2 Functions and Graphs (Q)

8/8/2019 2 Functions and Graphs (Q)

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2 Functions and Graphs

Extra Examples

(P01C02L01Q001)

Example 2.1R

Iff(x) = 4x2 + 3x, find the values of the function when

(a) x = 1,

(b) x =2

1.

(P01C02L01Q002)

Example 2.2R

Ifh(x) =1

12

+x

x , find the values of

(a) h(2),

(b) h(0),

(c) h(2).

(P01C02L01Q003)

Example 2.3R

Iff(x) = kx x2 andf(5) = 5, find the values of

(a) k,

(b) f(5).

(P01C02L01Q004)

Example 2.4R

Iff(x) = 2x2 +x, find the values of

(a) f

3

a,

(b) f(b 3).

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Question Bank

(P01C02L01Q005)

Example 2.5R

(a) Plot the graph of the function 12= xy .

(b) Hence, find thex-intercept and they-intercept of 12= xy .

(P01C02L01Q006)

Example 2.6R

The figure shows the graph ofy = ax + b.

(a) Find thex-intercept and they-intercept of the graph.

(b) Find the values ofa and b.

(P01C02L01Q007)

Example 2.7R

Determine the directions of opening and find they-intercepts of the graphs of the following functions.

(a) y =x2 3x + 1

(b) y =x(4 x) 2

(P01C02L01Q008)

Example 2.8R

(a) Plot the graph ofy = x2 + 4x 3 fromx = 0 tox = 4.

(b) State the following features of the graph:(i) Axis of symmetry

(ii) Coordinates of the vertex

(iii) y-intercept

(iv) Direction of opening

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(P01C02L01Q009)

Example 2.9R

Find (a) the direction of opening,

(b) the vertex,

(c) the axis of symmetry

of the graph ofy = 3(x + 1)2 + 2.

(P01C02L01Q010)

Example 2.10R

Giveny = 2(x 3)2 4, find

(a) its optimum value,

(b) the direction of opening and the axis of symmetry of its graph.

(P01C02L01Q011)

Example 2.11R

Find the optimum value for each of the following quadratic functions and the axis of symmetry of their

graphs.

(a) y =x2 + 6x 7

(b) y = 2x2 8x + 3

(P01C02L01Q012)

Example 2.12R

Given that the minimum value of the functiony =x2 6x + kis 17, find the value ofk.

(P01C02L01Q013)

Example 2.13R

A marble is projected vertically upwards to the ceiling of a house from the floor. After tseconds, its height

(h m) above the ground is given by:

h = 5t2

+ 5t+ 1(a) When will the marble attain its maximum height?

(b) If the ceiling is 3 m above the ground, will the marble hit the ceiling?

28

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NF

NF

NF

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Question Bank

(P01C02L01Q014)

Example 2.14R

The figure shows a garden in the shape of two squares alongside each other. IfMNis 12 m long, find the

minimum area of the garden.

(P01C02L01Q015)

Example 2.15R

Solve the following inequalities graphically.

(a) x2 4x + 1 > 2

(b) x2 4x + 1 < 2

(P01C02L01Q016)

Example 2.16R

Solve4

14 x graphically.

(P01C02L01Q017)

Example 2.17R

Solve 2x3 5x2 + 3x 2 graphically.

(P01C02L01Q018)

Example 2.18R

Given the graph ofy =x2 3x 1, solve the inequalityx2 3x + 2 > 0 by adding a suitable straight line on

the graph.

29

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(P01C02L01Q019)

Example 2.19R

It is given that f(x) = 2x2 + 3 and g(x) = 2x2 1. Iff(x) is transformed to g(x), describe the effect of the

transformation on the graph ofy =f(x).

(P01C02L01Q020)

Example 2.20R

In the figure, the graph ofy = g(x) is obtained by translating the graph ofy = x2 2x 3 in the positive

direction of they-axis. Find the symbolic representation ofg(x).

(P01C02L01Q021)

Example 2.21R

The following table shows the tabular representation of a functionf(x):

x 3 2 1 0 1 2 3 4f(x) 7 2 1 2 1 2 7 14

If the graph ofy =g(x) is obtained by translating the graph ofy =f(x) in the negative direction of the y-axis

by 3 units, complete the following tabular representation ofg(x):

x 3 2 1 0 1 2 3 4g(x)

(P01C02L01Q022)Example 2.22R

Iff(x) =x2 2 is transformed tog(x) =x2 6x + 7, describe the effect of the transformation on the graph of

y =f(x).

30

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NF

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Question Bank

(P01C02L01Q023)

Example 2.23R

(a) It is given thatf(x) = x2 + 2. Plot the graph ofy =f(x) for 4 x 4 and find the coordinates of the

vertex of the graph.

(b) Hence, ifg(x) =f(x 1), find the coordinates of the vertex of the graph ofy =g(x) without plotting the

graph.

(P01C02L01Q024)

Example 2.24R

In the figure, the graph ofy = g(x) is obtained by translating the graph ofy = (x + 3)2 in the negative

direction of thex-axis. Find the symbolic representation ofg(x).

(P01C02L01Q025)

Example 2.2X

If 1)( += xxf , find the values of

(a) f(0) andf(1),

(b)])]]0([[[

times100

fffff .

