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P2 Graphs Function

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The document contains 10 math problems involving graphing functions and inequalities on Cartesian planes. The problems involve sketching graphs of functions, finding coordinates that satisfy equations, drawing lines to solve equations, and shading regions defined by inequalities. Tables are used to list x and y values satisfying equations.

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ppr maths nbk Graphs function 1. On the diagrams, sketch the graphs of the following functions. (a) y = x + 1 (b) y = 2 - xΒ² y y x 0 x 0 2. On the diagrams, sketch the graphs of the following functions. 2 (a) y = xΒ³ + 1 (b) y = - x y y x x 0 0 3. (a) The diagram shows the graph of the function y = (x – 1)(x – 3) y P x 0 Q Find the coordinates of (i) P (ii) Q Answer : 1
ppr maths nbk (b) On the Cartesian plane below, sketch the graph of function y = x(x + 5) y x 0 4. (a) The diagram shows the graph of function y = xn – 8 . y x 0 a -8 State the value of (i) n (ii) a Answer : 8 (b) On the diagram, sketch the graph of the function y = for x β‰₯ 0 . x y x 0 5.Write the inequalities which define the shaded region in the graph. y x=5 Answer : y=x-5 x 0 y = -x 2
ppr maths nbk 6. State the three inequalities which define the shaded region in the graph. y x=6 y = 12 Answer : x 0 y + 2x = 12 7. State the three inequalities which define the shaded region in the graph. y y = x-3 Answer : 2 0 2y+x = 0 8. On the graph provided, shade the region which satisfies the two inequalities x – y β‰₯ 4 and x < 4. y x-y = 4 x 0 4 -4 3
ppr maths nbk 9. On the graph provided, shade the region which satisfies the three inequalities y ≀ 2x+6, y β‰₯ x and y < 6. y y=x x 0 y = 2x+ 6 10. 10.On the graph provided, shade the region which satisfies the three inequalities x + 2y β‰₯ 6 , x ≀ 4 and y ≀ x . y y=x x 0 y=4 x + 2y = 6 4
ppr maths nbk EXERCISE 2 1.(a) x -4 - 2.5 -1 0 1 2 2.5 3 4 y p 6 -3 -4 -1 q 11 17 32 Table 1 Table 1 shows the values of x and y which satisfy the equation y = 2x2 + x – 4. Calculate the value of p and of q. [2 marks] (b) Use graph paper to answer this part of the question. You may use a flexible curve ruler. Using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the graph of y = 2x2 + x – 4 for – 4 ≀ x ≀ 4. [4 marks] (c) From your graph in (b), find (i) the value of y when x = - 3, (ii) the values of x when y = 5. [2 marks] (d) Draw a suitable straight line on your graph to find all the values of x which satisfy the equation y = 2x2 + 5x – 12 for – 4 ≀ x ≀ 4. [4 marks] 2.(a) x -3 -2 -1 - 0.5 0.5 1 1.5 2 3 y -1 -3 6 3 1.5 1 Table 2 3 Copy and complete Table 2 for the equation y = . [3 marks] x (b) For this part of the question, use a graph paper. You may also use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph 3 of y = for – 3 ≀ x ≀ 3. [3 marks] x (d) From your graph, find (i) the value of y when x = - 2.5, (ii) the values of x when y =1.8. [3 marks] (d) Draw a suitable straight line on your graph to find all the values of x which satisfy the 3 equation - = - 2x + 3 for – 3 ≀ x ≀ 3. State these values of x. [3 marks] x 5
