3 or 4 correct (b) (2.5, k) where 6 < k ≤ 6.5 1 (c) 5.4 to 5.7 1FT –0.4 to –0.7 1FT (d) (i) correct line drawn 1 (ii) x = 2.5 1 (iii) 15 1

This question in 0580/31 May/June 2015

4 (a) (i) Complete the table of values for y = 8 + 3x – x2. x 0 1 2 3 4 5 y 8 10 8 4 [2] (ii) On the grid, draw the graph of y = 8 + 3x – x2 for 0 G x G 5 . y 12 11 10 9 8 7 6 5 4 3 2 1 0 x 1 2 3 4 5 –1 –2 –3 – 4 [3] (iii) Write down the co-ordinates of the highest point of the graph. Answer(a)(iii) ( … , … ) [1] 12 . (b) (i) Complete the table of values for y = x x 1 2 3 4 5 y 12 4 2.4 [2] 12 (ii) On the same grid, draw the graph of y = for 1 G x G 5 . [3] x 12 . (c) Use your graphs to write down the solutions of the equation 8 + 3 x - x 2 = x Answer(c) x = … or x = … [2] __________________________________________________________________________________________

13 marks

Mark scheme: 4 (a) (i) 10 −2 1, 1 (ii) 6 points correctly plotted 3 B2FT for 5 or 6 points correctly plotted correct smooth curve or B1FT for 3 or 4 points correctly plotted (iii) (1.4 to 1.6, 10.1 to 10.4) 1 (b) (i) 6 3 1, 1 (ii) 5 points correctly plotted 3 B2FT for 4 or 5 points correctly plotted correct curve or B1FT for 2 or 3 correct points (c) 1.1 to 1.3 4.1 to 4.3 1FT, 1FT

This question in 0580/32 May/June 2015

7 (a) The grid below shows the straight line L. y 15 10 5 x –3 –2 –1 0 1 2 3 4 5 6 –5 L The equation of the line L is y = 2x + c. Find the value of c. Answer(a) c = … [1] (b) (i) Complete the table of values for y = x 2 – 3x – 2. x –3 –2 –1 0 1 2 3 4 5 6 y 8 2 –4 –4 2 8 [2] (ii) On the grid, draw the graph of y = x 2 – 3x – 2 for –3 x 6. [4] (iii) Write down the values of x where the line L intersects the curve y = x 2 – 3x – 2. Answer(b)(iii) x = … and x = … [2]

9 marks

Mark scheme: 7 (a) –1 1 (b) (i) 16.…–2.…–2….16 2 B1 for 2 correct (ii) 10 points correctly plotted 4 B3FT for 9 or 10 points correctly plotted Correct smooth curve B2FT for 7 or 8 points correctly plotted B1FT for 5 or 6 points correctly plotted (iii) Strict FT their intersection 2FT B1 for one correct value

This question in 0580/33 May/June 2015

67 (a) The table shows some values of y = . x x –5 –4 –3 –2 –1 1 2 3 4 5 y –1.2 –1.5 6 2 1.5 1.2 (i) Complete the table. [2] 6 (ii) On the grid, draw the graph of y = for –5  x  –1 and 1  x  5. x y 6 5 4 3 2 1 x –5 –4 –3 –2 –1 0 1 2 3 4 5 –1 –2 –3 –4 –5 –6 [4] (iii) On the same grid, draw the line y = 4. [1] 6 (iv) Find the co-ordinates of the point where the line y = 4 crosses the graph of y = . x Answer(a)(iv) ( … , … ) [1] (b) y 5 4 3 2 1 x –5 –4 –3 –2 –1 0 1 2 3 4 5 –1 –2 –3 –4 –5 (i) On this grid, plot the point A (–1, –3). [1] (ii) Draw a line with gradient 2 through point A. [1] (iii) Write down the equation of your line in the form y = mx + c. Answer(b)(iii) y = … [2] __________________________________________________________________________________________

12 marks

Mark scheme: 7 (a) (i) –2, –3, –6, 3 2 B1 for 2 or 3 correct (ii) Correct curves 4 B3FT for 9 or 10 correctly plotted points or B2FT for 7 or 8 correctly plotted points or B1FT for 5 or 6 correctly plotted points (iii) Ruled line y = 4 1 (iv) (1.4 to 1.6, 4) 1 SC1 for (4, 1.4 to 1.6) from line x = 4 drawn (b) (i) (–1, –3) plotted 1 (ii) Correct ruled line 1FT FT line with gradient 2 through their A (iii) 2x – 1 2FT FT 2x + their y-intercept for 2 marks B1 for 2x + k or mx – 1 (m ≠ 0) or mx + their y-intercept (m ≠ 0)

This question in 0580/31 Oct/Nov 2015

9 y l 8 7 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 4 –1 –2 –3 –4 –5 –6 –7 –8 (a) Write down the equation of the line l in the form y = mx + c. Answer(a) y = … [3] 2 (b) Complete the table of values for y = . x x −4 −3 −2 −1 −0.5 −0.25 0.25 0.5 1 2 3 4 y −0.7 −4 4 0.7 [3] 2 (c) On the grid, draw the graph of y = for –4  x  –0.25 and 0.25  x  4. [4] x

10 marks

Mark scheme: 9 (a) [ y = ] 2 x + 4 3 B2 for 2 x + c or kx + 4 k ≠ 0 or 2 k rise M1 for gradient = ± or attempt at k run using a triangle or co-ordinates allowing one slip (b) –0.5, –1, –2, –8, 8, 2, 1, 0.5 3 B2 for any 6 or 7 correct or B1 for any 4 or 5 correct (c) Correct curve 4 B3FT for 11 or 12 points correctly plotted B2FT for 9 or 10 points correctly plotted B1FT for 7 or 8 points correctly plotted

This question in 0580/32 Oct/Nov 2015

8 y L 18 16 14 12 10 8 6 4 2 x –2 –1 0 1 2 3 4 5 –2 –4 –6 –8 (a) The line L is drawn on the grid. Find the equation of the line in the form y = mx + c. Answer(a) y = … [3] (b) (i) Complete the table of values for y = x2 – 4x – 2. x –2 –1 0 1 2 3 4 5 y –2 –6 –5 –2 3 [3] (ii) On the grid above, draw the graph of y = x2 – 4x – 2 for –2  x  5. [4] (iii) Use your graph to solve the equation x2 – 4x – 2 = 0. Answer(b)(iii) x = … or x = … [2] __________________________________________________________________________________________

12 marks

Mark scheme: 8 (a) 5x + 3 3 B2 for 5x + c or kx + 3 k not equal 0 Rise or M1 for attempt at Run (b) (i) 10, 3, −5 3 B1 for each correct (ii) Correct curve 4 B3FT for 7 or 8 points correctly plotted B2FT for 5 or 6 points correctly plotted B1FT for 3 or 4 points correctly plotted (iii) −0.5 to – 0.4 and 4.4 to 4.5 2FT B1FT for each correct

This question in 0580/33 Oct/Nov 2015

9 (a) Complete the table of values for y = x 2 - 3x - 1. x –2 –1 0 1 2 3 4 5 y 9 –1 [3] (b) On the grid, draw the graph of y = x 2 - 3x - 1 for – 2  x  5. y 9 8 7 6 5 4 3 2 1 x –2 –1 0 1 2 3 4 5 –1 –2 –3 –4 [4] (c) Write down the co-ordinates of the lowest point of the graph. ( … , … ) [1] (d) (i) On the grid, draw the line of symmetry of the graph. [1] (ii) Write down the equation of the line of symmetry of the graph. … [1]

10 marks

Mark scheme: 9 (a) (9), 3, (–1), –3, –3, –1, 3, 9 3 B2 for any 5 correct or B1 for any 3 or 4 correct (b) completely correct curve 4 B3FT for 7 or 8 correct plots B2FT for 5 or 6 correct plots B1FT for 3 or 4 correct plots (c) (1.5, k) where –3.5 ⩽ k < –3 1 (d) (i) ruled line x = 1.5 drawn 1 (ii) x = 1.5 oe 1

This question in 0580/32 Feb/March 2016

9 (a) Complete the table of values for y = 8 + 7x − x2. x 0 1 2 3 4 5 6 7 8 y 8 18 18 8 [3] (b) On the grid, draw the graph of y = 8 + 7x − x2 for 0 G x G 8. y 22 20 18 16 14 12 10 8 6 4 2 x 0 1 2 3 4 5 6 7 8 [4] (c) Write down the co-ordinates of the highest point of the curve. ( … , … ) [1] (d) (i) On the grid, draw the line y = 16. [1] (ii) Use your line to solve the equation 8 + 7x − x2 = 16. x = … or x = … [2]