(P01C02L01Q026)

Example 2.3X

It is given that 2)(2

++= kxxxf .(a) Find )( xf .

(b) If for any real numberx, )()( xfxf = , find the value ofk.

31

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NF


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(P01C02L01Q027)

Example 2.15X

Given the graph of 142 = xxy , solve the inequality 2142 xx by adding a suitable straight

line on the graph.

(P01C02L01Q028)

Example 2.18X

Given the graph ofx

y3= , solve the inequality 2

3>

x.

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Question Bank

Pre-requisite Questions

(P01C02L02Q001)

1. Write down the linear equation 12x 2y + 8 = 0 in the formy = ax + b.

(P01C02L02Q002)

2. Write down the linear equation 4(1 x) + 5y = 24 in the formy = ax + b.

(P01C02L02Q003)

3. (a) Giveny = 2x + 3, complete the following table.

x 2 0 2y

(b) Plot the graph ofy = 2x + 3 fromx = 2 tox = 2.

(P01C02L02Q004)

4. (a) Giveny = 4x 1, complete the following table.

x 1 0 1y

(b) Plot the graph ofy = 4x 1 fromx = 1 tox = 1.

(P01C02L02Q005)

5. IfP(a, 2) and Q(0, b) both lie on the graph ofx y + 4 = 0, find the values ofa and b.

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(P01C02L02Q006)

6. IfP(2, 7) and Q(1, 2) both lie on the graph ofy = ax + b, find the values ofa and b.

(P01C02L02Q007)

7. The figure shows the graph ofy =x2 +x 2.

(a) Using the graph, state the number of real roots ofx2 +x = 2.

(b) Solve the equationx2 +x = 2 graphically.

(P01C02L02Q008)

8. The figure shows the graph ofy =x2 + bx + c which cuts thex-axis atA(6, 0) andB(1, 0). Find the

values ofb and c.

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Question Bank

(P01C02L02Q009)

9. (a) Plot the graph ofy =x2 4x 5 fromx = 2 tox = 6.

(b) Hence, solve the equationx2 4x 5 = 0 graphically.

(P01C02L02Q010)

10. (a) Plot the graph ofy =x2 4x + 4 fromx = 1 tox = 5.

(b) Hence, solve the equationx2 + 2 = 2(2x 1) graphically.

(P01C02L02Q011)

11. Given that the following expressions are perfect squares, find the value ofp. Rewrite the expressions

in the form (x m)2.

(a) x2 + 6x +p (b) x2 10x +p

(P01C02L02Q012)

12. Given that the following expressions are perfect squares, find the value ofp. Rewrite the expressions

in the form (x m)2.

(a) x2 + 3x +p (b) px

x +2

2

35

NF

NF


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Level 1 Questions

(P01C02L03Q001)

1. Iff(x) = 2x + 5, find the values of the function when

(a) x = 0, (b) x = 5, (c) x = 3.

(P01C02L03Q002)

2. Ifg(x) = 1 (x + 1)(x 2), find the values ofg(x) when

(a) x = 2, (b) x = 0, (c) x = 4.

(P01C02L03Q003)

3. Ifh(x) = 5

5

+

x

x

, find the values of

(a) h(0),

(b) h(a + 5),

(c) h

b

1.

(P01C02L03Q004)

4. It is given thatf(x) = 3x2.

(a) Find the values off(1) andf(2).

(b) Does the relationf(1) +f(1) =f(2) hold?

(P01C02L03Q005)

5. Ifh(x) = 221

xax

+ and h(5) = 100, find the value ofa.

(P01C02L03Q006)

6. It is given thatf(x) = ax2

+ 2x + 1 andg(x) =x2

+ 3x.(a) Iff(2) = 1, find the value ofa.

(b) Ifh(x) =g(x 1), find the symbolic representation ofh(x).

(c) Solve the equationf(x) = h(x) + 2.

(P01C02L03Q007)

7. It is given thatf(x) = (x + 3)(kx 3) andg(x) = (x 1)(x + 6).

(a) Iff(3) = 18, find the value ofk.

(b) Ifh(x) =g(2x), write down the symbolic representation ofh(x).

(c) Solve the equation 2f(x) h(x) = 0.

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Question Bank

(P01C02L03Q008)

8. (a) Plot the graph of the functiony = 2x + 5.

(b) Hence, find thex-intercept and they-intercept of the graph.

(P01C02L03Q009)

9. The figure shows the graph ofax +y + 9 = 0.

(a) Find thex-intercept and they-intercept of the graph.

(b) Find the value ofa.

(P01C02L03Q010)

10. Determine the directions of opening and find they-intercepts of the graphs of the following functions.

(a) y =x2 4x + 6 (b) y = (1 2x)2 3

(P01C02L03Q011)

11. Determine the directions of opening and find they-intercepts of the graphs of the following functions.

(a) y = 3x2 + 10x 7 (b) y = (x 8)(1 + 2x) + 6

(P01C02L03Q012)

12. The figure shows the graph ofy = 2x2

+ 8x + 6. State the following features of the graph:(a) Axis of symmetry

(b) Coordinates of the vertex

(c) y-intercept

(d) Direction of opening

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(P01C02L03Q013)

13. The figure shows the graph ofy = 2x2 + 8x. State the following features of the graph:

(a) Axis of symmetry

(b) Coordinates of the vertex

(c) y-intercept

(d) Direction of opening

(P01C02L03Q014)

14. The figure shows the graph ofy =x2 4x + c.D(d, 16) is the minimum point of the graph.

(a) Find the values ofc and d.