ppr maths nbk 3.(a) Copy and complete the table below for the equation y = x2 - 3x + 5. x -2 - 1.5 -1 - 0.5 1 2 3.5 4 5 y 1.5 11.75 6.75 3 6.75 9 15 Table 3 [2 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph of y = x2 - 3x + 5 for – 2 ≀ x ≀ 5. [4 marks] (e) From the graph, find (i) the value of y when x = 1.5, (ii) the value of x when y = 8. [2 marks] (d) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation x2 + 5x – 2=0 for – 2 ≀ x ≀ 5. State these values of x. [4 marks] 4.(a) x -4 -3 - 2.5 -2 -1 0 1 2 3 y -60 -28 -15.75 -6 p 8 0 q -46 Table 4 Table 4 shows the values of x and y which satisfy the equation y = 8 – 3x – 5x2. Calculate the values of p and q. [2 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 10 units on y-axis, draw the graph of y = 8 – 3x – 5x2 for – 4 ≀ x ≀ 3. [4 marks] (f) From the graph, find (i) the value of y when x = 1.5, (ii) the value of x when y = -20. [3 marks] (d) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation 10 – 4x – 5x2 =0 for – 4 ≀ x ≀ 3. [3 marks] 6
ppr maths nbk 5.(a) Complete Table 5 for the equation y = 12 + 5x – 2x2 x -3 -2 -1 0 1 2.5 3 4 5 y -21 5 12 15 12 9 -13 Table 5 [2 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 50 units on y-axis, draw the graph of y = 12 + 5x – 2x2 for – 3 ≀ x ≀ 5. [4 marks] (c) From the graph, find (iii) the value of y when x = -2.5, (iv) the value of x when y = 14. [2 marks] (d) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation 15 + 3x – 2x2 = 0 for – 3 ≀ x ≀ 5. [4 marks] 6.(a) Complete Table 6 for the equation y = x3 – 3x + 6. x -3 -2 -1 0 1 1.5 2 2.5 3 y -12 8 6 4 14.1 24 Table 6 [3 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 50 units on y-axis, draw the graph of y = x3 – 3x + 6 for – 3 ≀ x ≀ 3. [4 marks] ( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation x3 – 6x + 4 = 0 for – 3 ≀ x ≀ 3. [3 marks] (d) Shade the region which represents the inequalities x β‰₯ 0, y β‰₯ - 8 and y ≀ - 4x. [2 marks] 7
ppr maths nbk 7.(a) Complete Table 7 for the equation y = 3x – x2. x 0 1 2 3 4 5 6 7 8 y 0 2 0 -4 - 18 - 28 Table 7 [3 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the graph of y = 3x – x2 for 0 ≀ x ≀ 8. [4 marks] ( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation 5x – x2 = 2 for 0 ≀ x ≀ 8. [3 marks] (d) Shade the region which represents the two inequalities and y ≀ 3x – x2 and y β‰₯ - 10. [2 marks] 6 8.(a) Complete Table 8 for the equation y = – . x x -4 - 2.5 -1 - 0.6 0.6 1 2 3 4 y 1.5 6 10 - 10 -6 -3 - 1.5 Table 2 [3 marks] (b) For this part of the question, use a graph paper. You may also use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph 6 of y = – for – 4 ≀ x ≀ 4. [3 marks] x (c) From your graph, find (v) the value of y when x = 1.3, (vi) the values of x when y =3.5. [3 marks] (d) Draw a suitable straight line on your graph to find all the values of x which satisfy the 3 equation + 1 = x for – 4 ≀ x ≀ 4. x [3 marks] 8