11 marks

Mark scheme: 9 (a) … 14 … 20 20 … 14 … 0 3 B2 for 3 or 4 correct B1 for 2 correct (b) Completely correct curve 4 B3FT for 8 or 9 points correctly plotted or B2FT for 6 or 7 points correctly plotted or B1FT for 4 or 5 points correctly plotted (c) (3.5, h) 1 20 < h ⩽ 20.4 (d) (i) Correct ruled line 1 (ii) 1.4 5.6 1, 1FT FT their graph and line

This question in 0580/31 May/June 2016

6 y 6 4 2 x 0 –5 –4 –3 –2 –1 1 2 –2 –4 –6 –8 –10 (a) On the grid, (i) draw the line y = 3, [1] (ii) draw the line that is perpendicular to the line y = 3 that passes through the point (1, −4). [2] (b) Complete the table of values for y = 2 − 3x − x2. x −5 −4 −3 −2 −1 0 1 2 y −2 2 2 −2 [2] (c) On the grid, draw the graph of y = 2 − 3x − x2 for - 5 G x G 2 . [4] (d) Write down the co-ordinates of the highest point of the graph of y = 2 − 3x − x2. ( … , … ) [1] (e) Use your graphs to solve the equation 2 − 3x − x2 = 3. x = … or x = … [2]

12 marks

Mark scheme: 6 (a) (i) Ruled continuous line y = 3 1 (ii) Ruled continuous line x = 1 2 B1 for (1, –4) plotted or B1 for any line perpendicular to their y = 3 drawn (b) –8, 4, 4, –8 2 B1 for 3 correct (c) Completely correct curve 4 B3FT for 7 or 8 points correctly plotted B2FT for 5 or 6 points correctly plotted B1FT for 3 or 4 points correctly plotted (d) (–1.5, 4.1 to 4.4) 1 (e) –2.5 to –2.7 and –0.3 to –0.5 2FT FT intersection of their (a)(i) with their curve B1FT for one correct

This question in 0580/32 May/June 2016

16 5 (a) (i) Complete the table of values for y = , x ! 0 . x x −16 −8 −4 −2 −1 1 2 4 8 16 y −1 −2 −8 16 4 2 [2] 16 (ii) On the grid, draw the graph of y = for - 16 G x G - 1 and 1 G x G 16 . x y 16 14 12 10 8 6 4 2 x 0 –16 –14 –12 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 16 –2 –4 –6 –8 –10 –12 –14 –16 [4] (b) Write down the order of rotational symmetry of your graph. … [1] (c) One line of symmetry crosses the graph twice. (i) Draw this line of symmetry on the grid. [1] (ii) Write down the equation of this line of symmetry. … [1] 16 (d) By drawing a suitable line on the grid, solve the equation = 7 . x x = … [2]

11 marks

Mark scheme: 5 (a) (i) −4 −16 8 1 2 B1 for 3 correct (ii) Completely correct curve 4 B3FT for 9 or 10 correctly plotted B2FT for 7 or 8 correctly plotted B1FT for 5 or 6 correctly plotted (b) 2 1 (c) (i) Ruled line y = x drawn 1 Must at least intersect the graph in two places (ii) y = x oe 1 (d) Continuous ruled line y = 7 1 Must intersect the graph drawn 2.1 to 2.5 1FT

This question in 0580/33 May/June 2016

7 y 20 18 L 16 14 12 10 8 6 4 2 x –2 –1 0 1 2 3 –2 –4 –6 –8 (a) The line L is drawn on the grid. Find the equation of the line in the form y = mx + c. y = … [3] (b) (i) Complete the table of values for y = x2 + 2x + 4. x −2 −1 0 1 2 3 y 4 4 7 19 [2] (ii) On the grid above, draw the graph of y = x2 + 2x + 4 for –2 G x G 3 . [4] (c) For –2 G x G 3 , write down the x co-ordinate of the point of intersection of the curve y = x2 + 2x + 4 with the line L.

10 marks

Mark scheme: 7 (a) −5x + 6 3 B2 for –5x (oe) + 6 or –5x + k or rise B1 for kx + 6 k ≠ 0 or [gradient = ] run k with correct values or [gradient =] ±5 k (b) (i) 3 12 1 , 1 (ii) Correct curve 4 B3FT for 5 or 6 correctly plotted points or B2FT for 3 or 4 correctly plotted points or B1FT for 1 or 2 correctly plotted points (c) 0.2 to 0.35 1 FT

This question in 0580/31 Oct/Nov 2016

4 (a) Complete the table of values for y = x 2 - 5x + 3 . x –1 0 1 2 3 4 5 y 3 –1 –1 3 [2] (b) On the grid, draw the graph of y = x 2 - 5x + 3 for -1 G x G 5 . y 10 8 6 4 2 x –1 0 1 2 3 4 5 –2 –4 [4] (c) Write down the equation of the line of symmetry of the graph of y = x 2 - 5x + 3 . … [1] (d) Write down the co-ordinates of the point where the line y = 4 - x (i) crosses the x-axis, ( … , … ) [1] (ii) crosses the y-axis. ( … , … ) [1] (e) On the grid, draw the line y = 4 - x . [1] (f) Write down the co-ordinates of the points of intersection of the graph of y = x 2 - 5x + 3 and the line y = 4 - x . ( … , … ) ( … , … ) [2]

12 marks

Mark scheme: 4 (a) 9, –3, –3 2 B1 for 9 or –3 and –3 (b) Correct curve 4 B3FT for 6 or 7 correctly plotted points or B2FT for 4 or 5 correctly plotted points or B1FT for 2 or 3 correctly plotted points (c) x = 2.5 1 (d) (i) (4, 0) 1 (ii) (0, 4) 1

This question in 0580/32 Oct/Nov 2016

8 (a) Complete the table of values for y = x 2 - 2x . x - 3 - 2 - 1 0 1 2 3 4 y 3 - 1 3 [3] (b) On the grid, draw the graph of y = x 2 - 2x for - 3 G x G 4 . y 16 14 12 10 8 6 4 2 x 0 –3 –2 –1 1 2 3 4 –2 [4] (c) On the grid, draw the line y = 6 . [1] (d) Use your graph to solve the equation x 2 - 2x = 6 . Give your answers correct to 1 decimal place. x = … or x = … [2] Question 9 is printed on the next page.

10 marks

Mark scheme: 8 (a) 15 8 … 0 … 0 … 8 3 B1 for 8 and 8 in the correct place B1 for 0 and 0 in the correct place B1 for 15 in the correct place (b) Correct curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points (c) Correct ruled line 1 (d) –1.8 or –1.7 or –1.6 2FT B1FT for one correct 3.6 or 3.7 or 3.8 or B1FT for both correct answers as co-ordinates or B1FT for both answers correct to more than 1dp

This question in 0580/33 Oct/Nov 2016

89 (a) Complete the table of values for y = . x x –8 –6 –4 –2 –1 1 2 4 6 8 y –1 –1.3 –8 2 1.3 1 [2] 8 (b) On the grid, draw the graph of y = for - 8 G x G - 1 and 1 G x G 8 . x y 8 7 6 5 4 3 2 1 x –8 –7–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 –3 –4 –5 –6 –7 –8 [4] 8 (c) The graph of y = has two lines of symmetry. x Write down the equation of each of these lines. … and … [2] 8 (d) Mark a point, P, on the graph of y = where the x and y co-ordinates are equal. [1] x

9 marks

Mark scheme: 9 (a) −2, −4, 8, 4 2 B1 for any 2 correct (b) completely correct curve 4 B3FT for 9 or 10 correct plots B2FT for 7 or 8 correct plots B1FT for 5 or 6 correct plots (c) y = x , y = − x oe 1,1 (d) point at (2.8, 2.8) or ( −2.8, − 2.8) 1FT FT a point on their curve lying on y = x

This question in 0580/32 Feb/March 2017

5 (a) Complete the table of values for y = x 2 + 2x - 1. x -5 -4 -3 -2 -1 0 1 2 3 y 14 2 -1 -1 2 [3] (b) On the grid, draw the graph of y = x 2 + 2x - 1 for -5 G x G 3 . y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 x –5–5 –4–4 –3–3 –2–2 –1–1 00 11 22 33 –1 –2 –3 –4 (c) (i) On the grid, draw the line of symmetry. [1] (ii) Write down the equation of the line of symmetry. … [1] (d) (i) On the grid, plot the points (- 5 , 7) and (0, - 3) and join them with a straight line, L. [2] (ii) Write down the x co-ordinate of each point where the line L crosses the graph of y = x 2 + 2x - 1. x = … and x = … [2] (iii) Work out the gradient of the line L. … [2]

