22[a math cd]

1 Penerbitan Pelangi Sdn. Bhd.

Paper 1

1. y = x + 1 is a linear function with y-intercept = 1 and gradient 0.Answer: A

2. y = x2 + 2 is a quadratic function with y-intercept = 2. Its shape is .

Answer: C

3. y = x2 + 1 is a quadratic function with y-intercept = 1. Its shape is .Answer: A

4. y = x3 is a cubic function with y-intercept = 0.Answer: A

5. y = 1x is a reciprocal function.Answer: D

6. The shaded region is above the line y = x. Hence, the shaded region is represented by y x.Answer: C

7. The shaded region is below the line x + y = 2. Hence, the shaded region is represented by x + y 2.Answer: C

8. x 2 means that the shaded region is to the right of the line x = 2.Answer: B

9. y 1 means that the shaded region is below the dashed line y = 1.Answer: A

Paper 2

1. (a) y = 3x 2 When x = 3, y = 3(3) 2 = 7

x 3y 7

(b)

012

4

6

8

1 2

2

4

6

8

y = 3x 2

x

y

32

2. (a) y = 2x2 x 8 When x = 3, y = 2(3)2 (3) 8 = 13 When x = 1, y = 2(1) (1) 8 = 7

x 3 1y 13 7

CHAPTER

22 Graphs of Functions IICHAPTER

2 Mathematics SPM Chapter 22

Penerbitan Pelangi Sdn. Bhd.

(b)

0 15

5.5

3.65 2 3 44 3 2 1

5

10

15

20

25

30

10

x

y = 2x2 x 8

y

(c) (i) y = 5.5 (ii) x = 3.65

3. (a) y = x3 3x + 6 When x = 2, y = (2)3 3(2) + 6 = 4 When x = 1.5, y = (1.5)3 3(1.5) + 6 = 4.875

x 2 1.5y 4 4.9

(b)

0123 321

5

10

1517.5

2.1

15

10

5

20

25

x

y

y = x3 3x + 6

(c) (i) y = 17.5 (ii) x = 2.1

4. (a) y = 2x 3 When x = 0, y = 2(0) 3 = 3 When y = 0, 0 = 2x 3 x = 32

y = 2x 3y

x0

3

32

(b) y = 4x + 1 When x = 0, y = 4(0) + 1 = 1 When y = 0, 0 = 4x + 1 x = 14

y = 4x + 1

y

x0

1

14

(c) y = x2 3 When x = 0, y = 0 3 = 3 When y = 0, 0 = x2 3 x2 = 3 x = 3 or 3

y = x 2 3

y

x0 333

(d) y = 4 x2 When x = 0, y = 4 0 = 4 When y = 0, 0 = 4 x2 x2 = 4 x = 2 or 2



y = 4 x2

y

x02 2

4

5. (a) y = 2x3 1 When x = 0, y = 2(0) 1 = 1

y = 2x3 1y

x01

(b) y = x3 + 2 When x = 0, y = 0 + 2 = 2

y = x3 + 2

y

x0

2

(c)

y =

y

x

x

0

5

(d)

y =

y

x

x

0

8

6. y = 2x2 x 8 ...................1 0 = 2x2 x 22 ..................21 2: y = 8 (22) y = 14The suitable straight line is y = 14.

7. The suitable straight line is y = 3x + 2.

8. y = x3 3x + 6 ...................1 0 = x3 6x + 3 ...................21 2: y = 3x + 3The suitable straight line is y = 3x + 3.