ppr maths nbk 9.(a) Complete Table 9 for the equation y = x3 – 4x – 6. x -4 -3 -2 -1 0 1 2 3 3.5 y - 54 - 21 -3 -6 -6 22.88 Table 9 [3 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 10 units on y-axis, draw the graph of y = x3 – 4x – 6 for – 4 ≀ x ≀ 3.5. [4 marks] ( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation x3 – 4x + 2 = 0 for – 4 ≀ x ≀ 3.5. [3 marks] (d) Shade the region which represents the three inequalities y ≀ x3 – 4x – 6, y β‰₯ 5x - 10 and ≀ 2.5. [2 marks] 10.(a) Complete Table 10 for the equation y = x3 – 10x + 5. x - 3.5 -3 -2 -1 0 1 2 3 3.5 y - 2.9 8 14 5 -7 12.88 Table 10 [3 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the graph of y = x3 – 10x + 5 for – 3.5 ≀ x ≀ 3.5. [4 marks] (c) From your graph, find the value of y when x = - 2.5. [2 marks] (d) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation x3 – 6x + 4 = 0 for – 3 ≀ x ≀ 3. [3 marks] 9
ppr maths nbk ANSWERS : EXERCISE 1 1. On the diagrams, sketch the graphs of the following functions. (a) y = x + 1 (b) y = 2 - xΒ² y y 2 1 x 0 x -1 0 2. On the diagrams, sketch the graphs of the following functions. 2 (a) y = x3 + 1 (b) y = - x y y 1 x 1 x 0 0 -2 3. (a) The diagram shows the graph of the function y = (x – 1)(x – 3) y P x 1 Q Find the coordinates of (i) P (ii) Q Answer : (i) When x = 0, y = 4 (ii) When y = 0, x = 1 or x = 3 P ( 0, 4 ) Q ( 3, 0 ) 10
ppr maths nbk (b) On the Cartesian plane below, sketch the graph of function y = x2 - 2 y 0 x -2 4. (a) The diagram shows the graph of function y = xn – 8 . y x 1 a -8 State the value of (i) n (ii) a Answer : (i) n = 3 (ii) When y = 0, x3 – 8 = 0 x3 = 8 x =38 x=a=2 8 (b) On the diagram, sketch the graph of the function y = for x β‰₯ 0 . x y 8 8 y= x x 0 1 5.Write the inequalities which define the shaded region in the graph. y x=5 Answer : y=x–5 x≀5 y≀x–5 x 0 y > -x y = -x 11
ppr maths nbk 6. State the three inequalities which define the shaded region in the graph. y x=6 y = 12 Answer : x<6 y ≀ 12 y + 2x β‰₯ 12 x 1 y + 2x = 12 7. State the three inequalities which define the shaded region in the graph. y y = x-3 Answer : 2 yβ‰₯x–3 y≀2 2y + x β‰₯ 0 0 2y+x = 0 8. On the graph provided, shade the region which satisfies the two inequalities x – y β‰₯ 4 and x < 4. y x-y = 4 x 1 4 x=4 -4 12
ppr maths nbk 9. On the graph provided, shade the region which satisfies the three inequalities y ≀ 2x+6, y β‰₯ x and y < 6. y y=x y=6 0 x y = 2x+ 6 10.On the graph provided, shade the region which satisfies the three inequalities x + 2y β‰₯ 6 , x ≀ 4 and y ≀ x . y y=x x 0 y=4 x + 2y = 6 13
ppr maths nbk ANSWERS : EXERCISE 2 1. (a) p = 24, q = 6 (b) y y = - 4x + 8 x 0 (c) (i) 11 (ii) - 2.4, 1.9 (d) - 4, 1.5 2. (a) x = - 2, y = - 1.5 x = - 0.5, y = - 6 x = 1.5, y = 2 (b) y y=2x - 3 x 0 (c) (i) x = - 2.5, y = - 1.2 (ii) y = 1.8, x = 1.7 14
ppr maths nbk (d) y = 2x – 3 x = 2.19, - 0.69 3. (a) y = 9, y = 3 (b) y x 0 y = - 8x + 7 (c) (i) y = 2.75 (ii) x = - 0.8, 3.8 (d) y = - 8x + 7, x = 0.38 4. (a) p = 6, q = - 18 (b) y x 0 y=x-2 (c) (i) y = - 7.75 (ii) x = 2.09, x = - 2.69 (d) x = - 1.83, x = 1.1, y = x - 2 15
ppr maths nbk 5. (a) x = - 2, y = - 6 x = 4, y = 0 (b) y y = 2x - 3 x 0 (c) (i) y = - 13 (ii) x = 0.5 , x = 2 (d) x = - 2.1, x = 3.6, y = 2x - 3 6. (a) x = - 2, y = 4 x = 1.5, y = 4.75 x = 2, y = 8 (b) , (d) y x 0 y = -8 y = - 4x (c) y = 3x + 2 x = - 2.8, 0.85, 2 16
ppr maths nbk 7. (a) x = 2, y = 2 x = 5, y = - 10 x = 8, y = - 40 (b), (d) y x 0 y = - 10 (c) (i) y = 2 – 2x (ii) x = 0.45, x = 4.55 8. (a) x = - 2.5, y = 2.4 x = 3, y = - 2 (b) y x 0 y=- 2x + 2 (c) (i) y = - 4.6 (ii) x = - 1.7 (d) y = - 2x + 2, x = - 1.3, 2.3 17