15 marks

Mark scheme: 5(a) 7 –2 7 14 3 B2 for 3 correct B1 for 2 correct 5(b) Correct smooth curve 4 B3FT for 8 or 9 correct plots or B2FT for 6 or 7 correct plots or B1FT for 4 or 5 correct plots 5(c)(i) Ruled line, x = –1, drawn 1 5(c)(ii) x = –1 oe 1 5(d)(i) Ruled line L drawn, joining 2 B1 for one of the points correct and line drawn, or (–5, 7) and (0, −3) both points correct and no or wrong line. 5(d)(ii) −3.3 to −3.5, −0.5 to −0.7 2FT B1FT for one correct. 5(d)(iii) −2 2 Rise y 2 − y1 M1FT for their from part (d)(i) or their Run x2 − x1 If zero scored, SC1 for answer 2

This question in 0580/31 May/June 2017

158 (a) Complete the table for y = . x x -5 -4 -3 -2 -1 1 2 3 4 5 y -3.75 -15 15 5 [3] 15 (b) On the grid, draw the graph of y = for - 5 G x G - 1 and 1 G x G 5 . x y 15 10 5 x –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –10 –15 [4] 15 (c) Use your graph to solve the equation = 8 . x x = … [1]

8 marks

Mark scheme: 8(a) –3, –5, –7.5, 7.5, 3.75, 3 3 B2 for 4 or 5 correct B1 for 2 or 3 correct 8(b) Correct curve drawn 4 B3FT for 9 or 10 points correctly plotted or B2FT for 7 or 8 points correctly plotted or B1FT for 5 or 6 points correctly plotted 8(c) 1.8 ⩽ x < 2 1 If zero scored, then FT their graph

This question in 0580/32 May/June 2017

6 (a) (i) Complete the table of values for y = 2x2 − 4x − 6. x –2 –1 0 1 2 3 4 y –6 –6 0 [2] (ii) On the grid, draw the graph of y = 2x2 − 4x − 6 for - 2 G x G 4 . y 12 10 8 6 4 2 x –2 –1 0 1 2 3 4 –2 –4 –6 –8 –10 [4] (b) (i) On the grid, draw the line y = 5. [1] (ii) Use your graph to solve the equation 2x2 − 4x − 6 = 5. x = … or x = … [2] (c) Explain why the equation 2x2 − 4x − 6 = −9 has no solutions. … … [1] (d) (i) Write down the equation of the line of symmetry of y = 2x2 − 4x − 6. … [1] (ii) Use the symmetry of the graph to complete this statement. When 2x2 − 4x − 6 = 64, there are two solutions for x, x = 7 or x = … [1]

12 marks

Mark scheme: 6(a)(i) 10, 0, −8, 10 2 B1 for 2 or 3 correct 6(a)(ii) Completely correct curve 4 B3FT for 6 or 7 correctly plotted points B2FT for 4 or 5 correctly plotted points B1FT for 2 or 3 correctly plotted points 6(b)(i) Ruled continuous line y = 5 1 6(b)(ii) 3.5 1FT FT their graph –1.5 1FT FT their graph 6(c) –9 is below –8 oe 1 6(d)(i) x = 1 1 6(d)(ii) –5 1

This question in 0580/33 May/June 2017

9 (a) (i) Complete the table of values for y = x2 + 3x − 4. x −3 −2 −1 0 1 2 3 y −4 −6 −4 0 [3] (ii) On the grid, draw the graph of y = x2 + 3x − 4 for - 3 G x G 3 . y 16 12 8 4 x –3 –2 –1 0 1 2 3 –4 –8 [4] (b) (i) On the same grid, draw the line y = 5. [1] (ii) Write down the co-ordinates of the point of intersection of the line y = 5 and the graph of y = x2 + 3x – 4 for - 3 G x G 3 . ( … , … ) [1]

9 marks

Mark scheme: 9(a)(i) −6, 6, 14 3 B1 for each 9(a)(ii) Correct curve 4 B3FT for 6 or 7 points correctly plotted or B2FT for 4 or 5 points correctly plotted or B1FT for 2 or 3 points correctly plotted 9(b)(i) Correct ruled line 1 9(b)(ii) 1.8 ⩽ x < 2.0, 5 1FT FT intersection of their curve with the line y = 5

This question in 0580/32 Oct/Nov 2017

4 (a) Complete the table of values for y = 5x − x2. x −1 0 1 2 3 4 5 6 y 0 6 6 −6 [2] (b) On the grid, draw the graph of y = 5x − x2 for - 1 G x G 6. y 8 7 6 5 4 3 2 1 x –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6 –7 –8 [4] (c) Write down the equation of the line of symmetry of the graph. … [1] (d) (i) Complete the table of values for y = 1.5x − 2. x 0 2 5 y [2] (ii) On the grid, draw the graph of y = 1.5x − 2 for - 1 G x G 6. [2] (iii) Use your graphs to write down the solutions to the equation 1.5x − 2 = 5x − x2. x = … or x = … [2]

13 marks

Mark scheme: 4(a) −6 4 4 0 2 B1 for 2 or 3 correct 4(b) Correct smooth curve 4 B3FT for 7 or 8 correct plots or B2FT for 5 or 6 correct plots or B1FT for 3 or 4 correct plots 4(c) x = 2.5 cao 1 4(d)(i) −2 1 5.5 2 B1 for 2 correct 4(d)(ii) Correct continuous ruled line from 2 B1FT for 2 or 3 correct plots x = −1 to x = 6 4(d)(iii) [x =] −0.6 to −0.4 and 3.9 to 4.1 2 B1FT for each

This question in 0580/32 May/June 2018

66 (a) Complete the table of values for y = , x =Y 0 . x x - 6 - 4 - 3 - 2 - 1 1 2 3 4 6 y - .15 - 3 3 1.5 [3] 6 (b) On the grid, draw the graph of y = for - 6 G x G - 1 and 1 G x G 6 . x y 6 5 4 3 2 1 x – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 – 1 – 2 – 3 – 4 – 5 – 6 [4] (c) On the grid, draw the line y =- 5 . [1] 6 (d) Use your graph to solve the equation =- 5 . x x = … [1]

9 marks

Mark scheme: 6(a) –1 … –2 … –6 … 6 … 2 … 1 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 6(b) correct smooth curves 4 B3FT for 9 or 10 points plotted correctly B2FT for 7 or 8 points plotted correctly B1FT for 5 or 6 points plotted correctly FT their table 6(c) correct continuous ruled line 1 6(d) –1.2 oe 1 or FT their line and their graph

This question in 0580/33 May/June 2018

4 (a) (i) Complete the table of values for y = x 2 - 5x . x –1 0 1 2 3 4 5 6 y –4 –6 –6 –4 0 [2] (ii) On the grid, draw the graph of y = x 2 - 5x for -1 G x G 6 . y 7 6 5 4 3 2 1 0 x –1 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6 –7 [4] (iii) Write down the co-ordinates of the lowest point of your graph. ( … , … ) [1] (iv) Use your graph to solve the equation x 2 - 5x = 3 . x = … or x = … [2] (b) y 5 L 4 3 2 1 –5 –4 –3 –2 –1 0 1 2 3 4 5 x –1 –2 –3 –4 –5 Line L is drawn on the grid. (i) Find the equation of line L in the form y = mx + c. y = … [3] (ii) Line P is parallel to line L and passes through the point (0, -1). On the grid above, draw line P for -5 G x G 5 . [2]

14 marks

Mark scheme: 4(a)(i) 6, 0, 6 2 B1 for two correct 4(a)(ii) Correct curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 4(a)(iii) (2.5, –6.4 to –6.1) 1 4(a)(iv) –0.7 to –0.4, 5.4 to 5.7 2 FT their curve B1 for each 4(b)(i) 1 3 1 y = − x + 2 oe M2 for gradient = − oe soi 2 2 1 or M1 for rise / run or gradient = 2 and B1 for y = mx + 2, m ≠ 0 4(b)(ii) Correct ruled line for –5 ⩽ x ⩽ 5 2 B1 for line through (0, –1) or line parallel to line L or correct short line at least from (–4, 1) to (4, –3)

This question in 0580/31 Oct/Nov 2018

67 (a) Complete the table of values for y = . x x –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 y –1 –2 –3 –6 6 3 2 1.2 1 [2] 6 (b) On the grid, draw the graph of y = for -6 G x G -1 and 1 G x G 6 . x y 6 5 4 3 2 1 0 x –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6 [4] 6 (c) Use your graph to solve the equation = 4. 5 . x x = … [1] (d) (i) On the grid, draw the line y = x. [1] 6 (ii) Write down the co-ordinates of the points of intersection of y = and y = x. x ( … , … ) and ( … , … ) [2]