9. y x 6 Region above the line y = x 6x 6 Region to the left of the dashed line x = 6

y

xy = x 6

x = 6

0

6

6

10. y x Region above the line y = xx + y 4 Region above the line x + y = 4

y

x0

x + y = 4

y = x

11. y x + 3 Region below the line y = x + 3y 3 Region above the dashed line y = 3x 5 Region to the left of the line x = 5

1

0 1 2 3 4 5 6 7

23456789

10

y

x

y = x +

3

x = 5

y = 3



Paper 1

1. y = x2 + x 6 is a quadratic function. Its shape is .When y = 0, x2 + x 6 = 0 (x + 3)(x 2) = 0 x = 3, x = 2The x-intercepts are 3 and 2.Answer: B

2. y = 3 x2 = x2 + 3y = 3 x2 is a quadratic function with y-intercept = 3. Its shape is .Answer: C

3. The graph is a reciprocal function with an equation y = 2x = 2x 1.Hence, m = 2 and n = 1Answer: B

4. The graph is a cubic function, y = ax3 + 16, with a , 0.Substitute x = 2 into y = 2x3 + 16. 0 = 2(2)3 + 16Hence, x = 2 satisfies the equation y = 2x3 + 16.Therefore, the graph is y = 2x3 + 16.Answer: C

5. y = x2 9 is a quadratic function with y-intercept = 9. Its shape is .Answer: C

6. y = 4 + x3 is a cubic function with y-intercept = 4.a . 0 (a = 1), its shape is .Answer: D

7. y = 5 2x3 is a cubic function with y-intercept = 5. Answer: A

Paper 2

1. y 2x + 6 Region below the line y = 2x + 6y 12 x Region above the line y =

12 xy 6 Region below the dashed line y = 6

y = 2x + 6

y = 6

y

x0

1y = x 2

2. (a) y = 8x

When x = 1, y = 8(1) = 8 When x = 2, y = 82 = 4

x 1 2y 8 4

(b)

x

y = 4x 5

y

01 2 3 44 3 2

2.10.6

0.9

1

5

10

15

20

20

15

10

5 8y = x



(c) (i) y = 10 (ii) x = 0.6

(d) 8x = 4x + 5 8x = 4x 5 The suitable straight line is y = 4x 5. The values of x which satisfy 8x = 4x + 5 are

x = 2.1 and x = 0.9.

3. (a) y = x3 4x + 8 When x = 3, m = (3)3 4(3) + 8 = 7 When x = 2, n = (2)3 4(2) + 8 = 8(b)

x

y = 9x + 18

y = x3 4x + 8

y

0 1 2 3 44

3.15 0.8

19

3 2 1

20

10

30

40

50

30

20

10

(c) (i) y = 19 (ii) x = 2.65

(d) y = x3 4x + 8 ............1 0 = x3 13x 10 .........2 1 2: y = 9x + 18 The suitable straight line is y = 9x + 18. The values of x which satisfy x3 13x 10 = 0

are x = 3.15 and x = 0.8.

4. (a) y = 3x2 + 2x 10 When x = 3, y = 3(3)2 +2(3) 10 = 11 When x = 2, y = 3(2)2 + 2(2) 10 = 6

x 3 2y 11 6

(b) y

x0 1 2 312345

10

20

30

40

50

60

3.8

12

2.5

y = 9x + 26y = 3x2 + 2x 10

10

(c) (i) y = 12 (ii) x = 3.8(d) y = 3x2 + 2x 10 ...............1 3x2 7x = 36 0 = 3x2 7x 36 ...............2 1 2: y = 9x + 26 The suitable straight line is y = 9x + 26. The value of x which satisfies 3x2 7x = 36 is

x = 2.5.



5. (a) y = 5 x 2x2 When x = 2, y = 5 (2) 2(2)2 = 1 When x = 3, y = 5 (3) 2(3)2 = 16

x 2 3y 1 16

(b)

x

y = 5 x 2x 2

y

0123 21 43 5

57

2.45 2.7 3.6

35

20

25

10

15

5

10

30

3y = x 8 2

(c) (i) y = 7 (ii) x = 3.6(d) y = 5 x 2x2 ...................1 0 = 13 + 12 x 2x

2 ............2

1 2: y = 8 32 x

The suitable straight line is y = 8 32 x.

The values of x which satisfy 13 + 12 x 2x2

= 0 are x = 2.45 and x = 2.7.