(b) State the following features of the graph:

(i) Axis of symmetry

(ii) Coordinates of the vertex

(iii) y-intercept

(iv) Direction of opening

(P01C02L03Q015)

15. For each of the following quadratic functions,

find (i) the direction of opening,

(ii) the coordinates of the vertex,

(iii) the axis of symmetry

of its graph.

(a) y = (x 1)2 + 3

(b) y = 3(x + 4)2 9

38

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Question Bank

(P01C02L03Q016)

16. Find the optimum values of the following quadratic functions.

(a) y = (x 5)2 17 (b) y = (x + 2)2 + 1

(P01C02L03Q017)

17. Find the optimum value for each of the following quadratic functions and the axis of symmetry of

their graphs.

(a) y =x2 10x + 27 (b) y = 2x2 16x + 1

(P01C02L03Q018)

18. For the quadratic functiony = 3x2 12x 20,

(a) find the optimum value of the function,

(b) state (i) the direction of opening,


(iii) the axis of symmetry

of its graph.

(P01C02L03Q019)

19. Given that the maximum value of the functiony = 3x2 + 6x +p is 11.

(a) Find the value ofp.

(b) State the coordinates of the vertex of its graph.

(P01C02L03Q020)

20. Given that the maximum value of the functiony = (4 2x)(x + k) + 2(k 10)x is 4.

(a) Find the value ofk.

(b) State the axis of symmetry of its graph.

(P01C02L03Q021)

21. If the sum of two numbers is 12, find the maximum value of the product of these two numbers.

(P01C02L03Q022)

22. Given that the difference between two numbers is 10, find the minimum value of the product of these

two numbers.

39

NF

NF

NF

NF

NF

NF

NF


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(P01C02L03Q023)

23. Describe the properties (number ofx-intercepts, axis of symmetry and maximum or minimum point)

of the following graphs.

(a) (b)

(P01C02L03Q024)



(a) (b)

(P01C02L03Q025)



(a) (b)

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Question Bank

(P01C02L03Q026)

26. The figure shows the graph ofy = x2 2x 3. Solve the inequality x2 2x 3 < 3 by drawing a

suitable straight line on the graph.

(P01C02L03Q027)

27. The figure shows the graph ofy =x2 x + 1. Solve the inequalityx2 x + 1 > 3 by drawing a suitable

straight line on the graph.

(P01C02L03Q028)

28. The figure shows the graph ofy =x3 2. Solve the inequalityx3 2 6 by drawing a suitable straight

line on the graph. (Give your answer correct to the nearest 0.2.)

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(P01C02L03Q029)

29. The figure shows the graph ofy = 2x + 3. Solve the inequality 2x + 3 7 by drawing a suitable straight

line on the graph. (Give your answer correct to 1 decimal place.)

(P01C02L03Q030)

30. It is given that the graphs ofy =x2 + 5x + 9 andy = 4. Suggest an inequality that can be solved.

(P01C02L03Q031)

31. In each of the following, iff(x) is transformed tog(x), describe the effect of the transformation on the

graph ofy =f(x).

(a) g(x) =f(x) 2 (b) g(x) =f(x + 2)

(P01C02L03Q032)

32. The following table shows the tabular representation of a functionf(x):

x 0 1 2 3 4f(x) 3 4 6 9 13

If the graph ofy =g(x) is obtained by translating the graph ofy =f(x) in the negative direction of the

y-axis by 7 units, complete the following table.

x 0 1 2 3 4g(x)

(P01C02L03Q033)

33. The following table shows the tabular representation of a functionf(x):

x 0 1 2 3 4 5f(x) 3 2 1 0 1 2

If the graph ofy =g(x) is obtained by translating the graph ofy =f(x) in the positive direction of thex-

axis by 1 unit, complete the following table.

x 1 2 3 4g(x)

(P01C02L03Q034)

34. Iff(x) =x2 x 6 is transformed tog(x) =x2 x + 36, describe the effect of the transformation on the

42

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Question Bank

graph ofy =f(x).

(P01C02L03Q035)

35. Iff(x) = (3x + 5)2 is transformed tog(x) = 9x2 + 12x + 4, describe the effect of the transformation on

the graph ofy =f(x).

(P01C02L03Q036)

36. It is given thatf(x) =x2 2x + 3. If the graph ofy =g(x) is obtained by translating the graph ofy =f(x)

in each of the following directions, find the symbolic representation ofg(x).

(a) Translated in the negative direction of they-axis by 2 units.

(b) Translated in the positive direction of thex-axis by 4 units.

(P01C02L03Q037)

37. It is given thatf(x) = (2 3x)2. If the graph ofy =g(x) is obtained by translating the graph ofy =f(x) in

each of the following directions, expressg(x) in the form ax2 + bx + c.