10 marks

Mark scheme: 7(a) −1.2, −1.5, 1.5 2 B1 for 2 correct 7(b) Correct curve 4 B3FT for 11 or 12 points correctly plotted or B2FT for 9 or 10 points correctly plotted or B1FT for 6, 7 or 8 points correctly plotted 7(c) 1.2 to 1.4 1 FT their (b) 7(d)(i) Correct ruled line 1 7(d)(ii) (−2.6 to −2.3, −2.6 to −2.3,) and 2 FT y = x drawn and their curve (2.3 to 2.6, 2.3 to 2.6) B1FT for one correct, or both x values correct or both y values correct

This question in 0580/33 Oct/Nov 2018

8 (a) y 4 L 3 2 1 –2 –1 0 1 2 3 4 5 6 7 8 x –1 Line L is drawn on the grid. Find the equation of line L. Give your answer in the form y = mx + c. y = … [3] - 7. (b) The points (9, a) and (b, 3) lie on the line y = 23 x Work out the value of (i) a, a = … [2] (ii) b. b = … [2] (c) (i) Complete the table of values for y = x (3 - x). x -4 -2 -1 0 1 2 4 y -10 0 2 -4 [3] (ii) On the grid, draw the graph of y = x (3 - x) for - 4 G x G 4. y 5 -4 -2 0 2 4 x -5 -10 -15 -20 -25 -30 [4] (iii) Write down the co-ordinates of the highest point of the graph for - 4 G x G 4. ( … , … ) [1]

15 marks

Mark scheme: 8(a) 1 3 1 [ y = ] − x + 3 B2 for [ y = ] − x + c 2 2 or rise 1 M1 for or m = ± oe run 2 and B1 for [ y = ] kx + 3 , k ≠ 0 or c = 3 8(b)(i) –1 2 2 M1 for [ a = ] × 9 − 7 or better 3 8(b)(ii) 15 2 2 M1 for 3 = b − 7 or better 3 8(c)(i) –28, –4, 2 3 B1 for each 8(c)(ii) correct smooth curve 4 B3FT for 6 or 7 correct plots or B2FT for 4 or 5 correct plots or B1FT for 2 or 3 correct plots 8(c)(iii) (1.5 , 2.25) 1 accept (x, y) where 1 < x < 2 and 2 < y < 4

This question in 0580/32 Feb/March 2019

4 (a) Complete the table of values for y = 5 + 2 x - x 2 . x -2 -1 0 1 2 3 4 y 2 5 6 -3 [2] (b) On the grid, draw the graph of y = 5 + 2x - x 2 for - 2 G x G 4 . y 7 6 5 4 3 2 1 0 –2 –1 1 2 3 4 x –1 –2 –3 [4] (c) (i) On the grid, draw the line of symmetry. [1] (ii) Write down the equation of the line of symmetry. … [1] (d) Use your graph to find the solutions of the equation 5 + 2x - x 2 = 4 . x = … or x = … [2] (e) (i) On the grid, draw a line from (- 1, 2) to (1, 6) . [1] (ii) Find the equation of this line in the form y = mx + c . y = … [3]

14 marks

Mark scheme: 4(a) −3 5 2 2 B1 for 2 correct 4(b) Correct curve 4 B3FT for 6 or 7 points correct B2FT for 4 or 5 points correct B1FT for 2 or 3 points correct 4(c)(i) Ruled line x = 1 drawn 1 4(c)(ii) x = 1 1 4(d) −0.5 to −0.3 and 2.3 to 2.5 2 B1 for each If 0 scored, B1 for y = 4 drawn 4(e)(i) Correct ruled continuous line 1 4(e)(ii) [y =] 2x + 4 3 B2 for [y =] 2x + k rise or M1 for run B1 for kx + 4 , k ≠ 0, or c = 4

This question in 0580/33 May/June 2019

6 (a) Complete the table of values for y = x 2 - 5x + 3 . x −1 0 1 2 3 4 5 6 y −1 −3 −3 −1 3 [2] (b) On the grid, draw the graph of y = x 2 - 5x + 3 for - 1 G x G 6 . y 10 9 8 7 6 5 4 3 2 1 x – 1 0 1 2 3 4 5 6 – 1 – 2 – 3 – 4 [4] (c) Use your graph to solve the equation x 2 - 5x + 3 = 0 . x = … or x = … [2]

8 marks

Mark scheme: 6(a) 9, 3, 9 2 B1 for two correct 6(b) Correct curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 6(c) 0.6 to 0.8, 4.2 to 4.4 2 FT their curve B1 for each

This question in 0580/32 Oct/Nov 2019

8 (a) (i) Complete the table of values for y = x 2 - 2x . x –2 –1 0 1 2 3 4 y 8 0 –1 0 8 [1] (ii) On the grid, draw the graph of y = x 2 - 2x for - 2 G x G 4 . y 9 8 7 6 5 4 3 2 1 x – 2 – 1 0 1 2 3 4 – 1 – 2 [4] (b) Here are the first four terms of a sequence. 3 9 15 21 (i) Find the next term. … [1] (ii) Write down the rule for continuing this sequence. … [1] (iii) Find the nth term of this sequence. … [2] Question 9 is printed on the next page.

9 marks

Mark scheme: 8(a)(i) 3 3 1 8(a)(ii) Correct curve 4 B3FT for 6 or 7 points correctly plotted or B2FT for 4 or 5 points correctly plotted or B1FT for 2 or 3 points correctly plotted 8(b)(i) 27 1 8(b)(ii) Add 6 oe 1 8(b)(iii) 6n – 3 oe final answer 2 B1 for 6n + a or bn – 3 (b ≠ 0)

This question in 0580/33 Oct/Nov 2019

4 (a) Complete the table of values for y = 7 + 2 x - x 2 . x -2 -1 0 1 2 3 4 y -1 8 7 -1 [2] (b) On the grid, draw the graph of y = 7 + 2x - x 2 for - 2 G x G 4 . y 9 8 7 6 5 4 3 2 1 – 2 – 1 0 1 2 3 4 x – 1 – 2 [4] (c) Write down the equation of the line of symmetry of the graph. … [1] (d) Use your graph to solve the equation 7 + 2x - x 2 = 0 . x = … or x = … [2]

9 marks

Mark scheme: 4(a) 4 7 4 2 B1 for one correct 4(b) Correct curve 4 B3FT for 6 or 7 points correct or B2FT for 4 or 5 points correct or B1FT for 2 or 3 points correct 4(c) x = 1 oe 1 4(d) −1.9 to −1.7 and 3.7 to 3.9 2 B1 for each

This question in 0580/31 May/June 2020

9 (a) Complete the table of values for y = x 2 - 3x - 6 . x -3 -2 -1 0 1 2 3 4 5 6 y 12 -2 -2 12 [3] (b) On the grid, draw the graph of y = x 2 - 3x - 6 for - 3 G x G 6 . y 14 12 10 8 6 4 2 -3 -2 -1 0 1 2 3 4 5 6 x -2 -4 -6 -8 -10 [4] (c) Write down the equation of the line of symmetry of the graph. … [1] (d) Use your graph to solve the equation x 2 - 3x - 6 = 0 . x = … or x = … [2] Question 10 is printed on the next page.

10 marks

Mark scheme: 9(a) 4 –6 –8 –8 –6 4 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 9(b) Correct curve 4 B3FT for 9 or 10 points correctly plotted or B2FT for 7 or 8 points correctly plotted or B1FT for 5 or 6 points correctly plotted 9(c) x = 1.5 1 9(d) −1.4 4.4 2 B1 for each

This question in 0580/32 May/June 2020

8 (a) Complete the table of values for y = - x 2 + x + 5 . x -3 -2 -1 0 1 2 3 4 y -1 3 3 [3] (b) On the grid, draw the graph of y = - x 2 + x + 5 for - 3 G x G 4 . y 6 5 4 3 2 1 -3 -2 -1 0 1 2 3 4 x -1 -2 -3 -4 -5 -6 -7 -8 [4] (c) Write down the coordinates of the highest point of the graph. ( … , … ) [1] (d) Write down the equation of the line of symmetry of the graph. … [1] (e) (i) On the grid, draw the line y = x for - 3 G x G 4 . [1] (ii) Write down the values of x where the line y = x crosses the curve y =- x 2 + x + 5 . x = … and x = … [2] Question 9 is printed on the next page.