6. (a) y = x3 6x + 7 When x = 3, y = (3)3 6(3) + 7 = 2 When x = 2, y = (2)3 6(2) + 7 = 3

x 3 2y 2 3

(b)

x

y

y = x3 6x + 7y = 2x + 11

4

30

25

20

15

10

5

5

0

10

15

33.05

12

0.552.553.2

211234

(c) (i) y = 12 (ii) x = 3.2(d) y = x3 6x + 7 ..........1 x3 = 8x + 4 ..................2 1 + 2: y + x3 = x3 + 2x + 11 y = 2x + 11 The suitable straight line is y = 2x + 11. The values of x which satisfy x3 = 8x + 4 are

x = 2.55, 0.55, 3.05.

7. y 8 2x Region above the line y = 8 2xy 8 Region below the line y = 8x 4 Region to the left of the dashed line x = 4

y = 8y

x0 4 y = 8 2x

x = 4



8. (a) y = 2x2 2x 5 When x = 1, y = 2(1)2 2(1) 5 = 1 When x = 3, y = 2(3)2 2(3) 5 = 7

x 1 3y 1 7

(b)

x

y

0 1 2 3 4

4.2 4.35

5

5

12

5

10

15

20

25

30

35

10

y = 2x2 2x 5

y = 2x + 13

(c) (i) y = 10 (ii) x = 4.35(d) y = 2x2 2x 5 ...............1 18 = 2x2 4x ......................2 1 2: y 18 = 2x 5 y = 2x + 13 The suitable straight line is y = 2x + 13. The value of x which satisfies 2x2 4x = 18 is

x = 4.2.

9. (a) y = 18x When x = 3, y = 183 = 6

When x = 0.5, y = 180.5 = 36

x 3 0.5y 6 36

(b)

x

y

1 1 2 3 45

2.55 1.65 1.8

7

5

10

15

20

25

30

35

40

10

15

20

234

y = 4x + 3

18xy =

0

(c) (i) y = 7 (ii) x = 1.65(d) 4x2 + 3x = 18 4x + 3 = 18x The suitable straight line is y = 4x + 3. The values of x which satisfy 4x2 + 3x = 18 are

x = 2.55 and x = 1.8.



Paper 1

1. y = 2x 3 is a linear function with y-intercept = 3 and gradient 0. Answer: B

2. y = x2 2 is a quadratic function with y-intercept = 2. Its shape is .Answer: A


4. y = 2x3 is a cubic function with y-intercept = 0. Its shape is .

Answer: C

5. y = x3 4 is a cubic function with y-intercept = 4.Its shape is .Answer: A

6. y = x(3 x) = 3x x2 It is a quadratic function with x-intercepts = 0 and 3. Its shape is .Answer: B


8. y = 2x2 + 1 is a quadratic function with y-intercept = 1. Its shape is .Answer: C

9. y = 4 xn is a cubic function. Hence, n = 3.Answer: C

10. y = 9 xn is a quadratic function. Hence, n = 2.Answer: A

11. y = 2xn is a reciprocal function. Hence, n = 1.Answer: D

12. y = xm + p is a quadratic function with y-intercept = 3. Hence, m = 2 and p = 3.Answer: D

13. y = 2xn + k is a cubic function with y-intercept = 4. Hence, n = 3 and k = 4.Answer: D

14. The graph is a quadratic function with y-intercept = 3.Answer: A

15. The graph is a cubic function with y-intercept = 2.Answer: C

16. The graph is a quadratic function with y-intercept = 5.Answer: B

17. x 0 Region to the right of the y-axis 2y x Region above the dashed line 2y = xy 2 x Region below the line y = 2 x

y

x0

2y = x

y = 2 x

Answer: D

18. y 8 2x Region above the line y = 8 2xy 2x Region above the line y = 2xy 8 Region below the dashed line y = 8

y = 2xy

x0

8

y = 8 2x

Answer: C



19. Region to the right of the y-axis x 0 Region to the left of the line x = 6 x 6Region above the x-axis y 0Region below the dashed line 2y x = 1 2y x 1Answer: B

20. Region below the line x + y = 6 x + y 6Region above the dashed line 2x + y = 6 2x + y 6Region above the line y = 3 y 3Answer: B

