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Chapter 3 Equations and Graphs 43

Chapter 3 Equations and Graphs

WARM-UP EXERCISE1. Solve the following equations.

(a) 2x 3 11 (b) 4x – 2 7

2. Solve the following equations.(a) 4(x – 2) 6 x (b) 3(8x – 1) 2(x 3)

3. Make x the subject of each of the following formulae.

(a) 4x 3y 8 (b)

4. Complete the table and draw the graph of each of the following equations.(a) y 3x – 2 (b) 5x – 2y 4

x 2 1 0 1 2 x 4 2 0 2 4y y

5. (a) If A(1, 3) is a point on the graph of ax + 2y 9, find the value of a.(b) If B(2, –3) is a point on the graph of 2x + 5y b, find the value of b.

6. Solve the following simultaneous equations graphically.[ Copies of the figures are provided in the Appendix. ]

(a) (b)

O

2

2

4

2x

y

1 3 4

1

3

1

3

6

5

O

2

2

4

2x

y

1 3 4

1

3

1

3

6

5

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44 New Trend Mathematics S4A — Supplement

7. Use the method of substitution to solve the following simultaneous equations.(a) (b)

8. Use the method of elimination to solve the following simultaneous equations.(a) (b)

9. Solve the following equations by factorization.(a) x2 – 4x – 5 0 (b) 2x2 3x – 14 0

10. Solve the following equations by using the quadratic formula.(a) x2 6x – 8 0 (b) 3x2 8x – 203 0

11. Determine the nature of the roots of each of the following equations.(a) x2 – 9x 21 0 (b) 5x2 8x 3 0

BUILD-UP EXERCISE[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set, Intermediate Set and Advanced Set. You may choose to complete any ONE set according to your need. ]

Exercise 3A Elementary Set

Level 11. According to the equation y x2 + 3x, complete the following table.

x 2 1 0 1 2y

2. Consider the equation y x2 2x – 8.(a) Complete the following table and plot the graph of the equation from x 5 to x 3.

x 5 4 3 2 1 0 1 2 3y

(b) Write down the x-intercepts and y-intercept of the graph.

3. Consider the equation y x2 – 4x 5.(a) Complete the following table and plot the graph of the equation from x to x 5.

x 1 0 1 2 3 4 5

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y

(b) Write down the x-intercept and y-intercept of the graph.

4. Consider the equation y x2 4x 4.(a) Complete the following table and plot the graph of the equation from x to x 6.

x 2 1 0 1 2 3 4 5 6y

(b) Write down the x-intercepts and y-intercept of the graph, correct your answers to 1 decimal place if necessary.

5. The figure shows the graph of the equation

. Solve the equation

graphically. (Correct your answers to 1 decimal place.)



graphically. (Correct your answer to 1 decimal place.)

7. Referring to each of the following graphs of the equations , complete the corresponding table.

O 1

1

1

2

2 3 4x

y

y x 2 4x 7

4

O 3

1

1

2

3

21x

y

y 2x 2 6x 9

2

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(a)

1

O

3

4

5

2

1 1 1

2

2 3 2 3 4x

yy ax b

(b)

1

O

3

4

2

1 1 1

2

3

4

2 2 3 4 5 6x

y

y ax b

x 2 2 x 2 0y 4 1 y 0 4

8. The figure shows the graph of the linear equation y x k. If the graph passes through P(2, 3), find the value of k.

9. The figure shows the graph of the linear equation y k – 3x. If the graph passes through Q(2, 9), find the value of k.

10. The figure shows the graph of the linear equation y kx 5. If the graph passes through R(1, 2), find the value of k.

11. The figure shows the graph of the equation y ax + b. The graph passes through P(–

xO 3

y

y ax b

P ( 4, 2)

xO

y

y x k

P (2, 3)

xO

y

y k 3x

Q (2, 9)

xO

yy kx 5

R (1, 2)


4, 2) and its x-intercept is –3.(a) Find the values of a and b.(b) Find the y-intercept of the graph.