12 marks

Mark scheme: 8(a) −7, 5, 5, −1, −7 3 B2 for 3 or 4 correct B1 for 2 correct 8(b) Correct curve 4 B3FT for 7 or 8 correct plots or B2FT for 5 or 6 correct plots or B1FT for 3 or 4 correct plots. 8(c) (0.5, k) where 5 < k < 6 1 8(d) x = 0.5 oe 1 8(e)(i) y = x ruled from 1 ( −3, −3) to (4, 4) 8(e)(ii) −2.2, 2.2 2 B1 for each accept −2.4 to −2.1 and 2.1 to 2.4

This question in 0580/33 May/June 2020

159 (a) Complete the table of values for y = . x x - 5 - 3 - 2 - 1 1 2 3 5 y - 15 15 [3] 15 (b) On the grid, draw the graph of y = for - 5 G x G - 1 and 1 G x G 5 . x y 16 14 12 10 8 6 4 2 0 x – 5 – 4 – 3 – 2 – 1 1 2 3 4 5 – 2 – 4 – 6 – 8 – 10 – 12 – 14 – 16 [4] (c) On the grid, draw the line y = 6 . [1] 15 (d) Use your graph to solve = 6 . x x = … [1]

9 marks

Mark scheme: 9(a) −3 −5 −7.5 7.5 5 3 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 9(b) Correct curve 4 B3FT for 7 or 8 points plotted correctly or B2FT for 5 or 6 points plotted correctly or B1FT for 3 or 4 points plotted correctly 9(c) Correct ruled line 1 9(d) 2.5 or 2.4 to 2.6 1 FT their line (y = k) and their curve

This question in 0580/31 Oct/Nov 2020

10 (a) Complete the table of values for y = x 2 - 4x - 3 . x - 2 - 1 0 1 2 3 4 5 y 2 - 3 - 6 - 6 - 3 2 [2] (b) On the grid, draw the graph of y = x 2 - 4x - 3 for - 2 G x G 5 . y 10 8 6 4 2 – 2 – 1 0 1 2 3 4 5 x – 2 – 4 – 6 – 8 [4] (c) Use your graph to solve the equation x 2 - 4 x - 3 = 0 . x = … or x = … [2]

8 marks

Mark scheme: 10(a) 9 –7 2 B1 for each 10(b) Correct curve 4 B3FT for 7 or 8 points correctly plotted B2FT for 5 or 6 points correctly plotted B1FT for 3 or 4 points correctly plotted 10(c) – 0.8 to –0.5 and 4.5 to 4.8 2 B1 for each

This question in 0580/32 Oct/Nov 2020

3 (a) Complete the table of values for y = 1 + 5 x - x 2 . x - 1 0 1 2 3 4 5 y 1 5 7 1 [2] (b) On the grid, draw the graph of y = 1 + 5x - x 2 for - 1 G x G 5 . y 8 6 4 2 0 x – 1 1 2 3 4 5 – 2 – 4 – 6 [4] (c) (i) On the grid, draw the line y = 3 . [1] (ii) Use your line to solve the equation 1 + 5x - x 2 = 3 . x = … or x = … [2]

9 marks

Mark scheme: 3(a) –5, 7, 5 2 B1 for 2 correct 3(b) Correct curve 4 B3FT for 6 or 7 points correctly plotted or B2FT for 4 or 5 points correctly plotted or B1FT for 2 or 3 points correctly plotted 3(c)(i) Ruled line y = 3 1 3(c)(ii) 0.3 to 0.6 4.4 to 4.7 2 FT their y = k and their (b) B1 for one correct or B1 for both correct answers as coordinates

This question in 0580/33 Oct/Nov 2020

5 (a) Complete the table of values for y =- x 2 - x + 14 . x –5 –4 –3 –2 –1 0 1 2 3 4 y 8 12 12 8 [3] (b) On the grid, draw the graph of y =- x 2 - x + 14 for - 5 G x G 4 . y 16 14 12 10 8 6 4 2 x – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 – 2 – 4 – 6 – 8 [4] (c) (i) Write down the equation of the line of symmetry of the graph. … [1] (ii) Find the coordinates of the highest point on the graph. ( … , … ) [1] (d) Use your graph to solve the equation - x 2 - x + 14 =- 2 . x = … or x = … [2]

11 marks

Mark scheme: 5(a) −6 2 14 14 2 −6 3 B2 for 4 or 5 correct B1 for 2 or 3 correct 5(b) Completely correct curve 4 B3FT for 9 or 10 correctly plotted points B2FT for 7 or 8 correctly plotted points B1FT for 5 or 6 correctly plotted points 5(c)(i) x = −0.5 oe 1 5(c)(ii) ( −0.5, k) oe 1 where 14 < k  14.8 5(d) 3.3 to 3.7, −.74 to −4.3 2 B1FT for each

This question in 0580/32 Feb/March 2021

5 y 88 66 44 22 x –– 88 –– 66 –– 44 –– 22 0 22 44 66 88 –– 22 –– 44 –– 66 –– 88 k The diagram shows the graph of y = for 1 G x G 8 . x (a) Use the graph to find the value of x when y = 4 . x = … [1] (b) (i) Show that k = 8 . [1] (ii) Calculate the value of y when x = 250 . y = … [1] 8 (c) (i) Complete this table of values for y = . x x - 8 - 4 - 2 - 1 y [2] 8 (ii) On the grid, draw the graph of y = for - 8 G x G - 1. [3] x (d) Write down the equation of each line of symmetry of the graph. … and … [2]

10 marks

Mark scheme: 5(a) 2 1 5(b)(i) k 1 accept use of any correct co-ordinates e.g. 4 = 2 leading to k = 8 5(b)(ii) 0.032 oe 1 5(c)(i) −−−−1, 2, 4, 8 2 B1 for 2 correct 5(c)(ii) Correct curve 3 B2FT for 3 or 4 correct plots B1FT for 1 or 2 correct plots 5(d) y = x oe y = − x oe 2 B1 for each

This question in 0580/31 May/June 2021

185 (a) Complete the table of values for y = . x x -8 -6 -4 -3 -2 2 3 4 6 8 y -3 -6 6 3 [3] 18 (b) On the grid, draw the graph of y = for -8 G x G - 2 and 2 G x G 8 . x y 1010 88 66 44 22 x –– 88 –– 66 –– 44 – 2 0 22 44 66 88 – 2 – 4 – 6 – 8 – 10 (c) Write down the order of rotational symmetry of the graph. … [1] (d) (i) On the grid, plot and join the points (-8, -3) and (6, 4). [2] 18 (ii) Write down the values of x where this line intersects the graph of y = . x x = … and x = … [2] (iii) Find the equation of this line in the form y = mx + c . y = … [2]

14 marks

Mark scheme: 5(a) −2.25 −4.5 −9 9 4.5 2.25 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 5(b) Correct curve 4 B3FT for 9 or 10 points correctly plotted or B2FT for 7 or 8 points correctly plotted or B1FT for 5 or 6 points correctly plotted 5(c) 2 1 5(d)(i) (−8, −3) and (6, 4) plotted and joined in a 2 B1 for one point correctly plotted or both ruled line correctly plotted but not joined, or ruled 5(d)(ii) −7.3 to −6.9 and 4.9 to 5.3 2 B1FT for each 5(d)(iii) 1 2 1 [y =] x + 1 oe final answer B1 for x + c (c ≠ +1) or 2 2  1  kx + 1  k ≠ 0 or   2  or B1FT for (their m)x + c or kx + their intercept (k ≠ 0)

This question in 0580/32 May/June 2021

9 The table shows some values for y = x 2 + x - 5 . x -4 -3 −2 −1 0 1 2 3 y 7 -3 -5 -5 7 (a) Complete the table. [2] (b) Draw the graph of y = x 2 + x - 5 for - 4 G x G 3 . y 8 7 6 5 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 x – 1 – 2 – 3 – 4 – 5 – 6 – 7 [4] (c) Write down the equation of the line of symmetry of this graph. … [1] (d) Use the graph to solve the equation x 2 + x - 5 = 0 . x = … or x = … [2]

9 marks

Mark scheme: 9(a) 1 −3 1 2 B1 for 2 correct 9(b) Correct curve 4 B3FT for 7 or 8 points correctly plotted or B2FT for 5 or 6 points correctly plotted or B1FT for 3 or 4 points correctly plotted 9(c) x = −[0].5 oe 1 9(d) −2.9 to −2.7 1.7 to 1.9 2 B1FT for each