Paper 2

1. (a) y = 24 + 3x 2x2 When x = 1, y = 24 + 3(1) 2(1)2 = 19 When x = 4, y = 24 + 3(4) 2(4)2 = 4

x 1 4y 19 4

(b)

y

y = 10

x

5

0 1 2 33 2 1 4

5

10

15

20

25

3.50.7

19

2

y = 24 + 3x 2x 2

(c) (i) x = 0.7 (ii) y = 19(d) y = 24 + 3x 2x2 .....................1 0 = 2x2 3x 14 .....................2 1 + 2: y = 10 The suitable straight line is y = 10. The values of x which satisfy 2x2 3x 14 = 0

are x = 2 and x = 3.5.

2. (a) y = 4 x x2 When x = 4, y = 4 ( 4) ( 4)2 = 8 When x = 2, y = 4 (2) (2)2 = 2

x 4 2y 8 2

(b)

0 1

10

2 3 44 3 2 1

8

6

4

2

22.8

3.32.73.7

4

6

12

x

y = 6

y = 4 x x2

y

(c) (i) y = 2.8 (ii) x = 3.3(d) 4 x x2 = 6 The suitable straight line is y = 6. The values of x which satisfy 4 x x2 = 6 are

x = 3.7 and x = 2.7.

10

Mathematics SPM Chapter 22


3. (a) y = x2

2 2x

When x = 1, y = (1)2

2 2(1) = 2.5 When x = 3, y = (3)

22 2(3)

= 1.5x 1 3y 2.5 1.5

(b)

y

x2y = 2x 2

x

1

0 1 2 31 4 5 6

1

2

2

3

xy = 2 2

4

5

6

0.6

1.65

3.4

y = 1

(c) (i) y = 1.65 (ii) x = 0.6 or x = 3.4

(d) x2

2 2x = 2 x2

The suitable straight line is y = 2 x2 .

The values of x which satisfy x2

2 2x = 2 x2

are x = 1 and x = 4.

4. (a) y = 12 x2(3 x)

When x = 1, y = 12 (1)2[3 (1)]

= 2 When x = 1.5, y = 12 (1.5)

2[3 (1.5)] = 1.7

x 1 1.5y 2 1.7

(b) y

x

1

0

1y = x 2(3 x) 2

1 2 312 4

1

2

3

2

3

4

5

6

2.71

.2

0.7

5

y = 1

(c) x = 1.2(d) x2(3 x) = 2 12 x

2(3 x) = 1 The suitable straight line is y = 1. The values of x which satisfy x2(3 x) = 2 are

x = 0.75, x = 1 and x = 2.7.

11



5. (a) y = x(x2 4) When x = 1.5, y = 1.5[(1.5)2 4] = 2.6 When x = 0.5, y = 0.5[(0.5)2 4] = 1.9

x 1.5 0.5y 2.6 1.9

(b)

y

x

2

0 1 2 3123

2

4

4

6

8

10

12

14

16

y = x 2

2.4

2.35 0.45

y = x (x 2 4)

(c) x = 2.35(d) x(x2 4) = x 2 The suitable straight line is y = x 2. The values of x which satisfy x(x2 4) = x 2

are x = 2.4, x = 0.45 and x = 2.

6. (a) y = x2 3x + 10 When x = 4, y = ( 4)2 3( 4) + 10 = 38 When x = 2, y = (2)2 3(2) + 10 = 8

x 4 2y 38 8

(b)

0 1 2 3 4 54 3 23.3 0.7 3.7 4.3

26

1

15

20

5

10

25

30

35

40

x

y = x 2 3x + 10

y = 2x + 24

y

(c) (i) y = 26 (ii) x = 0.7 or x = 3.7(d) y = x2 3x + 10 ........................1 0 = x2 x 14 ..........................2 1 2: y = 2x + 24 The suitable straight line is y = 2x + 24. The values of x which satisfy x2 x 14 = 0 are

x = 3.3 and x = 4.3.

