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H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College

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JC2 Revision Package 1

H2 Mathematics (9740)

Graphing Techniques

l1

l2

y

x


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3. 2011 CJC Prelim P2/Q1(ii)

(i)

A curve C has parametric equations tx sin2= and ty cos3= where π≤≤ 20 t .

Find the Cartesian equation of C and sketch C.

[3]

4. 2011 DHS Prelim/P1/7

(a)

(b)

The diagram shows the graph of f ( ),y x= which has turning points at ( 2,4)A − and

(2,3).B The horizontal and vertical asymptotes are y = 2 and x = −1 respectively.

Sketch, on separate diagrams, the graphs of

(i) f ( ),y x= −

(ii) 2

f ( ),y x=

showing clearly all relevant asymptotes, intercepts and turning point(s), where

possible

The graph of g( )y x= above intersects the x-axis at ( ,0)α and ( ,0),β where

1 and 1.α β> − > It has a turning point (0, 1)− and a vertical asymptote 1.x = −

[2]

[3]

y

O x

x = 1

y = 2

x

y

x = 1


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g( )y x= undergoes two transformations in sequence: a translation of 1 unit in the

positive y-direction, followed by a scaling of factor 2 parallel to the -axis.x The

resulting graph is h( ).y x=

Sketch, on separate diagrams, the graphs of h( )y x= and1

,g( )

yx

= showing clearly

all relevant asymptotes, intercepts and turning point(s), where possible.

[5]

5. 2011 DHS Prelim/P2/3(iii)

(i)

A curve C has parametric equations 1 1

, , 0.x at y bt tt t

= − = + >

For 1 and 1,a b= = the curve C has two oblique asymptotes y x= and y x= − . By

considering the curve of C, sketch the graph of f '( ).y x=

[3]

6. 2011 HCI Prelim/P1/10

(a)

(b)

The curve C has the equation ( )22 3 1 1y x= − + .

(i) Draw a sketch of C , indicating clearly the axial intercepts, the equations of

the asymptotes and the coordinates of the stationary points.

(ii) It is given that the curve ( )23 1 1y x= − + intersects another curve

( )2

21 1

yx

h

+ − =

at exactly 2 points. Find the range of values of h .

The diagram below shows a sketch of the curve of ( )fy x= with a maximum point

at ( )1, 0 . The lines 2y = − , 2y = and 0x = are asymptotes of the curve.

Sketch on separate diagrams, the curves of

(i) 1

f ( )y

x= ,

(ii) 2 f ( )y x= ,

stating the equations of any asymptotes and the coordinates of any intersections

with the axes.

[4]

x 1 O

y


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7. 2011 JJC Prelim/P1/12

(a)

(b)

The curve D has the equation 2( )

1

x ay

x

+=

+ where a is a constant such that 1 < a ≤ 3

and 1x ≠ − .

(i) Find the equations of the asymptotes of D.

(ii) Show that the stationary points of D are ( ,0)a− and ( 2, 4 4)a a− − .

(iii) Draw a sketch of D, which should include the asymptotes, turning points

and points of intersection with the axes.

(iv) Hence state the set of values of k for which the line y = k does not intersect

D.

The curve G given below has equation y = f(x). Sketch, on separate diagrams, the

graphs of

(i) f (3 )y x= −

(ii) 1

f ( )y

x=

[2]

[2]

[3]

[1]

[3]

[3]

8. 2011 MJC Prelim/P1/10

(a)

(b)

State a sequence of transformations which transform the graph of 2 2 1x y+ = to the

graph of ( )2 21 4x y− + = .

It is given that ( )22 7

f1

x axx

x

+ −=

−, where a is a constant, 5a ≠ . Find the range of

values of a such that ( )fy x= has no turning points.

Using 1a = , in separate diagrams, sketch

(i) ( )f ,y x=

(ii) ( )1

.f

yx

=

[3]

[4]

[3]

[3]

y

2

(3, −2)

−2 x

y = −1

0

y = f(x)


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9. 2011 NJC Prelim/P2/4(ii)

Sketch the curve C, with equation ( )

( )2

223 1

4

yx

+− − = , indicating clearly the

equations of any asymptotes.

[2]

10. 2011 NYJC Prelim/P2/4(a)(b)

(a)

(b)

The diagrams below show the graphs of | f ( ) |y x= and f '( )y x= .

Sketch the graph of f ( )y x= , stating the equations of any asymptotes and the

coordinates of any axial intercepts and turning points.

Hence, find the range of values of k if there is exactly 1 real root to the equation

f ( ) 0x k− = .

The diagram below shows the graph of f (2 1)y x= − . The curve passes through the

point A( −1, 0) and B(1, 1− ). The asymptotes are x = 0 and y = 0 and 3y = .

[3]

[2]

x = 3

y = 2

3

1 2

y = |f(x)|

y = f (x)

x

y

y

x = 3

y = 0


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y

0 1 2

2

( 3 , 3 )

x

Sketch, on separate clearly labelled diagrams, the graphs of

(i) f (2( | |) 1)y x= − − ,

(ii) f ( )y x= , describing the sequence of transformations involved.