This question in 0580/33 May/June 2021

10 (a) Complete the table of values for y = 4 + 3 x - x 2 . x -2 -1 0 1 2 3 4 y 0 4 6 0 [2] (b) On the grid, draw the graph of y = 4 + 3x - x 2 for - 2 G x G 4 . y 8 6 4 2 - 2 - 1 0 1 2 3 4 x - 2 - 4 - 6 - 8 [4] (c) The line y = 2 x - 1 is drawn on the grid. Use your graph to solve the equation 4 + 3x - x 2 = 2x - 1. x = … or x = … [2]

8 marks

Mark scheme: 10(a) –6, 6, 4 2 B1 for 2 correct 10(b) Correct curve 4 B3FT for 6 or 7 points correctly plotted or B2FT for 4 or 5 points correctly plotted or B1FT for 2 or 3 points correctly plotted 10(c) 2.7 to 2.9 and –1.9 to –1.7 2 FT their curve B1 for one correct If 0 scored, SC1 for both correct or FT answers as coordinates

This question in 0580/31 Oct/Nov 2021

10 (a) Complete the table of values for y = x 2 - 5x - 2 . x - 2 - 1 0 1 2 3 4 5 6 y 4 - 2 - 8 - 8 - 2 4 [2] (b) On the grid, draw the graph of y = x 2 - 5x - 2 for - 2 G x G 6 . y 14 12 10 8 6 4 2 – 2 – 1 0 1 2 3 4 5 6 x – 2 – 4 – 6 – 8 – 10 [4] (c) On the grid, draw the line y = 2 . [1] (d) Use your graph to solve the equation x 2 - 5 x - 2 = 2 . x = … or x = … [2]

9 marks

Mark scheme: 10(a) 12 −6 −6 2 B1 for 1 or 2 correct 10(b) Correct curve 4 B3FT for 8 or 9 points correctly plotted or B2FT for 6 or 7 points correctly plotted or B1FT for 4 or 5 points correctly plotted 10(c) Ruled line y = 2 drawn 1 10(d) −0.9 to −0.5 2 FT y = 2 and their curve B1 for each 5.5 to 5.9

This question in 0580/32 Oct/Nov 2021

8 (a) Line L is shown on the grid. y 25 20 L 15 10 5 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 x – 5 – 10 Find the equation of line L in the form y = mx + c . y = … [3] (b) (i) Complete the table of values for y = x 2 + 4x . x -6 -5 -4 -3 -2 -1 0 1 2 3 y 12 5 0 -3 -3 0 5 12 [2] (ii) On the grid, draw the graph of y = x 2 + 4x for - 6 G x G 3 . y 25 20 15 10 5 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 x – 5 – 10 [4] (iii) Use your graph to solve the equation x 2 + 4x = 10 . x = … or x = … [2]

11 marks

Mark scheme: 8(a) 2.5x + 10 final answer 3 B2 for 2.5x + c OR M1 for a correct rise over run or for a right-angled triangle marked on grid with rise =10 and run = 4 oe B1 for [y =] kx +10 (k ≠ 0) 8(b)(i) –4 21 2 B1 for each 8(b)(ii) Correct curve 4 B3FT for 9 or 10 points correctly plotted or B2FT for 7 or 8 points correctly plotted or B1FT for 5 or 6 points correctly plotted 8(b)(iii) –5.8 to –5.6 and 1.6 to 1.8 2 FT their curve B1 for each

This question in 0580/33 Oct/Nov 2021

8 (a) (i) Complete the table of values for y = x 2 + 6x - 160 . x -20 -15 -10 -5 0 5 10 15 y 120 -120 -165 -160 -105 [3] (ii) On the grid, draw the graph of y = x 2 + 6x - 160 for - 20 G x G 15 . y 200 150 100 50 – 20 – 15 – 10 – 5 0 5 10 15 x – 50 – 100 – 150 – 200 [4] (iii) (a) Write down the equation of the line of symmetry of the graph. … [1] (b) Find the coordinates of the lowest point on the graph. ( … , … ) [1] (iv) Use your graph to solve the equation x 2 + 6x - 160 = 0 . x = … or x = … [2] (b) Rearrange the formula y = mx + c to make x the subject. x = … [2]

13 marks

Mark scheme: 8(a)(i) −25 0 155 3 B1 for each 8(a)(ii) Completely correct curve 4 B3FT for 7 or 8 correctly plotted points B2FT for 5 or 6 correctly plotted points B1FT for 3 or 4 correctly plotted points 8(a)(iii)(a) x = −3 oe 1 8(a)(iii)(b) ( −3, k) oe 1 FT their graph where − 180  k < − 165 8(a)(iv) 10 −16 2 B1 FT for each 8(b) y − c 2 M1 for correct first step [ x = ] oe final answer m y c y − c = mx or = x + m m

This question in 0580/32 Feb/March 2022

8 The grid shows a line L. y 6 5 L 4 3 2 1 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 x – 1 – 2 – 3 – 4 – 5 – 6 (a) Find the equation of line L. Give your answer in the form y = mx + c . y = … [2] (b) (i) Complete the table of values for y = 2x + 5 . x −5 −3 0 y −5 5 [1] (ii) On the grid, draw the graph of y = 2x + 5 . [1] (c) Write down the coordinates of the point which lies on both line L and the graph of y = 2x + 5 . ( … , … ) [1] (d) Write down the equation of the line that is parallel to y = 2x + 5 and passes through the point (0, 18). … [1]

6 marks

Mark scheme: 8(a) 1 2 M1 for rise ÷ run [y =] – + 2 or for [y =] kx + 2 (k ≠ 0) 2x 1 or [y =] – 2x + j oe 8(b)(i) −1 1 8(b)(ii) Correct ruled line on grid 1 8(c) −1.2, 2.6 1 FT Line L and their (b)(ii) 8(d) y = 2x + 18 1

This question in 0580/31 May/June 2022

129 (a) Complete the table of values for y = , x ! 0 . x x -6 -4 -3 -2 -1 1 2 3 4 6 y -3 -6 6 3 [3] 12 (b) On the grid, draw the graph of y = for - 6 G x G - 1 and 1 G x G 6 . x y 12 10 8 6 4 2 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 x – 2 – 4 – 6 – 8 – 10 – 12 [4] (c) On the grid, draw the line y = 5 . [1] 12 (d) Use your graph to solve the equation = 5 . x x = … [1]

9 marks

Mark scheme: 9(a) −2 … −4 … −12 12 … 4 … 2 3 B2 for 4 or 5 correct B1 for 2 or 3 correct 9(b) Correct curve 4 B3FT for 9 or 10 points plotted correctly B2FT for 7 or 8 points plotted correctly B1FT for 5 or 6 points plotted correctly 9(c) Correct ruled line drawn 1 9(d) 2.4 1 FT their graph and y = 5

This question in 0580/31 May/June 2022

157 (a) Complete the table of values for y = , x ≠ 0. x x - 15 - 10 - 5 - 3 - 2 - 1 1 2 3 5 10 15 y - .15 - 5 - 15 15 5 [3] 15 (b) On the grid, draw the graph of y = for - 15 G x G - 1 and 1 G x G 15 . x y 16 14 12 10 8 6 4 2 x – 16 – 14 – 12 – 10 – 8 – 6 – 4 – 2 0 2 4 6 8 10 12 14 16 – 2 – 4 – 6 – 8 – 10 – 12 – 14 – 16 [4] (c) Write down the order of rotational symmetry of the graph. … [1] (d) (i) On the grid, draw the lines of symmetry of the graph. [2] (ii) Write down the equation of the line of symmetry that does not intersect the graph. … [1] 15(e) Use your graph to solve the equation =- 6 . x x = … [1]

12 marks

Mark scheme: 7(a) −1 −3 −7.5 7.5 3 1.5 1 3 B2 for 5 or 6 correct B1 for 3 or 4 correct 7(b) Correct curve 4 B3FT for 11 or 12 points correctly plotted B2FT for 9 or 10 points correctly plotted B1FT for 6, 7 or 8 points correctly plotted 7(c) 2 1 7(d)(i) Lines y = x and y = −x drawn 2 B1 for each 7(d)(ii) y = −x oe 1 7(e) −2.5 1 FT their intersection of y = –6 with their graph

This question in 0580/33 May/June 2022

8 (a) (i) Complete the table of values for y = x 2 + 2x - 4 . x −5 −4 −3 −2 −1 0 1 2 3 y 11 −1 −1 11 [3] (ii) On the grid, draw the graph of y = x 2 + 2x - 4 for - 5 G x G 3 . y 12 11 10 9 8 7 6 5 4 3 2 1 – 5 – 4 – 3 – 2 – 1 0 1 2 3 x – 1 – 2 – 3 – 4 – 5 – 6 [4] (iii) Write down the equation of the line of symmetry of the graph. … [1] (iv) Use your graph to solve the equation x 2 + 2x - 4 = 0 . x = … or x = … [2] (b) A line L has an equation of y = 5x + 7 . Write down the equation of the line parallel to L that passes through (0, −2). … [2]