12



7. (a) y = x3 5x 8 When x = 2, y = (2)3 5(2) 8 = 6 When x = 1, y = (1)3 5(1) 8 = 12

x 2 1y 6 12

(b)

01

5

5

10

10

15

20

15

20

25

30

35

2 33 2 1 4x

y = 9

y = x3 5x 8

y = 15 5x

y

2.35 0.2 2.1x = 2.85

(c) x = 2.1, x = 0.2 or x = 2.35(d) y = x3 5x 8 .........................1 0 = x3 23 ...............................2 1 2: y = 5x + 15 The suitable straight line is y = 5x + 15. The value of x which satisfies x3 23 = 0 is

x = 2.85.

8. (a) y = x3 13x + 18 When x = 3, y = (3)3 13(3) + 18 = 30 When x = 2, y = (2)3 13(2) + 18 = 0

x 3 2y 30 0

(b)

x

y = 2x + 20

y = x 3 13x + 18y

0 1

5

10

15

20

25

30

35

2 3

3.44

0.2

0.9

3.0

5

3.2

44 3 2 1

y = 29

(c) (i) y = 4 (ii) x = 3.05 or x = 0.9(d) y = x3 13x + 18 .....................1 0 = x3 11x 2 ........................2 1 2: y = 2x + 20 The suitable straight line is y = 2x + 20. The values of x which satisfy x3 11x 2 = 0 are

x = 3.2, x = 0.2 and x = 3.4.

13



9. (a) y = 20x 2x2 When x = 2, y = 20(2) 2(2)2 = 32 When x = 5, y = 20(5) 2(5)2 = 50

x 2 5y 32 50

(b)

5

10

15

20

25

30

35

40

45

50

y

x0 1 2 3 4 5 6 7 8 9

y = 28

y = 4x + 25y = 20x 2x2

1.7

2.15

5.9

8.3

(c) (i) y = 50 (ii) x = 1.7 or x = 8.3(d) y = 20x 2x2 ...........................1 0 = 16x 2x2 25 ....................2 1 2: y = 4x + 25 The suitable straight line is y = 4x + 25. The values of x which satisfy 16x 2x2 25 = 0

are x = 2.15 and x = 5.9.


x 1 5y 0 6

(b)

5

10

15

20

25

25

20

15

10

50 11 2 3 4 5 6 7

x

y

y = 5

y = 2x + 10

y = 14 + 11x 3x 2

0.7

0.4 3.

4

4.35

(c) x = 0.7 or x = 4.35(d) y = 14 + 11x 3x2 ....................1 0 = 4 + 9x 3x2 ........................2 1 2: y = 10 + 2x The suitable straight line is y = 10 + 2x. The values of x which satisfy 4 + 9x 3x2 = 0

are x = 0.4 and x = 3.4.

14



11. (a) y = 2x2 + x 5 When x = 3, y = 2(3)2 + (3) 5 = 10 When x = 2, y = 2(2)2 + (2) 5 = 5

x 3 2y 10 5

(b)

0 1 2 3 44 3 2

2.4 1.92.51

5

5

10

15

20

25

30

35

y

y = 4

y = 2x 2 + x 5

x

(c) y = 10(d) y = 2x2 + x 5 ....................1 9 = 2x2 + x ..........................2 1 2: y 9 = 5 y = 4 The suitable straight line is y = 4. The values of x which satisfy 2x2 + x = 9 are

x = 2.4 and x = 1.9.

12. (a) y = 4x When x = 4, y = 4( 4) = 1 When x = 0.8, y = 40.8 = 5

x 4 0.8y 1 5

(b)

2

2

4

6

8

10

4

0.85

2.9

6

8

10

0 1 2 3 445 3 2 1 5

y

x

y = x

4y = x

4y = x

(c) (i) y = 2.9 (ii) x = 0.85(d) The equation of the straight line is y = x. The values of x are x = 2 and x = 2.