ppr maths nbk 9. (a) x = - 2, y = - 6 x = 1, y = - 9 x = 3, y = 9 (b) , (d) x = -2.5 y y = 5x - 10 x 0 (c) y =-8 x = - 2.2, 0.5, 1.5 10. (a) x = - 2, y = 17 x = 1, y = - 4 x = 3, y = 2 (b) y 5 x 0 (c) y = 13.5 (d) y = 11, x = - 2.75, - 0.65, 3.45 18
ppr maths nbk DIAGNOSTIC TEST 1. Complete the following table for the equation y = 5 + 7x – 2x2. x -3 -2 -1 0 1 2 3 4 5 y - 34 -4 5 11 8 - 10 6 2. Complete the following table for the equation y = – . x x -4 -2.5 -1 -0.6 0.6 1 2 3 4 y 1.5 6 10 -10 -6 -3 - 1.5 3. Use the graph paper to answer this question. Table 11 shows the values of two variables, x and y, of a function. x -4 -3 -2 -1 0 1 2 3 y 9 0 -5 -6 -3 4 15 30 Table 11 The scale on the x-axis is 2cm to 1 unit. The scale on the y-axis is 2cm to 5 units. By using the scale given, draw the graph of the function. 4. Use the graph paper to answer this question. Table 12 shows the values of two variables, x and y, of a function. x -4 -3.5 -3 -2.5 -2 -1 0 1 2 y -56 -35 -19 -8 0 7 8 9 16 Table 12 The scale on the x-axis is 2cm to 1 unit. The scale on the y-axis is 2cm to 10 units. By using the scale given, draw the graph of the function. 5. Use the graph paper to answer this question. Table 13 shows some values of x and y which satisfy the equation y = 2x2 – 4x – 5. x -2 -1 0 1 2 3 4 5 y 11 m -5 -7 n 1 11 25 Table 13 (a) Calculate the values of m and n. (b) Using a scale of 2cm to 1 unit on the x-axis and 2cm to 5 units on the y-axis, draw the graph of y = 2x2 – 4x – 5 for – 2 ≀ x ≀ 5. 19
ppr maths nbk ANSWERS – Diagnostic Test 1. x = - 2, y = - 17; x = 1, y = 10; x = 4, y = 1 2. x = - 2.5, y = 2.4; x = 3, y = - 2 3. y x 0 4 y x 0 20
ppr maths nbk 5. (a) m = 1, n= - 5 (b) y x 0 21

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P2 Graphs Function

  • 1. ppr maths nbk Graphs function 1. On the diagrams, sketch the graphs of the following functions. (a) y = x + 1 (b) y = 2 - xΒ² y y x 0 x 0 2. On the diagrams, sketch the graphs of the following functions. 2 (a) y = xΒ³ + 1 (b) y = - x y y x x 0 0 3. (a) The diagram shows the graph of the function y = (x – 1)(x – 3) y P x 0 Q Find the coordinates of (i) P (ii) Q Answer : 1
  • 2. ppr maths nbk (b) On the Cartesian plane below, sketch the graph of function y = x(x + 5) y x 0 4. (a) The diagram shows the graph of function y = xn – 8 . y x 0 a -8 State the value of (i) n (ii) a Answer : 8 (b) On the diagram, sketch the graph of the function y = for x β‰₯ 0 . x y x 0 5.Write the inequalities which define the shaded region in the graph. y x=5 Answer : y=x-5 x 0 y = -x 2
  • 3. ppr maths nbk 6. State the three inequalities which define the shaded region in the graph. y x=6 y = 12 Answer : x 0 y + 2x = 12 7. State the three inequalities which define the shaded region in the graph. y y = x-3 Answer : 2 0 2y+x = 0 8. On the graph provided, shade the region which satisfies the two inequalities x – y β‰₯ 4 and x < 4. y x-y = 4 x 0 4 -4 3
  • 4. ppr maths nbk 9. On the graph provided, shade the region which satisfies the three inequalities y ≀ 2x+6, y β‰₯ x and y < 6. y y=x x 0 y = 2x+ 6 10. 10.On the graph provided, shade the region which satisfies the three inequalities x + 2y β‰₯ 6 , x ≀ 4 and y ≀ x . y y=x x 0 y=4 x + 2y = 6 4