(a) Translated in the positive direction of they-axis by 4 units.

(b) Translated in the negative direction of thex-axis by 3 units.

43

NF

NF

NF

NF


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Level 2 Questions

(P01C02L04Q001)

1. If 221

)(x

xxf = , show that )(1 xfxf =.

(P01C02L04Q002)

2. It is given thatg(x) = 4x 9. Find the values ofx if 01

)( =

x

gxg .

(P01C02L04Q003)

3. Iff(x) =x2 + 3x 15, find the values of

(a) f(3), (b) f(a), (c) f(2b + 1).

(P01C02L04Q004)

4. It is given thatf(x) = (x + k)(x 1) 2x andf(k) = k2 3. Find the value(s) ofk.

(P01C02L04Q005)

5. It is given thatf(x) = (x + 2k)2 andg(x) = 5k+x.

(a) Iff(0) 3g(3) = k 25, find the value ofk.

(b) Solve the equationf(x) 2g(x) = 3.

(P01C02L04Q006)

6. Let2

1

1)(

+=

x

xxg .

(a) Findg

+1

1

x

x.

(b) Hence, find the value ofg

7

5.

(P01C02L04Q007)

7. It is given thatf(x) =x2 3x + 2.

(a) Find the values off(2a) andf(a + 2).

(b) Iff(2a) =f(a + 2) + 2f(a), find the value(s) ofa.

(P01C02L04Q008)

8. It is given thatf(x) = (x a)(x b) + 5. Iff(a) = b andf(2b) = 3b, find the values ofa and b.

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Question Bank

(P01C02L04Q009)

9. It is given thatf(x) = ax2 + bx + 3.

(a) Iff(1) = 1 andf(3) = 33, find the values ofa and b.

(b) If )(2

xfxg =, find the symbolic representation ofg(x).

(c) Hence, find the values ofg(5) andg(p 1).

(P01C02L04Q010)

10. (a) Plot the graph ofy = 3x2 12x + 9 fromx = 0 tox = 4.

(b) State (i) the axis of symmetry,


(iii) they-intercept,

(iv) the direction of opening

of the graph.

(P01C02L04Q011)

11. (a) Plot the graph ofy = x2 2x + 15 fromx = 6 tox = 4.





of the graph.

(P01C02L04Q012)

12. (a) Plot the graph ofy = 9 (x + 5)(x 3) fromx = 6 tox = 4.




(iv) the direction of openingof the graph.

(P01C02L04Q013)

13. (a) Plot the graph ofy = (2x 3)2 + 2(1 2x) fromx = 0 tox = 4.





of the graph.

45

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(P01C02L04Q014)

14. For each of the following graphs ofy = ax2 + bx + c, state the signs ofa and c.

(a) (b)

(P01C02L04Q015)

15. The figure shows the graph ofy = ax2 2x + c. Show that 0 < ac < 1.

(P01C02L04Q016)

16. For the quadratic functiony = (x + 3)2 2(5 + 2x),


(b) state (i) the coordinates of the vertex,

(ii) the axis of symmetry,

(iii) the direction of openingof its graph.

(P01C02L04Q017)

17. For the quadratic functiony = (3x 2)(4 x) 2(x 2),


(b) state (i) the coordinates of the vertex,

(ii) the axis of symmetry,

(iii) the direction of opening

of its graph.

46

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NF


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Question Bank

(P01C02L04Q018)

18. Given that the axis of symmetry of the graph ofy =x2 4qx + 6q2 isx = 6.

(a) Find the value ofq.

(b) Find the optimum value ofy.

(P01C02L04Q019)

19. Given that the minimum value of the functiony = (x k)2 + 6k(x 1) is 3.

(a) Find the value ofk.

(b) State the axis of symmetry of its graph.

(P01C02L04Q020)

20. If the optimum value of the function y = x2 3px + 21 is 6p, where p is positive, find the axis of

symmetry of its graph.

(P01C02L04Q021)

21. The profit ($P) of holding a party withx tickets sold is given by:

P= 2400x 80x2

(a) How many tickets are sold when the profit is maximum?

(b) What is the maximum profit of holding a party?

(P01C02L04Q022)

22. It is given that the total length of all the sides of the two cubes is 12 cm.

(a) Find the minimum value of the total volume of the two cubes.

(b) Find the lengths of a side of the two cubes when the total volume of the two cubes is minimum.

(P01C02L04Q023)

23. In the figure, a rectangular pictureABCD of perimeter 160 cm is hung

by a piece of ropeAED at the peg E, whereAE=ED. It is given that

the shortest distance between the peg and the picture is 15 cm.(a) Find the maximum possible area of the picture.

(b) Find the length of the rope when the area of the picture is

maximum.

(P01C02L04Q024)

24. Suggest a quadratic function such that the following conditions are satisfied:

(1) The maximum value of the function is 5.

(2) The axis of symmetry of its graph isx = 1.

(3) They-intercept of its graph is a positive integer.

47

NF

NF

NF

NF

NF

NF

NF


23/41


(P01C02L04Q025)

25. The figure shows the graph of the function 123 = xxxy .

(a) Find the minimum value of the function for 20 x .