15



13. (a) y = 6x When x = 2, y = 6(2) = 3

When x = 2.5, y = 62.5 = 2.4

x 2 2.5y 3 2.4

(b)

x

y

y = 2x

0 1

2

45

0.85 1.751.75

6

8

10

12

12

10

8

6

4

2 3 4 54 3 2 12

6y = x

(c) (i) y = 5 (ii) x = 0.85(d) The equation of the straight line is y = 2x. The values of x are x = 1.75 and x = 1.75.

14. (a) y = x3 4x 9 When x = 2, y = (2)3 4(2) 9 = 9 When x = 3, y = (3)3 4(3) 9 = 6

x 2 3y 9 6

(b)

x

30

40

50

60

20

10

10

20

30

40

1 1 2 3 4234 0

y

y = 10

y = x 3 4x 9

0.2 1.9 3.32.1

(c) (i) y = 20 (ii) x = 3.3(d) y = x3 4x 9 .........................1 0 = x3 4x + 1.........................2 1 2: y = 10 The suitable straight line is y = 10. The values of x which satisfy x3 4x + 1 = 0 are

x = 2.1, x = 0.2 and x = 1.9.

16



15. (a) y = 2x2 + 3x + 7 When x = 1.5, y = 2(1.5)2 + 3(1.5) + 7 = 2 When x = 1, y = 2(1)2 + 3(1) + 7 = 8 When x = 4, y = 2(4)2 + 3(4) + 7 = 13

x 1.5 1 4y 2 8 13

(b)

5

5

10

10

15

20

25

30

0 1 2 33 2 1 4 5

3.35

y

x

y = 6x 25

y = 2x2+ 3x + 7

(c) (i) y = 5 (ii) 2x2 3x = 7 2x2 + 3x + 7 = 0 When y = 0, x = 1.25 or x = 2.75. The

values of x which satisfy 2x2 3x = 7 are x = 1.25 or x = 2.75.

(d) y = 2x2 + 3x + 7 .....................1 0 = 2x2 3x + 32 ...................2 1 2: y = 6x 25 The suitable straight line is y = 6x 25. The value of x which satisfies 2x2 3x + 32 = 0

is x = 3.35.


x 2 4y 24 6

(b)

01

5

10

15

15

10

25

20

2 21 43 655

x

y = 4x 5

y = 2 + 9x 2x 2

y

3.51.6

(c) (i) x = 1.6 (ii) y = 10(d) y = 2 + 9x 2x2.......................1 0 = 2x2 5x 7 .......................2 1 + 2: y = 4x 5 The suitable straight line is y = 4x 5. The values of x which satisfy 2x2 5x 7 = 0 are

x = 1 and x = 3.5.

17. y 2x Region below the line y = 2xy 8 2x Region above the line y = 8 2xx 4 Region to the left of the dashed line x = 4

y

x0

y = 8 2x

y = 2xx = 4

8

4

17



18. y 4x Region below the line y = 4x3y x Region above the line 3y = xx 2 Region to the left of the dashed line x = 2

y

x03y = x

y = 4xx = 2

2

19. x + y 8 Region above the line x + y = 83y 2x Region above the line 3y = 2xy 8 Region below the dashed line y = 8

y

x0

3y = 2x

y = 8

x + y = 8

20. y x 4 Region above the line y = x 42y 12 3x Region below the line 2y = 12 3xx 2 Region to the right of the dashed line x = 2

y

x0

6

4

4

2y = 12 3x

y = x 4x = 2

21. y 3x + 9 Region above the line y = 3x + 9x 3 Region to the left of the dashed line x = 3y 9 Region below the line y = 9

y

x0 3

y = 3x + 9

x = 39 y = 9

22. y 2x 5 Region below the line y = 2x 5y x 5 Region above the line y = x 5y 4 Region below the dashed line y = 4

y

x0

4

5

y = 2x 5

y = x 5

y = 4

23. x 0 Region to the right of the y-axis2y x Region above the line 2y = x2x + 3y 6 Region above the line 2x + 3y = 6x + y 6 Region below the line x + y = 6

1

0 1 2 3 4 5 6 7

234567

y

x2x + 3y = 6

x + y = 62y = x

22[a math cd]

Documents