  • 5. ppr maths nbk EXERCISE 2 1.(a) x -4 - 2.5 -1 0 1 2 2.5 3 4 y p 6 -3 -4 -1 q 11 17 32 Table 1 Table 1 shows the values of x and y which satisfy the equation y = 2x2 + x – 4. Calculate the value of p and of q. [2 marks] (b) Use graph paper to answer this part of the question. You may use a flexible curve ruler. Using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the graph of y = 2x2 + x – 4 for – 4 ≀ x ≀ 4. [4 marks] (c) From your graph in (b), find (i) the value of y when x = - 3, (ii) the values of x when y = 5. [2 marks] (d) Draw a suitable straight line on your graph to find all the values of x which satisfy the equation y = 2x2 + 5x – 12 for – 4 ≀ x ≀ 4. [4 marks] 2.(a) x -3 -2 -1 - 0.5 0.5 1 1.5 2 3 y -1 -3 6 3 1.5 1 Table 2 3 Copy and complete Table 2 for the equation y = . [3 marks] x (b) For this part of the question, use a graph paper. You may also use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph 3 of y = for – 3 ≀ x ≀ 3. [3 marks] x (d) From your graph, find (i) the value of y when x = - 2.5, (ii) the values of x when y =1.8. [3 marks] (d) Draw a suitable straight line on your graph to find all the values of x which satisfy the 3 equation - = - 2x + 3 for – 3 ≀ x ≀ 3. State these values of x. [3 marks] x 5
  • 6. ppr maths nbk 3.(a) Copy and complete the table below for the equation y = x2 - 3x + 5. x -2 - 1.5 -1 - 0.5 1 2 3.5 4 5 y 1.5 11.75 6.75 3 6.75 9 15 Table 3 [2 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph of y = x2 - 3x + 5 for – 2 ≀ x ≀ 5. [4 marks] (e) From the graph, find (i) the value of y when x = 1.5, (ii) the value of x when y = 8. [2 marks] (d) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation x2 + 5x – 2=0 for – 2 ≀ x ≀ 5. State these values of x. [4 marks] 4.(a) x -4 -3 - 2.5 -2 -1 0 1 2 3 y -60 -28 -15.75 -6 p 8 0 q -46 Table 4 Table 4 shows the values of x and y which satisfy the equation y = 8 – 3x – 5x2. Calculate the values of p and q. [2 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 10 units on y-axis, draw the graph of y = 8 – 3x – 5x2 for – 4 ≀ x ≀ 3. [4 marks] (f) From the graph, find (i) the value of y when x = 1.5, (ii) the value of x when y = -20. [3 marks] (d) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation 10 – 4x – 5x2 =0 for – 4 ≀ x ≀ 3. [3 marks] 6
  • 7. ppr maths nbk 5.(a) Complete Table 5 for the equation y = 12 + 5x – 2x2 x -3 -2 -1 0 1 2.5 3 4 5 y -21 5 12 15 12 9 -13 Table 5 [2 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 50 units on y-axis, draw the graph of y = 12 + 5x – 2x2 for – 3 ≀ x ≀ 5. [4 marks] (c) From the graph, find (iii) the value of y when x = -2.5, (iv) the value of x when y = 14. [2 marks] (d) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation 15 + 3x – 2x2 = 0 for – 3 ≀ x ≀ 5. [4 marks] 6.(a) Complete Table 6 for the equation y = x3 – 3x + 6. x -3 -2 -1 0 1 1.5 2 2.5 3 y -12 8 6 4 14.1 24 Table 6 [3 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 50 units on y-axis, draw the graph of y = x3 – 3x + 6 for – 3 ≀ x ≀ 3. [4 marks] ( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation x3 – 6x + 4 = 0 for – 3 ≀ x ≀ 3. [3 marks] (d) Shade the region which represents the inequalities x β‰₯ 0, y β‰₯ - 8 and y ≀ - 4x. [2 marks] 7