(b) Is the value obtained in (a) a minimum value of the function?

(P01C02L04Q026)

26. The figure shows the graph of the function baxxy ++= 23 .

(a) Find the values ofa and b.

(b) IfC(0, c) andD(2, d) lie on the graph, find the values ofc and d.

(P01C02L04Q027)

27. Given the graph ofy =x2

+ 7x + 5. Find the equation of the straight line that should be added on thegraph in order to solvex2 + 7x < 1.

(P01C02L04Q028)

28. Given the graph ofy =x3 2x 1. Find the equation of the straight line that should be added on the

graph in order to solve 2x3 4x 8.

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Question Bank

(P01C02L04Q029)

29. The figure shows the graph ofy = 3x.

(a) Solve 3x 9 < 0 graphically.

(Give your answer correct to the nearest 0.2.)

(b) Find the smallest integerx that satisfies 3x + 10 < 5.

(P01C02L04Q030)

30. The figure shows the graph ofy = x2 2x + 1.

(a) Solve (x 1)2

+ 4x < 0 graphically.(b) Find the largest integerx that satisfiesx(x + 2) 4.

(P01C02L04Q031)

31. Solve the inequalityx2 + 5x 4 > 10 graphically.

(P01C02L04Q032)

32. Solve the inequality 3x2 + 6x + 2 2 graphically.

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(P01C02L04Q033)

33. (a) Plot the graph ofy =x2 + 3x 4.

(b) Hence, find the largest integerksuch that the inequality (x + 2)(x 2) > k 3x is always true.

(P01C02L04Q034)

34. It is given that f(x) = x2 + 4x + 1 andg(x) = x2 + 10x + 22. Iff(x) is transformed tog(x), describe the

effect of the transformation on the graph ofy =f(x).

(P01C02L04Q035)

35. Given thatf(x) = 2x23x4 is transformed tog(x) = 2x23x + 1 first, then to h(x) = 2x211x + 15.

Describe the effect of the transformation on the graph ofy =f(x) whenf(x) is transformed to h(x).

(P01C02L04Q036)

36. It is given thatf(x) =x2 + 5,g(x) =f(x) 8 and h(x) =g(x + 2).

(a) (i) Describe the effect of the transformation on the graph ofy =f(x) whenf(x) is transformed

tog(x).

(ii) Find the symbolic representation ofg(x).

(b) (i) Describe the effect of the transformation on the graph ofy =g(x) wheng(x) is transformed

to h(x).

(ii) Find the symbolic representation ofh(x).

(P01C02L04Q037)

37. It is given thatf(x) = x2 + 3x29, g(x) =f(x3) and h(x) =g(x) + 3.

(a) (i) Describe the effect of the transformation on the graph ofy =f(x) whenf(x) is transformed

tog(x).


(b) (i) Describe the effect of the transformation on the graph ofy =g(x) wheng(x) is transformed

to h(x).

(ii) Find the symbolic representation ofh(x).

(P01C02L04Q038)

38. (a) Given that the functionf(x) =x23x + 10, find they-intercept of its graph.

(b) Hence, ifg(x) =f(x) 2, find they-intercept of the graph ofy =g(x).

(P01C02L04Q039)

39. (a) Given that the functionf(x) =x2 + 8x + 12, find thex-intercept(s) of its graph.

(b) Hence, if the graph ofy = g(x) is obtained by translating the graph ofy = f(x) in the negative

direction of thex-axis by 2 units, find thex-intercepts of the graph ofy =g(x).

50

NF

NF

NF

NF

NF

NF


26/41

Question Bank

(P01C02L04Q040)

40. (a) It is given that f(x) = x2 + 4x 5. Plot the graph ofy = f(x) for 6 x 2 and find the

coordinates of the vertex of the graph.

(b) It is given thatg(x) =f(x + 1).

(i) Describe the effect of the transformation on the graph ofy =f(x) whenf(x) is transformed

tog(x).

(ii) By the result obtained in (a), find the coordinates of the vertex of the graph ofy =g(x).

(P01C02L04Q041)

41. In the figure, the graph ofy =g(x) is obtained by translating the graph ofy = x2 + 2x + 15.

(a) Describe the effect of the transformation on the graph ofy = x2 + 2x + 15.

(b) Find the symbolic representation ofg(x).

(P01C02L04Q042)

42. In the figure, the graph ofy =g(x) is obtained by translating the graph ofy = 2x28x + 10.

(a) Describe the effect of the transformation on the graph ofy = 2x28x + 10.

(b) Find the symbolic representation ofg(x).

51

NF

NF

NF


27/41


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Question Bank

(P01C02L04Q049)

49. The following tables show the tabular representations of the functionsp(x) and q(x).

x 3 2 1 0 1 2 3

p(x) 35 21 11 5 3 5 11

x 3 2 1 0 1 2 3

q(x) 2 2 2 2 2 2 2

(a) (i) Find the symbolic representation ofq(x).

(ii) Given thatp(x) is a quadratic function. Find the symbolic representation ofp(x).

(b) (i) Ifr(x) =p(x) + q(x), describe the effect of the transformation on the graph ofy =p(x) when

p(x) is transformed to r(x).