12 marks

Mark scheme: 8(a)(i) 4 –4 –5 –4 4 3 B2 for 3 or 4 correct B1 for 1 or 2 correct 8(a)(ii) Correct curve 4 B3FT for 8 or 9 points correctly plotted B2FT for 6 or 7 points correctly plotted B1FT for 4 or 5 points correctly plotted 8(a)(iii) x = −1 oe 1 8(a)(iv) −3.3 to −3.1 1.1 to 1.3 2 FT their graph B1FT for each 8(b) y = 5x – 2 final answer 2 B1 for y = 5x + k oe or y = kx − 2 oe or y = 5x + – 2 oe or y = kx + – 2 oe or 5x − 2 oe or 5x + − 2 oe

This question in 0580/32 Oct/Nov 2022

6 y 10 L 9 8 7 6 5 4 3 2 1 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 x – 1 – 2 – 3 – 4 (a) Find the equation of line L in the form y = mx + c. y = … [2] (b) Write down the coordinates of the point where line L crosses the x-axis. ( … , … ) [1] (c) (i) Complete the table of values for y = x 2 + 5x + 3 . x −6 −5 −4 −3 −2 −1 0 1 y 9 −1 −1 [3] (ii) On the grid, draw the graph of y = x 2 + 5x + 3 for -6 G x G 1. [4] (d) (i) On the grid, draw the line y = 6 . [1] (ii) Use your graphs to solve the equation x 2 + 5x + 3 = 6 . x = … or x = … [2]

13 marks

Mark scheme: 6(a)  y = 2 x + 7 2 B1 for 2 x + c, c  7 or B1 for mx + 7 where m is their gradient and m  2 6(b) ( −3.5, 0 ) 1 6(c)(i) 3, −3, −,3 3, 9 3 B2 for 3 or 4 correct or B1 for 1 or 2 correct 6(c)(ii) Completely correct curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 6(d)(i) Correct ruled line drawn 1 6(d)(ii) 0.4 to 0.7, −5.7 to −5.4 2 FT their graph and their line B1FT for each

This question in 0580/32 Feb/March 2023

6 (a) Complete the table of values for y = 5 + 3 x - x 2 . x - 2 - 1 0 1 2 3 4 5 y 1 7 –5 [3] (b) On the grid, draw the graph of y = 5 + 3x - x 2 for - 2 G x G 5 . y 12 11 10 9 8 7 6 5 4 3 2 1 – 2 – 1 0 1 2 3 4 5 x – 1 – 2 – 3 – 4 – 5 – 6 [4] (c) Write down the equation of the line of symmetry of the graph. … [1] (d) (i) Complete the table of values for y = 2x + 1. x - 1 0 2 y [2] (ii) On the grid, draw the graph of y = 2x + 1 for - 2 G x G 5 . [1] (e) Write down the coordinates of the two points where the two graphs intersect. ( … , … ) and ( … , … ) [3]

14 marks

Mark scheme: 6(a) −5 5 7 5 1 3 B2 for 3 or 4 correct or B1 for 1 or 2 correct 6(b) Correct and accurate curve 4 B3FT for 7 or 8 points correctly plotted or B2FT for 5 or 6 points correctly plotted or B1FT for 3 or 4 points correctly plotted 6(c) x = 1.5 oe 1 6(d)(i) −1 1 5 2 B1 for 2 correct 6(d)(ii) Correct ruled line 1 6(e) (−1.7 to −1.4, −2.4 to −1.8) 3 FT their curve and their line (2.4 to 2.7, 5.8 to 6.4) B2FT for 3 values correct or B1FT for 2 values correct

This question in 0580/31 May/June 2023

6 y 8 L 7 6 5 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 x – 1 – 2 – 3 – 4 – 5 – 6 – 7 (a) Write down the equation of line L in the form y = mx + c . y = … [2] (b) (i) Complete the table of values for y = x 2 - 3x - 3 . x - 2 - 1 0 1 2 3 4 5 y 1 - 5 - 5 1 [2] (ii) On the grid, draw the graph of y = x 2 - 3x - 3 for - 2 G x G 5 . [4] (c) (i) Write down the coordinates of the lowest point of the graph of y = x 2 - 3x - 3 . ( … , … ) [1] (ii) On the grid, draw the line of symmetry of the graph of y = x 2 - 3x - 3 . [1] (iii) Write down the equation of the line of symmetry. … [1] (d) Write down the coordinates of the point where line L intersects the graph of y = x 2 - 3x - 3 for x 2 0 . ( … , … ) [1]

12 marks

Mark scheme: 6(a)  y  2 x  3 final answer 2 B1 for –2x + c as final answer or B1 for mx + 3, m  0, as final answer 6(b)(i) 7, 3, ,3 7 2 B1 for 2 or 3 correct 6(b)(ii) Correct and accurate curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 6(c)(i) 1.5, k  1 where –5.5 ⩽ k < –5 6(c)(ii) Line x  1.5 drawn accurately 1 6(c)(iii) x  1.5 oe 1 6(d)  3,  3  1 FT their curve with line L

This question in 0580/33 May/June 2023

4 (a) Complete the table of values for y = x 2 - 4x - 2 . x -2 -1 0 1 2 3 4 5 y 3 -2 -5 -5 -2 3 [2] (b) On the grid, draw the graph of y = x 2 - 4x - 2 for - 2 G x G 5 . y 10 9 8 7 6 5 4 3 2 1 x -2 -1 0 1 2 3 4 5 -1 -2 -3 -4 -5 -6 [4] (c) Use your graph to solve the equation x 2 - 4x - 2 = 0 . x = … or x = … [2]

8 marks

Mark scheme: 4(a) 10 –6 2 B1 for each 4(b) Correct curve 4 B3FT for 7 or 8 points correctly plotted or B2FT for 5 or 6 points correctly plotted or B1FT for 3 or 4 points correctly plotted 4(c) –0.6 to –0.3, 4.3 to 4.6 2 FT their curve B1 for each

This question in 0580/32 Oct/Nov 2023

10 y NOT TO SCALE x (–1, 0) (3, 0) The sketch shows the graph of y = x 2 - 2x - 3 . The graph crosses the x-axis at ( - 1, 0) and (3, 0). (a) Find the equation of the line of symmetry of the graph. … [1] (b) (i) The point A with coordinates (6, k) lies on the graph. Show that the value of k is 21. [1] (ii) The point B with coordinates (p, 21) also lies on the graph. Find the value of p. p = … [1] (c) Write down the y-coordinate of the point where the graph crosses the y-axis. … [1]

4 marks

Mark scheme: 10(a) x = 1 oe 1 10(b)(i) 62 – 2 × 6 – 3 [leading to 21] 1 10(b)(ii) –4 1 10(c) –3 1

This question in 0580/33 Oct/Nov 2023

5 (a) (i) Complete the table of values for y =- x 2 + 5x + 7 . x -1 0 1 2 3 4 5 6 y 11 11 1 [3] (ii) On the grid, draw the graph of y =- x 2 + 5x + 7 for - 1 G x G 6 . y 14 13 12 11 10 9 8 7 6 5 4 3 2 1 – 1 0 1 2 3 4 5 6 x [4] (iii) (a) Write down the equation of the line of symmetry of the graph. … [1] (b) The points ( -8, -97) and ( t, -97) also lie on the graph of y =- x 2 + 5 x + 7 . Use symmetry to find the value of t. t = … [1] (b) Write down the gradient of the line y = 9x - 4 . … [1] (c) Write down the equation of a line parallel to y =-5x + 19 . y = … [1] (d) y 7 6 L 5 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 Find the equation of line L in the form y = mx + c . y = … [2] (e) Make x the subject of the formula y = mx + c . x = … [2]

15 marks

Mark scheme: 5(a)(i) 1, 7, 13, 13, 7 3 B2 for 3 or 4 correct or B1 for 1 or 2 correct 5(a)(ii) Completely correct curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 5(a)(iii)(a) x = 2.5 oe 1 5(a)(iii)(b) 13 1 5(b) 9 1 5(c) y = −5 x + k where k  19 1 5(d)  y =  − x + 2 final answer 2 B1 for  y =  − x + c or  y =  mx + 2 where m is their gradient 5(e) − c 2 M1 for a correct first step  x = y oe final answer m y c y −=c mx or = x + m m