  • 8. ppr maths nbk 7.(a) Complete Table 7 for the equation y = 3x – x2. x 0 1 2 3 4 5 6 7 8 y 0 2 0 -4 - 18 - 28 Table 7 [3 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the graph of y = 3x – x2 for 0 ≀ x ≀ 8. [4 marks] ( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation 5x – x2 = 2 for 0 ≀ x ≀ 8. [3 marks] (d) Shade the region which represents the two inequalities and y ≀ 3x – x2 and y β‰₯ - 10. [2 marks] 6 8.(a) Complete Table 8 for the equation y = – . x x -4 - 2.5 -1 - 0.6 0.6 1 2 3 4 y 1.5 6 10 - 10 -6 -3 - 1.5 Table 2 [3 marks] (b) For this part of the question, use a graph paper. You may also use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph 6 of y = – for – 4 ≀ x ≀ 4. [3 marks] x (c) From your graph, find (v) the value of y when x = 1.3, (vi) the values of x when y =3.5. [3 marks] (d) Draw a suitable straight line on your graph to find all the values of x which satisfy the 3 equation + 1 = x for – 4 ≀ x ≀ 4. x [3 marks] 8
  • 9. ppr maths nbk 9.(a) Complete Table 9 for the equation y = x3 – 4x – 6. x -4 -3 -2 -1 0 1 2 3 3.5 y - 54 - 21 -3 -6 -6 22.88 Table 9 [3 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 10 units on y-axis, draw the graph of y = x3 – 4x – 6 for – 4 ≀ x ≀ 3.5. [4 marks] ( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation x3 – 4x + 2 = 0 for – 4 ≀ x ≀ 3.5. [3 marks] (d) Shade the region which represents the three inequalities y ≀ x3 – 4x – 6, y β‰₯ 5x - 10 and ≀ 2.5. [2 marks] 10.(a) Complete Table 10 for the equation y = x3 – 10x + 5. x - 3.5 -3 -2 -1 0 1 2 3 3.5 y - 2.9 8 14 5 -7 12.88 Table 10 [3 marks] (b) For this part of the question, use a graph paper. You may use a flexible curve ruler. By using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the graph of y = x3 – 10x + 5 for – 3.5 ≀ x ≀ 3.5. [4 marks] (c) From your graph, find the value of y when x = - 2.5. [2 marks] (d) Draw a suitable straight line on the graph to find all the values of x which satisfy the equation x3 – 6x + 4 = 0 for – 3 ≀ x ≀ 3. [3 marks] 9
  • 10. ppr maths nbk ANSWERS : EXERCISE 1 1. On the diagrams, sketch the graphs of the following functions. (a) y = x + 1 (b) y = 2 - xΒ² y y 2 1 x 0 x -1 0 2. On the diagrams, sketch the graphs of the following functions. 2 (a) y = x3 + 1 (b) y = - x y y 1 x 1 x 0 0 -2 3. (a) The diagram shows the graph of the function y = (x – 1)(x – 3) y P x 1 Q Find the coordinates of (i) P (ii) Q Answer : (i) When x = 0, y = 4 (ii) When y = 0, x = 1 or x = 3 P ( 0, 4 ) Q ( 3, 0 ) 10
  • 11. ppr maths nbk (b) On the Cartesian plane below, sketch the graph of function y = x2 - 2 y 0 x -2 4. (a) The diagram shows the graph of function y = xn – 8 . y x 1 a -8 State the value of (i) n (ii) a Answer : (i) n = 3 (ii) When y = 0, x3 – 8 = 0 x3 = 8 x =38 x=a=2 8 (b) On the diagram, sketch the graph of the function y = for x β‰₯ 0 . x y 8 8 y= x x 0 1 5.Write the inequalities which define the shaded region in the graph. y x=5 Answer : y=x–5 x≀5 y≀x–5 x 0 y > -x y = -x 11
  • 12. ppr maths nbk 6. State the three inequalities which define the shaded region in the graph. y x=6 y = 12 Answer : x<6 y ≀ 12 y + 2x β‰₯ 12 x 1 y + 2x = 12 7. State the three inequalities which define the shaded region in the graph. y y = x-3 Answer : 2 yβ‰₯x–3 y≀2 2y + x β‰₯ 0 0 2y+x = 0 8. On the graph provided, shade the region which satisfies the two inequalities x – y β‰₯ 4 and x < 4. y x-y = 4 x 1 4 x=4 -4 12