(ii) If the graph ofy = s(x) is obtained by translating the graph ofy = r(x) in the positive

direction of thex-axis by 6 units, find the symbolic relation between r(x) ands(x).

(c) Hence, find the symbolic representations ofr(x) ands(x).

53

NF


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Level 2+ Questions

(P01C02L05Q001)

1. It is given thatp(x) =x2 + 1, q(x) = 2x 5 and r(x) = 4x2 12x + 10.

(a) Show that r(x) =p[q(x) + 2].

(b) Find the symbolic representation ofq[p(x) + 2].

(c) Find the range of possible values ofx ifr(x) 2q[p(x) + 2].

(d) For the minimum integral value ofx in (c), find the value ofr[p(x) + q(x)].

(P01C02L05Q002)

2. (a) Given thatf(x) = (xa)(xb) and ab = k. Prove that the minimum value off(x) is4

2k

.

(b) Letg(x) =x26x + 5 and h(x) = (x3)(xb), where b < 3.

(i) By the result obtained in (a), ifg(x) and h(x) have the same minimum value, find the value

ofb.

(ii) Hence, ifh(x) =g(x + m) + n, find the values ofm and n.

(P01C02L05Q003)

3. Let 622

1)(

2 ++= xxxf . In the figure, the graph ofy =g(x) is obtained by translating the graph

of

y =f(x) in the direction of thex-axis. The graphs ofy =f(x) andy =g(x) intersect atP(0,p). The graph

ofy =f(x) cuts thex-axis at Q(q, 0) andR(r, 0). The graph ofy =g(x) cuts thex-axis at S(s, 0) and

T(t, 0). It is given thatA(a, 6) lies on the graph ofy =f(x).

(a) Find the values ofa,p, q and r.

(b) (i) Find the symbolic relation betweenf(x) andg(x).

(ii) Hence, find the values ofs and t.

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Question Bank

(c) (i) Find the area ofPQT.

(ii) Iff(x) is transformed tog (x) such that the area ofPQTformed is smaller than that in (c)

(i), suggest a possible functiong (x).

(P01C02L05Q004)

4. (a) Let f(x) = a(xh)2 + kand g(x) = b(xm)2 + n. If the graph ofy =g(x) is obtained by translating

the graph ofy =f(x), prove that a = b.

(b) LetF(x) = c(xs)2 + tand G(x) = 3x212x +16. It is given that the graph ofy = G(x) is obtained

by translating the graph ofy =F(x) in the positive direction of the x-axis by 3 units, and then in

the negative direction of they-axis by 2 units. Find the values ofc,s and t.

(c) Suggest quadratic functionsf(x) andg(x) such that the following conditions are satisfied:

(1) The graph ofy =f(x) passes through (1, 2).

(2) The graph of y = g(x) is obtained by translating the graph ofy = f(x) in the negative

direction of thex-axis, and then in the positive direction of they-axis.

(P01C02L05Q005)

5. The figure shows the graphs ofy =x2 and2

4

1xy = .

(a) (i) Solvex2

< 1 by adding a suitable straight line on the graph.(ii) Hence, solve (x2)(x + 2) < 0.

(Give your answers correct to 1 decimal place.)

(b) (i) Suggest a quadratic functionf(x) such that the following conditions are satisfied:

(1) The coordinates of the vertex of its graph are (0, 0).

(2) f(x) < 1 fork


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(c) For each of the following quadratic function g(x), determine whether the inequality g(x) > 1 is

always true.

(i) g(x) =x22x + 6

(ii) g(x) = x2 + 2x + 4

(d) Suggest a quadratic function h(x) such that the inequality h(x) < 4 is always true.

(P01C02L05Q006)

6. (a) It is given that 27x2 66x + 40 A(3x 2)2 +B(3x 2) + C. Find the values ofA,B and C.

(b) Hence, iff(3x 2) = 27x2 66x + 40, find the symbolic representation off(x).

(c) If the graph ofy =f(x) is obtained by translating the graph ofy =g(x) in the positive direction of

they-axis by 3 units, find

(i) the symbolic relation betweeng(x) andf(x),

(ii) the symbolic representation ofg(x).

(d) If the graph ofy =g(x) is obtained by translating the graph ofy = h(x) in the positive direction of

thex-axis by 2 units, find

(i) the symbolic relation between h(x) andg(x),

(ii) the symbolic representation ofh(x).

(P01C02L05Q007)

7. (a) It is given thaty = ax2 + kax + c, where a 0. Express the coordinates of the vertex of the graph

ofy = ax2 + kax + c in terms ofa, c and k.

(b) Letf(x) = 2x26x + 3,g(x) =x23x2 and h(x) =f(x) +g(x).

(i) By the result obtained in (a), find the minimum values off(x) andg(x).

(ii) Show that the minimum value ofh(x) is equal to the sum of the minimum values off(x) and

g(x).

(c) Suggest two quadratic functions such that the result obtained in (b)(ii) is not true.

(P01C02L05Q008)

8. (a) Sandy throws a ball Pupwards. Aftertpseconds, the height (hp m) of the ballPabove the groundis given by hp= 5tp2 + 20tp + 13.