12. The figure shows the graph of the equation .

(a) Referring to the given graph, complete the following table.

x 1 0 1 2y

(b) Find the values of x when(i) y 1.5.(ii) y 2.5.(Correct your answers to 1 decimal place.)

13. The figure shows the graph of the equation y ax2 bx c. From the graph,(a) find the corresponding value of y when

x 1.7.(b) find the corresponding values of x when

y 3.(Correct your answers to 1 decimal place.)

14. Find the number of x-intercept(s) of the graph of each of the following equations.(a) y x2 – 10x 3(b) y 3x2 – x 6

y

xO

1

2

3

4

5

6

7

8

3

2

11 2 1 3

y ax 2 bx c

O

1

1

2

3

21 1x

y

y ax 2 bx c

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(c) y x2 2x 5(d) y –x2 x 9

Level 215. If the graph of the equation cuts the x-axis at two distinct points, find the

range of values of k.

16. If the graph of the equation touches the x-axis, find the values of k.

17. The figure shows the graph of the equation from x 4 to x 3. Find the

range of values of x of the given graph such that(a) y < 0.(b) y 2.

18. The figure shows the graph of the equation y ax2 – 2x c.(a) From the graph, find the y-intercept.

Hence find the value of c.(b) Write down the coordinates of A. Hence

find the value of a.(c) (i) Find the x-intercepts of the graph.

(ii) When y = 1, find the corresponding values of x.

(Correct your answers to 1 decimal place.)

19. The figure shows the graph of the equation y x2 5x 7 from x 5 to x 0.(a) Solve the equation

graphically.(b) When y 6, find the corresponding

values of x, correct your answers to 1 decimal place.

(c) Find the range of values of x of the given graph such that y 3.

20. The figure shows the graph of the equation y 4 2x – x2 from x –2 to x 4.(a) Solve the equation

graphically.(b) Find the range of values of x of the

y

xO

1

2

3

121 3 4 2 3 1

y ax b

y

5

4

3

2

1

xO

2 3

4

11 2 3 4 1 2

y 4 2x x 2

O 3 1

5

4

3

2

1

2

21 1x

y

y ax 2 2x c

A

y

5

4

7

6

3

2

1

xO 5 1 2 3 4

y x 2 5x 7

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given graph such that y 0.(Correct your answers to 1 decimal place.)

Intermediate Set

Level 121. According to the equation y x2 – 2x 5, complete the following table.

x 3 1 1 3 5y

22. Consider the equation y 2x2 + x – 6.(a) Complete the following table and plot the graph of the equation from x 3 to x 2.

x 3 2 1 0 1 2y

(b) Write down the x-intercepts and y-intercept of the graph, correct your answers to 1 decimal place if necessary.

23. Consider the equation y x2 – x + 3.(a) Complete the following table and plot the graph of the equation from x 2 to x 3.

x 2 1 0 1 2 3y

(b) Write down the x-intercept and y-intercept of the graph.



graphically. (Correct your answers to 1 decimal place.)



graphically. (Correct your answer to 1 decimal place.)

y

4

3

2

1

xO

2

11 2 3 1

y x 2 2x 5

4

y

4

3

2

7

8

6

5

1

xO 1 2 1

y 2x 2 2x 1

2

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26. Referring to each of the following graphs of the equations , complete the corresponding table.(a)

O 2

6

4

2

4

6

42 31 1x

y

y ax b

(b) y

xO

1

4

5

3

2

121 4 2 1 3 5 7 8 9 10 6

y ax b

x 1 2 x 10 4y 4 4 y 0 1

27. The figure shows the graph of the linear equation y 2x k. If the graph passes through P(1, 4), find the value of k.

28. The figure shows the graph of the linear equation y k – 3x. If the graph passes through Q(3, 1), find the value of k.

29. The figure shows the graph of the equation . The graph passes through P(1, 2)

and its x-intercept is .