Your sketch should show clearly the equations of any asymptotes and the

coordinates of the points corresponding to A and B.

[2]

[5]

11. 2011 PJC Prelim/P1/10(a)(b)

(a)

The diagram below shows the graph of f ( )y x= . The graph crosses the x-axis at

0x = , 2x = and has a turning point at ( )3,3 . The asymptotes of the graph are 1x =

and 2y = .

A ( 1,0)

B (1, 1)

y = 3

x

y

x = 0

y = f(2x-1)


Page 7 of 15

(b)


(i) ( )1

fy

x= ,

(ii) ( )f 1y x= + .

A graph with the equation ( )fy x= undergoes, in succession, the following

transformations:

A: A translation of 1 unit in the direction of the x-axis.

B: A stretch parallel to the x-axis by a scale factor2

1.

C: A reflection in the y-axis.

The equation of the resulting curve is 2

4

4 4 1y

x x=

+ +. Determine the equation of

the graph ( )fy x= , giving your answer in the simplest form.

[3]

[3]

[4]

12. 2011 RJC Prelim/P1/10(i),(ii),(iii)

(i)

(ii)

(iii)

The curves 1C and 2

C have equations

2 2 21 1

( 39 399) and ( 39 399)10 10

y x x x y x x x= − + = − +

respectively.

Find, by differentiation, the coordinates of the turning points of and determine

their nature.

Sketch the curve 1,C indicating clearly any relevant features.

Hence sketch, on a separate diagram, the curve 2.C

[3]

[2]

[2]

13. 2011 RVHS Prelim/P1/10

(i)

(ii)

(iii)

The curve C has equation 2 5ax bx

yx c

+ −=

+ where a, b, c are constants and x c≠ − .

Given that 1x = is an asymptote of C and C has a turning point on the y-axis,

determine the values of b and c.

Given also that C has no x-intercept, show that 5

4a < − .

Sketch the curve C for 5 5

2 4a− < < − , stating clearly the coordinates of any

stationary point, point of intersection with the axes, and the equations of any

asymptotes.

[3]

[2]

[3]

1C


Page 8 of 15

(iv)

(v)

By adding an additional line on the same diagram, determine in terms of a, the set of

values of x which satisfies the inequality2 5

1ax bx

axx c

+ −> +

+ for

5 5

2 4a− < < − .

Sketch on a separate diagram, the graph of f ( )y x′= , where ( )2 5

fax bx

xx c

+ −=

+, for

.

5 5

2 4a− < < − .

[3]

[2]

14. 2011 RVHS Prelim/P2/5

(a)

(b)

Describe a sequence of transformations which transforms the graph of 2 21x y+ = to

that of 2 2(2 2) 4x y+ + = .

The diagram below shows the graph of ( )fy x= where 0a < .


(i) ( )fy x= − ,

(ii) ( )2

fy

x= .

Your sketch should include the axial intercepts, coordinates of turning points, and

equations of asymptotes.

[3]

[3]

[4]

15. 2011 SAJC Prelim/P1/10

(i)

The curve C has equation ( )

( )( ) x a x by

x c

+ +=

+, where a, b, c are constants and it is

given that 0 a b c< < < .

By expressing y in the form y xx c

βλ= + +

+, state the equations of the asymptotes

x O ●

●


Page 9 of 15

(ii)

(iii)

of C in terms of a, b and c.

Show that ( ) 0ab c a b c− + − > for C to have two stationary points.

Given that 1, 2, 3a b c= = = , sketch C. Show, on your diagram, the equations of the

asymptotes and the coordinates of the turning points in three significant figures.

Hence find the set of values of k for which the equation 2( 3) +3 2 k x x x+ = + has

exactly two real roots.

[3]

[2]

[6]

16. 2011 SRJC Prelim/P1/6 Find the equations of the asymptotes of the hyperbola .

Hence sketch the hyperbola, stating clearly the asymptotes.

Hence find the range of values of k, such that the equation

has no real solutions.

[2]

[3]

17. 2011 SRJC Prelim/P1/7 (a)

(b)

A graph with equation undergoes in succession, the following

transformations:

A: A reflection about the x − axis

B: A translation of 1 unit in the direction of the positive y − axis

C: Scaling parallel to the x − axis by a factor of 3

The equation of the resulting curve is given by . Find the equation

.

Given the curves of f ( )y x= and f ( )y x= − below, sketch the graph of f ( )y x=

stating clearly the turning points, asymptotes and axial intercepts (if any).

[3]

[3]

2 24 24 9 36 36x x y y− − + =

( ) ( )224 24 9 4 36 4 36 0x x kx kx− − + + + − =

g( )y x=

12

2 9

xy

x

−=

−g( )y x=

y

X

0

x = 2

y = 1

(1, 3 )


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18. 2011 TJC Prelim/P1/11

(a)

(b)

The graph of y = f(x) is shown below. The curve cuts the y-axis at 1

0,2

−

and the

x-axis at (1, 0) and (7, 0). Sketch the graph of 1

f ( )y

x= , showing clearly the main

relevant features of the curve.