(i) When will the ballPattain its maximum height?

(ii) What is the maximum height reached by the ballP?

(b) When the ballPattains its maximum height, Janis throws another ball Q. AftertQseconds, the

height (hQ m) of the ball Q above the ground is given by hQ= 5tQ2 + 15tQ + 13.

(i) Find the maximum height that can be reached by the ball Q.

(ii) Find the height of the ballPwhen the ball Q attains its maximum height.

(c) Who will receive her ball first? After how many seconds the other one will receive her ball?

56


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Question Bank

(P01C02L05Q009)

9. The figure shows the graph ofy = x2 + 5x + c which cuts the y-axis at P(0, 8), and passes through

Q(1, k). It is given that Q is reflected about the axis of symmetry of the graph toR.

(a) Find the values ofc and k.

(b) (i) Find the axis of symmetry of the graph.

(ii) Hence, find the coordinates ofR.

(c) Solvex(x5) ck.

(d) (i) Plot the graph ofy = x2 + 5x + c fromx = 2 tox = 7.

(ii) Solvex25xc + 4k 0 by adding a suitable straight line on the graph.

(e) Using the result obtained in (d), find the possible values of integerx that satisfy

k+ 6 < 3(x + 3) (1 x)2 < 7k.

(P01C02L05Q010)

10. In the figure, the graph ofy =g(x) is obtained by translating the graph ofy =f(x). It is given that

f(x) = 4x2 8x + 3.

(a) (i) Find the symbolic relation betweenf(x) andg(x).


57


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(b) If the graph ofy = h(x) is obtained by translating the graph ofy =g(x) in the positive direction of

thex-axis by 2 units,

(i) find the symbolic relation betweeng(x) and h(x),

(ii) find the symbolic representation ofh(x).

(c) (i) Solve the inequalityg(x) > 12 by adding a suitable straight line on the graph.

(ii) Hence, solve the inequality (x 3)(x 1) > 3 without adding any straight lines on the

graph.

58


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Question Bank

Multiple Choice Questions

(P01C02L06Q001)

1. Which of the following is NOT a function of

x?

A. y = 5 x

B. y =x2 + 9x 12

C.x

xy13 +=

D. y2 = 4x

(P01C02L06Q002)

2. Iff(x) = (x 1)(2x + 3), find the value off(5).

A. 7

B. 5

C. 52

D. 20

(P01C02L06Q003)

3. Iff(x) = ax + b, thenf(ax + b) =

A. ax + b.

B. a2x + ab + b.

C. a2x2 + 2ax + b2.

D. 2ax + 2b.

(P01C02L06Q004)

4. Ifg(x + 1) = 2x2 + 4x + 2, theng(x) =

A. x2.

B. 2x2

.C. 2x2 + 4x + 2.

D. 2x2 + 8x + 8.

(P01C02L06Q005)

5. Iff(x) =x3 + kx2 + kx + 1 andf(1) +f(1) = 8,

find the value ofk.

A. 0

B. 1

C. 3

D. 8

(P01C02L06Q006)

6. The figure shows the graph ofy = ax + b. If

f(x) = ax + b, find the value off(3).

A.2

3

B. 0

C. 9

D. 9

(P01C02L06Q007)

7. The figure shows the graph of

y = ax2 + bx + c. Determine the signs ofa and

c.

A. a < 0, c < 0

B. a > 0, c < 0

C. a < 0, c > 0

D. a > 0, c > 0

59

NF


35/41


(P01C02L06Q008)

8. Which of the following may represent the

graph ofy = ax2 + bx + c, where ac < 0?

A.

B.

C.

D.

(P01C02L06Q009)


y = x2 + 3x + 4. Find the area of the

rectangle OABC.

A. 9

B. 12

C. 16

D. 6

(P01C02L06Q010)


y = ax2 + bx + c. The coordinates of the vertex

of the graph is (2, 5). Which of the

following must be true?

I. The axis of symmetry isx = 5.

II. The graph opens downwards.

III. ac > 0

A. I only

B. II only

C. III only

D. I and III only

60


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Question Bank

(P01C02L06Q011)

11. In the figure, the axis of symmetry of the

graph ofy = (x h)2 + k is x = 3. Find the

optimum value ofy.

A. 4

B. 8

C. 12

D. 14

(P01C02L06Q012)

12. Which of the following functions has a

maximum value of 2?

A. f(x) = (x + 1)2 2

B. f(x) = (x 2)2 + 1

C. f(x) = (x + 1)2 2

D. f(x) = (x 2)2 + 1

(P01C02L06Q013)

13. If the axis of symmetry of the graph of

y = 2x2 + kx + 9 is2

3=x , find the value ofk.

A. 3B. 6

C. 3

D. 6

(P01C02L06Q014)

14. It is given that the function

y = (x + 1)2 + 4(x + 1) + 5. Which of the

following about its graph must be true?

I. The axis of symmetry isx = 3.

II. The graph opens downwards.

III. The coordinates of the vertex are (1, 9).

A. II only

B. I and II only

C. I and III only

D. II and III only

(P01C02L06Q015)

15. It is given thatf(x) = 4x2 4x + 13. Which of

the following must be true?