(a) Find the values of a and b.(b) Find the y-intercept of the graph.

xO

y

y 2x k

P (1, 4)

xO

y

y k 3x

Q ( 3, 1)

xO

y

21

y ax b

P ( 1, 2)

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y 1.5.(Correct your answers to 1 decimal place.)

Level 231. If the graph of the equation cuts the x-axis at two distinct points, find

the range of values of k.

32. Find the least integral value of k if the graph of the equation does not intersect the x-axis.

33. If the graph of the equation touches the x-axis at one point, find the values of k.

34. The figure shows the graph of the equation from to . Find the range of values of x of the given graph such that(a) y 2.(b) y > 4.

35. The figure shows the graph of the equation from to . Find the range of values of x of the given graph such that(a) y < 0.(b) y > 2.5.(Correct your answers to 1 decimal place if necessary.)

y

2

3

1

xO 1 2 3 1 2

y ax 2 bx c

y

5

4

6

3

2

1

xO 1 2 3 4 5 6 7 8 9 10 11 12

y ax b

O 1

2

1

1

3

3 4 5 2x

y

y ax b

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36. The figure shows the graph of the equation y x2 4x 4 from x 5 to x 1.(a) Solve the equation x2 + 4x + 4 0

graphically.(b) When y = 2, find the corresponding

values of x.(c) Find the range of values of x of the

given graph such that y > 2.(Correct your answers to 1 decimal place if necessary.)

37. The figure shows the graph of y x3 3x2 x – 2 from x 3 to x 1.(a) Find the x-intercepts of the graph.(b) Find the range of values of x of the given graph

such that y 0.(Correct your answers to 1 decimal place if necessary.)

38. The figure shows the graph of the equation A(2, 9) is a point on the graph

and –3 is one of the x-intercepts of the graph.(a) Find the values of a and b.(b) Find the other x-intercept.

39. The figure shows the graph of the equation y –2x2 – 12x + c. The graph touches the x-axis at (b, 0) and the y-intercept is 18.(a) Find the value of c.(b) Hence find the value of b.

y

5

4

7

8

9

6

3

2

1

xO 5 1 1 2 3 4

y x 2 4x 4

y

xO

1

2

3

4

5

3

2

11 2 1 3

y x 3 3x

2 x 2

xO

y

A ( 2, 9)

y ax 2 bx 9

3

xO

y

y 2x 2 12x c

b

18

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Advanced Set

Level 140. According to the equation y 3x2 6x – 2, complete the following table.

x 4 2 0 2 4y

41. Consider the equation .

(a) Complete the following table and plot the graph of the equation by completing the following table.

x 9 7 5 3 1 1 3y

(b) Write down the x-intercept and y-intercept of the graph, correct your answers to 1 decimal place if necessary.


Solve the equation

graphically. (Correct your

answers to 1 decimal place.)

43. The figure shows the graph of the equation Solve the equation

graphically.

44. Referring to the graph of the equation y ax b, complete the following table.

y

2

1

3

xO

2

1 1 2 1

3

y x 218

x 2

2

y

xO

2

1 1 2

3

4

5

6

1 3

y x 2 3x 5

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y ax b

O

1

1 2

2

3

1 2 3 4 5 6 7 8 1 2 3

y

x

x 8 0y 0 3

45. The figure shows the graph of the linear

equation . If the graph passes

through P(2, 3), find the value of k.

46. The figure shows the graph of the equation . The graph passes through P(4, 6)

and its x-intercept is 2.(a) Find the values of a and b.(b) Find the y-intercept of the graph.



y 1.(Correct your answers to 1 decimal place.)