This question in 0580/32 Feb/March 2024

8 The table shows some values for y = x 2 - x - 3 . x - 3 - 2 - 1 0 1 2 3 4 y - 1 - 1 9 (a) Complete the table. [3] (b) On the grid, draw the graph of y = x 2 - x - 3 for - 3 G x G 4 . y 9 8 7 6 5 4 3 2 1 x – 3 – 2 – 1 0 1 2 3 4 – 1 – 2 – 3 – 4 [4] (c) Write down the coordinates of the lowest point on the graph. ( … , … ) [1] (d) Use your graph to solve the equation x 2 - x - 3 = 7 . x = … or x = … [2]

10 marks

Mark scheme: 8(a) 9, 3, 3, 3, 3 3 B2 for 3 or 4 correct or B1 for 1 or 2 correct 8(b) Completely correct curve 4 B3FT for 7 or 8 correctly plotted points or B2FT for 5 or 6 correctly plotted points or B1FT for 3 or 4 correctly plotted points 8(c)  0.5,k  3.6  k 3 1 8(d) 3.7  2.7 2 B1FT for each

This question in 0580/31 May/June 2024

6 (a) (i) Complete the table of values for y =- x 2 + 8x + 1. x 0 1 2 3 4 5 6 7 8 y 13 17 13 1 [3] (ii) On the grid, draw the graph of y =- x 2 + 8x + 1 for 0 G x G 8 . y 18 16 14 12 10 8 6 4 2 0 x 0 2 4 6 8 [4] (iii) Write down the equation of the line of symmetry of the graph. … [1] 1(b) A straight line has a gradient of and passes through the point (2, 7). 2 (i) On the grid, draw this line for 0 G x G 8 . [2] (ii) Write down the equation of this line in the form y = mx + c . y = … [2] (iii) Write down the coordinates of the points where this line intersects the graph of y =- x 2 + 8x + 1. ( … , … ) and ( … , … ) [2]

14 marks

Mark scheme: 6(a)(i) 1 8 16 16 8 3 B2 for 3 or 4 correct or B1 for 1 or 2 correct 6(a)(ii) correct curve 4 B3FT for 8 or 9 correctly plotted points or B2FT for 6 or 7 correctly plotted points or B1FT for 4 or 5 correctly plotted points 6(a)(iii) x = 4 oe 1 6(b)(i) Correct line 2 1 B1 for a line with gradient 2 or a line through (2,7) 6(b)(ii) 1 2 B1 for [y =] ½ x + j oe or [y =] kx + 6 oe y = x + 6 oe k ≠ 0 2 6(b)(iii) (0.6 to 0.9 , 6.2 to 6.5) 2 B1FT for each (6.6 to 6.9 , 9.2 to 9.5)

This question in 0580/32 May/June 2024

9 (a) Line L has a gradient of 4 and passes through the point (0, 3). Write down the equation of line L in the form y = mx + c . y = … [1] (b) Line G has the equation y = 2 - 6x . Line G passes through the point (a, 5). Find the value of a. a = … [3] (c) (i) Complete the table of values for y = x 2 - 6 . x -4 -3 -2 -1 0 1 2 3 4 y 10 -2 -5 -5 -2 10 [2] (ii) On the grid, draw the graph of y = x 2 - 6 for - 4 G x G 4 . y 10 9 8 7 6 5 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 – 4 – 5 – 6 – 7 [4] (iii) Write down the equation of the line of symmetry of the graph. … [1] (iv) Use your graph to solve the equation x 2 - 6 = 0 for x 2 0 . x = … [1]

12 marks

Mark scheme: 9(a) [y =] 4x + 3 1 9(b) – 12 oe 3 B1 for 5 = 2 – 6a M1 for rearranging their linear equation to ra = s 9(c)(i) 3 –6 3 2 B1 for 2 correct 9(c)(ii) correct curve 4 B3FT for 8 or 9 points plotted accurately or B2FT for 6 or 7 points plotted accurately or B1FT for 4 or 5 points plotted accurately 9(c)(iii) x = 0 1 9(c)(iv) 2.3 to 2.6 1 FT their point of intersection

This question in 0580/33 May/June 2024

8 (a) (i) Complete the table of values for y = 4x - x 2 . x -1 0 1 2 3 4 5 6 y -5 0 3 4 3 -5 [2] (ii) On the grid, draw the graph of y = 4x - x 2 for -1 G x G 6 . y 5 4 3 2 1 – 1 0 1 2 3 4 5 6 x – 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10 – 11 – 12 – 13 [4] (iii) Use your graph to solve the equation 4x - x 2 = 1. x = … or x = … [2] (b) The line L is shown on the grid. y 5 L 4 3 2 1 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 – 4 Find the equation of line L. … [3] Question 9 is printed on the next page.

11 marks

Mark scheme: 8(a)(i) 0 –12 2 B1 for each 8(a)(ii) Correct curve 4 B3FT for 7 or 8 points correctly plotted or B2FT for 5 or 6 points correctly plotted or B1 FT for 3 or 4 points correctly plotted 8(a)(iii) 0.2 to 0.4 , 3.6 to 3.8 2 FT their curve B1 for each 8(b) y = 1.5x – 1 oe final answer 3 B2 for 1.5x – 1 oe or y = 1.5x + c oe or y = mx – 1 oe m is their gradient m ≠0 or B1 for 1.5x + c or mx – 1 where m is their gradient m ≠0

This question in 0580/33 Oct/Nov 2024

28 (a) (i) Complete the table of values for y = x 2 - x - 5 . x −3 −2 −1 0 1 2 3 4 y 1 −3 −3 1 [2] (ii) On the grid, draw the graph of y = x 2 - x - 5 for –3 G x G 4 . y 8 7 6 5 4 3 2 1 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 – 4 – 5 – 6 [4] (b) Write down the equation of the line of symmetry of the graph. … [1] (c) Use the graph to solve the equation x 2 - x - 5 = 0 . x = … or x = … [2]

9 marks

Mark scheme: 28(a)(i) 7, [1], [−3], –5, –5, [−3], [1], 7 2 B1 for 2 correct 28(a)(ii) Correct graph 4 B3FT for 7 points accurately plotted or B2FT for 5 points accurately plotted or B1FT for 3 points accurately plotted 28(b) 1 1 x = oe 2 28(c) –1.9 to –1.7 2.7 to 2.9 2 FT their curve B1 for each

This question in 0580/31 May/June 2025

20 19 (a) Complete the table of values for y = . x x 1 2 3 4 5 y 20 10 4 [2] 20 (b) On the grid, draw the graph of y = for 1 G x G 5 . x y 25 20 15 10 5 0 x 0 1 2 3 4 5 6 [3] 20 (c) Use your graph to solve the equation = 8 . x x = … [1]

6 marks

Mark scheme: 19(a) [20], [10], 6.67, 5, [4] 2 B1 for each 19(b) Correct graph 3 B2FT for 4 points correctly plotted or B1FT for 2 points correctly plotted 19(c) 2.45 to 2.55 1 FT their curve

This question in 0580/33 May/June 2025

12 27 (a) Complete the table of values for y = . x x -4 -3 -2 -1 1 2 3 4 y -4 -6 6 4 [2] 12 (b) On the grid, draw the graph of y = for - 4 G x G -1 and 1 G x G 4 . x y 12 10 8 6 4 2 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 2 – 4 – 6 – 8 – 10 – 12 [4] 12 (c) Use your graph to write down the solution of the equation = 10 . x x = … [1] Question 28 is printed on the next page.

7 marks

Mark scheme: 27(a) −3 −12 12 3 2 B1 for 2 correct 27(b) Correct curve 4 B3FT for 7 points correctly plotted B2FT for 5 points correctly plotted B1FT for 3 points correctly plotted 27(c) 1.1 to 1.3 1 FT their curve

This question in 0580/32 Oct/Nov 2025

22 (a) Complete the table of values for y = x 2 - 4 . x –4 –3 –2 –1 0 1 2 3 4 y 12 0 –3 –3 0 12 [2] (b) On the grid, draw the graph of y = x 2 - 4 for - 4 G x G 4 . y 12 11 10 9 8 7 6 5 4 3 2 1 – 4 – 3 – 2 – 1 0 1 2 3 4 x – 1 – 2 – 3 – 4 – 5 [4]

6 marks

Mark scheme: 22(a) 5 – 4 5 2 B1 for 2 correct 22(b) Correct curve 4 B3FT for 8 points correctly plotted B2FT for 6 points correctly plotted B1FT for 4 points correctly plotted

This question in 0580/33 Oct/Nov 2025