I. The minimum value off(x) is 12.

II. The axis of symmetry of the graph of

y =f(x) isx =2

1 .

III. The coordinates of the vertex of the

graph ofy =f(x) is (2

1 , 12).

A. I and II only

B. I and III only

C. II and III only

D. I, II and III

(P01C02L06Q016)

16. Find the optimum value of

y =6)4(

22 + x

.

A. maximum value ofy = 6

B. minimum value of3

1=y

C. maximum value ofy = 2

D. minimum value ofy = 3

61

NF

NF

NF

NF

NF


37/41


(P01C02L06Q017)

17. It is given that the perimeters of two

rectangles are 36 m and 44 m respectively.

What is the difference between their

maximum areas?

A. 40 m2

B. 64 m2

C. 81 m2

D. 121 m2

(P01C02L06Q018)

18. Which of the graphs of the following

functions has the maximum number ofx-

intercepts?

A. y = 5x2 6x + 8

B. y = 9 2x

C. y =x3

D. y = 10x x2

(P01C02L06Q019)

19. The figure shows the graph ofy = x2 + 5x.

Solvex25x + 6 < 0.

A. 2 3

C. 1 6

(P01C02L06Q020)

20. Which of the following inequality can be

solved by adding the straight line y = 3 on

the graph ofy = 3x3 + 2x2 5?

A. 3x3 + 2x2 8 > 0

B. 2x2 3x3 + 2

C. 3 2x2 < 3x3

D. 3x3 2x2 2

(P01C02L06Q021)

21. The figure shows the graph ofy =x2 4x 5.

Find the largest integerx that satisfies

x2 < 4(x + 2).

A. 2

B. 0

C. 4

D. 5

62

NF


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Question Bank

(P01C02L06Q022)


y = 2x2 + 5x + 8. Find the number of integral

solutions of the inequality 5x > 2(x2 + 1).

A. 0

B. 1

C. 2

D. 3

(P01C02L06Q023)

23. The figure shows the graph ofy = x2 4x

which has the minimum value ofk. Which of

the following inequality can be solved by

adding the straight line y = k + 1 on the

graph?

A. x(x2) 2x + 3

B. x(x2) 2x3

C. x(x3) x + 5

D. x(x3) x5

(P01C02L06Q024)

24. In the figure, the graphs ofy = 2x2 + 3x 1

and y = 1 intersect at A(a, 1) and B(b, 1).

Which of the following is true?

A.

+

>+

bxaxx

bxaxxx

w h e r e,232

o rw h e r e,232

2

2

B.

+

>bxaxx

bxaxxx

w h e r e,232

o rw h e r e,2322

2

C.

+

>


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25. Iff(x) =x27x4 is transformed to

g(x) = x27x + 4, find the symbolic relation

betweenf(x) andg(x).

A. g(x) =f(x) 8

B. g(x) =f(x) + 8

C. g(x) =f(x8)

D. g(x) =f(x + 8)

64

NF


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Question Bank

(P01C02L06Q026)

26. If the graph ofy = g(x) is obtained by

translating the graph of y = f(x) in the

negative direction of the x-axis by 2 units,

find the symbolic relation between f(x) and

g(x).

A. g(x) =f(x) 2

B. g(x) =f(x) + 2

C. g(x) =f(x2)

D. g(x) =f(x + 2)

(P01C02L06Q027)

27. In the figure, the graph ofy =g(x) is obtained

by translating the graph ofy = 2x2 2x + 4

downwards. Which of the following is the

symbolic representation ofg(x)?

A. g(x) = 2x2 2x + 7

B. g(x) = 2x2 2x + 1

C. g(x) = 2x2 + 10x + 16

D. g(x) = 2x2

14x + 28

(P01C02L06Q028)

28. The following tables show the tabular

representations of functions

f(x) = (1 2x)2 +x(x + 5) andg(x).

x 0 1 2 3 4f(x) 1 7 23 49 85

x 1 2 3 4 5g(x) 1 7 23 49 85

If f(x) is transformed to g(x), find the

symbolic representation ofg(x).

A. g(x) = 5x2 +x + 2

B. g(x) = 5x2 +x

C. g(x) = 5x2 + 11x + 7

D. g(x) = 5x2 9x + 5

(P01C02L06Q029)

29. It is given thatf(x) =x2 7x + 10. If the graph

ofy = g(x) is obtained by translating the

graph ofy = f(x) in the negative direction of

the y-axis by 6 units, find the coordinates of

the vertex of the graph ofy =g(x).

A. (2

19,

4

9 )

B. (2

5 ,

4

9 )

C. (2

7,4

15)

D. (

2

7,

4

33 )

65

NF

NF

NF

NF


41/41


(P01C02L06Q030)

30. In the figure, the graph ofy =g(x) is obtained

by translating the graph ofy = x2 2x in the

direction of the x-axis. IfA(0, 3) lies on the

graph of y = g(x), find the symbolic

representation ofg(x).

A. g(x) =x2 1

B. g(x) =x2 + 4x + 3

C. g(x) =x2 8x + 15

D. g(x) =x2 4x + 3

NF

2 Functions and Graphs (Q)

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Transcript of 2 Functions and Graphs (Q)