Level 2

xO

y

P ( 2, 3)

y x k12

xO

y

P (4, 6)

y ax b

2

y

2

1

xO

1

1 2 1

y ax 2 bx c


48. If the graph of the equation cuts the x-axis at two distinct points, find the range of values of k.

49. If the graph of the equation does not intersect the x-axis, find the range of values of k.

50. Find the greatest integral value of k if the graph of the equation does not intersect the x-axis.

51. If the graph of the equation y (m – 1)x2 + mx 2 touches the x-axis, find the values of m.

52. The figure shows the graph of the equation from to Find the

range of values of x of the given graph such that(a) y 1.(b) y < 3.6.(Correct your answers to 1 decimal place if necessary.)

53. The figure shows the graph of the equation from to Find the range

of values of x of the given graph such that(a) y 2.(b) y > 1.(Correct your answers to 1 decimal place if necessary.)

54. The figure shows the graph of the equation y x2 3x –1 from x 0.5 to x 3.5.(a) Solve the equation

graphically.(b) When find the corresponding

values of x.(c) Find the range of values of x of the given

graph such that y 0.5.(Correct your answers to 1 decimal place.)

55. The figure shows the graph of y x3 – x2 – 2x 2 from x 2 to x 2.(a) Find the x-intercepts of the graph.

O 3

1

1

2

21 1x

y

y ax b

y

xO

1

2

3

4

5

6

3

2

11 2 2 1

y x 3 x

2 2x 2

y

5

4

3

2

1

xO 2 1 3 5 7 9 4 6 8

y ax b

O 3

1

2

1

2

21x

y

y x 2 3x 1

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(b) Find the range of values of x of the given graph such that(i) y 0.(ii) y < 2.

(Correct your answers to 1 decimal place if necessary.)

56. The figure shows the graph of the equation . A(4, 6) is a point on the graph

and –5 is one of the x-intercepts of the graph.(a) Find the values of a and b.(b) Find the other x-intercept.

57. The figure shows the graph of the equation . The x-intercepts are 2 and r. The y-

intercept is –12.(a) Find the values of a and c.(b) When x 1, find the corresponding value of y.(c) Find the value of r.(d) Hence find the range of values of x such that

y > 0.

58. The figure shows the graph of the equation y ax2 – 60x c. The graph touches the x-axis at (b, 0) and the y-intercept is 450.(a) Find the value of c.(b) Find the value of a.(c) Hence find the value of b.

Exercise 3B[ In this exercise, correct your answers to 1 decimal place if necessary. ]

Elementary Set

xO 2 r

12

y

y ax 2 8x c

xO b

450

y

y ax 2 60x c

xO 5

y

A ( 4, 6)

y ax 2 bx 10


Level 1

Solve the following simultaneous equations algebraically. (1 – 12)1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. The figure shows the graph of from to . Solve the following

simultaneous equations graphically.[ A copy of the figure is provided in the Appendix. ]

(a)

(b)

(c)

14. The figure shows the graph of

from to . Solve the following simultaneous equations graphically.[ A copy of the figure is provided in the Appendix. ]

(a)

(b)

(c)

y 2x 2 x

O 1 1

2

3

4

5

6

1

1

2

2 2x

y

O 1 1

3

1

2

2x

2

1

y

y 1 5x 3x 2

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Level 2

Solve the following simultaneous equations algebraically. (15 – 18)15. 16.

17. 18.

Intermediate Set

Level 1


22. 23. 24.

25. 26. 27.

28. The figure shows the graph of

from to . Solve the following simultaneous equations graphically.[ A copy of the figure is provided in the Appendix. ]

(a)

(b)

(c)

29. The figure shows the graph of from to .

Solve the following simultaneous equations graphically.[ A copy of the figure is provided in the Appendix. ]

(a)(b)(c)

Level 23y x

2 5x

O 1

2

1

1

2

2 3 4 5x

y

y x 2x 2

O 1 1

1

2

3

4

5

6

2x

y

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33. 34. 35.

36. 37.

38. It is given that the perimeter of rectangle ABCD is 20 cm.(a) If , express the length of AD in terms of x.(b) If the area of rectangle ABCD is S cm2, express S in

terms of x.(c) The figure shows the graph of . If the area

of the rectangle is 16 cm2, find the values of x from the given graph.