  • 13. ppr maths nbk 9. On the graph provided, shade the region which satisfies the three inequalities y ≀ 2x+6, y β‰₯ x and y < 6. y y=x y=6 0 x y = 2x+ 6 10.On the graph provided, shade the region which satisfies the three inequalities x + 2y β‰₯ 6 , x ≀ 4 and y ≀ x . y y=x x 0 y=4 x + 2y = 6 13
  • 14. ppr maths nbk ANSWERS : EXERCISE 2 1. (a) p = 24, q = 6 (b) y y = - 4x + 8 x 0 (c) (i) 11 (ii) - 2.4, 1.9 (d) - 4, 1.5 2. (a) x = - 2, y = - 1.5 x = - 0.5, y = - 6 x = 1.5, y = 2 (b) y y=2x - 3 x 0 (c) (i) x = - 2.5, y = - 1.2 (ii) y = 1.8, x = 1.7 14
  • 15. ppr maths nbk (d) y = 2x – 3 x = 2.19, - 0.69 3. (a) y = 9, y = 3 (b) y x 0 y = - 8x + 7 (c) (i) y = 2.75 (ii) x = - 0.8, 3.8 (d) y = - 8x + 7, x = 0.38 4. (a) p = 6, q = - 18 (b) y x 0 y=x-2 (c) (i) y = - 7.75 (ii) x = 2.09, x = - 2.69 (d) x = - 1.83, x = 1.1, y = x - 2 15
  • 16. ppr maths nbk 5. (a) x = - 2, y = - 6 x = 4, y = 0 (b) y y = 2x - 3 x 0 (c) (i) y = - 13 (ii) x = 0.5 , x = 2 (d) x = - 2.1, x = 3.6, y = 2x - 3 6. (a) x = - 2, y = 4 x = 1.5, y = 4.75 x = 2, y = 8 (b) , (d) y x 0 y = -8 y = - 4x (c) y = 3x + 2 x = - 2.8, 0.85, 2 16
  • 17. ppr maths nbk 7. (a) x = 2, y = 2 x = 5, y = - 10 x = 8, y = - 40 (b), (d) y x 0 y = - 10 (c) (i) y = 2 – 2x (ii) x = 0.45, x = 4.55 8. (a) x = - 2.5, y = 2.4 x = 3, y = - 2 (b) y x 0 y=- 2x + 2 (c) (i) y = - 4.6 (ii) x = - 1.7 (d) y = - 2x + 2, x = - 1.3, 2.3 17
  • 18. ppr maths nbk 9. (a) x = - 2, y = - 6 x = 1, y = - 9 x = 3, y = 9 (b) , (d) x = -2.5 y y = 5x - 10 x 0 (c) y =-8 x = - 2.2, 0.5, 1.5 10. (a) x = - 2, y = 17 x = 1, y = - 4 x = 3, y = 2 (b) y 5 x 0 (c) y = 13.5 (d) y = 11, x = - 2.75, - 0.65, 3.45 18
  • 19. ppr maths nbk DIAGNOSTIC TEST 1. Complete the following table for the equation y = 5 + 7x – 2x2. x -3 -2 -1 0 1 2 3 4 5 y - 34 -4 5 11 8 - 10 6 2. Complete the following table for the equation y = – . x x -4 -2.5 -1 -0.6 0.6 1 2 3 4 y 1.5 6 10 -10 -6 -3 - 1.5 3. Use the graph paper to answer this question. Table 11 shows the values of two variables, x and y, of a function. x -4 -3 -2 -1 0 1 2 3 y 9 0 -5 -6 -3 4 15 30 Table 11 The scale on the x-axis is 2cm to 1 unit. The scale on the y-axis is 2cm to 5 units. By using the scale given, draw the graph of the function. 4. Use the graph paper to answer this question. Table 12 shows the values of two variables, x and y, of a function. x -4 -3.5 -3 -2.5 -2 -1 0 1 2 y -56 -35 -19 -8 0 7 8 9 16 Table 12 The scale on the x-axis is 2cm to 1 unit. The scale on the y-axis is 2cm to 10 units. By using the scale given, draw the graph of the function. 5. Use the graph paper to answer this question. Table 13 shows some values of x and y which satisfy the equation y = 2x2 – 4x – 5. x -2 -1 0 1 2 3 4 5 y 11 m -5 -7 n 1 11 25 Table 13 (a) Calculate the values of m and n. (b) Using a scale of 2cm to 1 unit on the x-axis and 2cm to 5 units on the y-axis, draw the graph of y = 2x2 – 4x – 5 for – 2 ≀ x ≀ 5. 19
  • 20. ppr maths nbk ANSWERS – Diagnostic Test 1. x = - 2, y = - 17; x = 1, y = 10; x = 4, y = 1 2. x = - 2.5, y = 2.4; x = 3, y = - 2 3. y x 0 4 y x 0 20
  • 21. ppr maths nbk 5. (a) m = 1, n= - 5 (b) y x 0 21