Advanced Set

Level 1


42. 43. 44.

45. The figure shows the graph of y 3x2 + x 3 from x 1.5 to x 1. Solve the following simultaneous equations graphically.[ A copy of the figure is provided in the Appendix. ]

(a)

(b)

(c)

D C

x cmA B

O 5 10x

25

20

15

10

5

y

S 10x x 2

y 3x 2 x 3

O 1 1

2

1

1

2

3

x

yE

x.3B A

dvanced Set

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46. The figure shows the graph of from to . Solve the following simultaneous equations graphically.[ A copy of the figure is provided in the Appendix. ]

x 2 7x 3y 6

O 1

2

1

1

2

2 3 4 5 6x

y

7

(a) (b) (c)

Level 2


50. 51. 52.

53. 54. 55.

56. The figure shows the graphs of

and A is one of the points of intersection of the two graphs.(a) Referring to the figure, find a solution

of .(b) Find the values of p and q.(c) Based on the answer to (b), find another

solution of .

57. Rectangle ABCD is 30 cm long and 50 cm wide. The length of rectangle PQRS is x cm longer than the length of rectangle ABCD and the width of rectangle PQRS is x cm shorter than the width of rectangle ABCD.(a) Express the area of rectangle PQRS in

terms of x.(b) The figure shows the graph of

O 20

500

40x

500

1 000

1 500

y

y x 2 20x 1 500

3010 10 20 30 50

O 2

2

4x

2

1

y

y x 2 6x q

y 4x p

A

1

3

31 5

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If the area of rectangle PQRS is , find the length and the width of rectangle PQRS.

58. A rectangular cardboard is 18 cm long and 15 cm wide. A square with sides x cm is cut away from each corner of the rectangular cardboard. Then the cardboard is folded up to form an open box.

x cm

18 cm

x cm

x cm

15 cm

(a) Express the base area of the open box in terms of x.(b) The figure shows the graph of y 2x2 – 33x + 135. If the base area of the open box is

40 cm2, find the capacity of the open box.

y 2x 2 33x 135

O 5

80

60

10

20

30

40

50

70

10 15x

y

CHAPTER TEST (Time allowed: 1 hour)

Section A1. The figure shows the graph of y = 4x2 – 12x + 5 from

x = 0 to x = 3. Find the range of values of x of the given graph such that(a) y 3. (1 mark)(b) y < 4. (1 mark)

y 4x 2 12x 5

O 1

4

5

3

1

2

2

1

3

4

2 3x

y


(c) y > 1. (2 marks)(Correct your answers to 1 decimal place if necessary.)

2. The figure shows the graph of y = 2x2 – 5x from x = 0 to x = 2.5.(a) Draw the graph of y 2x – 6 on the figure. (2 marks)

[ A copy of the figure is provided in the Appendix. ]

(b) Solve the simultaneous equations

graphically. (Correct your answers to 1 decimal place if necessary.) (2 marks)

3. Express the simultaneous equations into the form of

Hence solve the simultaneous equations. (5 marks)

4. In the figure, the x-intercept and the y-intercept of the graph of y ax b are –4 and 3 respectively.(a) Find the values of a and b. (3 marks)(b) Find the value of x when y = 10. (2 marks)

5. The y-intercept of the graph of the quadratic equation in two unknowns is

12. One of the x-intercepts of the graph of the equation is 6.(a) Find the values of a and h. (5 marks)(b) Find the other x-intercept of the graph. (2 marks)

Section B6. The figure shows the graph of a quadratic equation in two

unknowns y a(x + 4)(x – 6). It is known that the highest point of the graph is (k, 5) and the x-intercepts of the graph are –4 and 6.(a) By considering the symmetric property of parabola,

y 2x 2 5x

O 1

2

1

3

2 3x

y

y ax b

O 4

3

x

y

y a(x 4)(x 6)

O 4 6k

5

x

y


find the value of k. Hence find the value of a.(3 marks)

(b) Find the y-intercept of the graph. (2 marks)(c) If the graph of y = px q intersects the parabola at

P(1, y1) and Q(7, y2),(i) find the values of y1 and y2.(ii) find the values of p and q. (5 marks)

7. In the figure, the two graphs of the equations y = 4x + 20 and y = a(x – 1)(x – 5) intersect at A(0, 20) and B. C(3, k) is a point on the graph of y = a(x – 1)(x – 5).(a) Find the value of a. (2 marks)(b) Find the coordinates of B. (2 marks)(c) Find the value of k. (2 marks)(d) Find the area of ABC. (4 marks)

Multiple Choice Questions (3 marks each)8. The figure shows the graph of the

equation . P(1, 8) is a point on the graph. Find the x-intercept of the graph.

y ax 5

Ox

y

P (1, 8)

A. 5

B. 3

C.

D.

9. The figure shows the graphs of

and The roots of the quadratic equation is/are

O 2 4x

4

3

2

1

y

y x 2 px

y 3

1 3

A. 0 and 4.B. 1 and 3.C. 1 and 4.D. 2 only.

10. The figure shows the graph of the equation Find the range of values of x such that y 3.

y a(x 1)(x 5)

y 4x 20

O 1

A (0, 20)

B

C (3, k)

5x

y


y ax 2 bx c

O 1

1

2

2

4

1

3

42 3x

y

A. 0 x 4B. 1 x 3C. x 1 or x 3D. x 0 or x 4

11. Which of the following pairs of simultaneous equations has no real solutions?A.

B.

C.

D.

12. If , then x A. 1 or 15.B. 3 or 5.C. 3 or 5.D. 1 or 15.

13. In the figure, the graphs of the equations and intersect at O and P.

Find the coordinates of P.

y x

O

P

x

y

y x 2 4x

A. (3, 3)B. (4, 4)C. (5, 5)D. (6, 6)

14. In the figure, the area of POQ

y 6x

Ox

y

P

Q

y 2x 2 3x 5

A. 6.25.

B. 12.5.

C. 37.5.

D. 75.

15. The figure shows the graph of . The y-intercept of the

graph is 8 and the graph passes through P(6, 14). Find the values of a and b.

P (6, 14)

8

x

y

y x 2 ax b

O


A. a 8, b 8B. a 5, b 8C. a 5, b 8D. a 5, b 8

16. If the graph of y ax2 – 4x + 3 cuts the x-

axis at two distinct points, then the range of values of a is

A. .

B. .

C. .

D. .

17. In the figure, the parabola and the straight line intersect at two points A(1, 1) and B(3, 1). The equation of the parabola may be

A ( 1, 1) B (3, 1)

Ox

y

A. y 2x2 – 4x – 5.

B. y 2x2 + 4x + 3.

C. y 2x2 – 12x + 19.

D. y 2x2 + 12x – 53.

H INTS (for questions with in the textbook)

Revision Exercise 322. (a) key Information

AB AC AD 20 cm BC 20 cm PS x cm ED y cm All vertices of rectangle PQRS lie on the sides of ABC.AnalysisTo show that x y 20, we need to find the relation between x and y. Since we can find some similar triangles and the lengths of sides of them are found, the relation between x and y can be found by applying the properties of similar triangles.MethodAccording to the information given, we can find many pairs of similar triangles such as ABD ~ PBQ, ACD ~ SCR, ABC ~ APS. Choose an appropriate pair of similar triangles and use ‘corr. sides, ~ s’ to find the relation between x and y and then show that x y 20.

25. (d) key Information The cannonball will explode when its height above the ground is 14 m or below. The graph of the equation y 15 20x 5x2 obtained in (a).Analysis


Since the graph obtained in (a) has shown the change of the height of the cannonball above the ground after firing, the answer can be found from the graph.MethodFrom the graph, find the corresponding range of values of x when the height of the cannonball is 14 m or below.

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