The curve C has equation ( 3)

x py

x x

+=

+, where p is a non-zero constant.

(i) State the equations of the asymptotes.

(ii) Show that if C has 2 stationary points, then p < 0 or p > 3.

(iii) Given p = 4, sketch the curve C, showing clearly the equations of the

asymptotes and the coordinates of the axial intercepts and stationary points.

[3]

[2]

[4]

[2]

y

x

0

x = 2

y = –1

y = f(x)

1 3 5 7

2

x

y

O


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19. 2011 TPJC Prelim/P2/1

By expressing the equation 4

12

−

+=

x

xy in the form

4

By A

x= +

−, where A and B are

constants, state a sequence of transformations which transform the graph of x

y1

=

to the graph of4

12

−

+=

x

xy .

Sketch the graph of4

12

−

+=

x

xy , giving the equations of any asymptotes and the

coordinates of any points of intersection with the x- and y-axes.

Hence find the least value of k where k is a positive integer, such that the equation

kxx

x=

−

+2

4

12 has 3 real roots.

[3]

[3]

[2]

20. 2011 VJC Prelim/P1/13

(i)

(ii)

(iii)

(iv)

The diagram shows the graph of f (2 )=y x which has asymptotes 2= −x and 0=y .

The curve passes through the origin and has a minimum point (2, 4)− .

Sketch, on separate diagrams, the graphs of

f (2 4)= −y x ,

f ( 2 | |)= −y x ,

1

f ( )=y

x,

f '(2 )y x= .

21.

2011 VJC Prelim/P2/2

(i)

(ii)

(iii)

The equation of a curve C is 2

2

( ), 0 1.

4

x ay a

x

+= < <

−

Write down the equations of any asymptotes of C.

Find, in terms of a, the coordinates of any stationary points of C.

Sketch C.

[2]

[4]

[2]

y

x


Page 12 of 15

22. 2011 YJC Prelim/P1/6

(a)

(b)

Describe a sequence of transformations which transform the graph of 3 4

2

xy

x

+=

− to

the graph of 1

yx

= .

The diagram shows the graph of ( )fy x= . The curve passes through the points A(0,

0), B(3, 0), C(4,3), and D(6, 2) with a gradient of 2 and 3 at A and B respectively.

On separate diagrams, sketch the graphs of

(i) ( )fy x= − ,

(ii) ( )fy x′= ,

labelling the coordinates of the corresponding points of A, B, C, and D (if

applicable) and the equations of any asymptotes.

[3]

[3]

[3]

× ×

×

×

y = 4

D(6,

2)

C(4, 3)

A(0, 0) B(3, 0) x

y x = 2


Page 13 of 15

Section B [Term 3 Revision]

1. 2011 ACJC Prelim/P2/5

(a)

(b)

The curves 1C and 2C have equations 2 29 ( ) 9y x k= + − and 2

2

21

xy

k+ =

respectively, where k is a real constant such that 3k > .

Describe a sequence of transformations which transforms the graph of 2 2 1x y− = to

the graph of 1C .

(i) On the same diagram, sketch the graphs of 1C and 2C , stating clearly the

coordinates of any points of intersection with the axes and the equations of

any asymptotes.

(ii) Find the range of values of the positive constant a such that the equation

2 2

2

( ) 91

9

x x k

a

+ −+ =

has two real roots.

[2]

[5]

[2]

2. 2011 IJC Prelim/P1/2

(i)

(ii)

The diagrams show the graphs of f ( )y x= and ( )2 fy x= . On separate diagrams,

sketch the graphs of

( )fy x= ,

12f

2y x

= +

,

showing clearly the asymptotes and the coordinates of the intersections with the

axes.

[2]

[2]

2

−1 x

y

O −1

x

y

O


Page 14 of 15

3. 2011 NJC Prelim/P1/12

(a)

(b)

The diagram shows a sketch of the curve , with asymptotes

.

The curve cuts the x-axis at the point (3, 0) and has a turning point (5, 4).

Sketch on separate diagrams, the graph of

(i) ,

(ii) ,

showing clearly any asymptotes and the coordinates of any turning points and axial

intercepts.

The curve C has equation 2 4

2

ax xy

x

+ −=

+, where a is a constant.

(i) Find d

d

y

x in terms of a, and deduce the range of values of a if C has two

distinct stationary points.

(ii) Given that the asymptotes of C intersect at the point (−2, −7), show that a =

2.

(iii) Sketch C, indicating clearly the asymptotes and stationary points.

(iv) For 0r ≥ , find the range of values of r, correct to 3 decimal places, such that

the equation 2

2ln( )

2 4

xx r

x x

+− =

+ − has exactly one real root.

[2]

[3]

[4]

[2]

[2]

[2]

4. 2011 NYJC Prelim/P1/5

(i)

(ii)

It is given that 2

f ( )ax bx c

xx d

+ +=

+ , for non-zero constants a, b, c and d.

Given that 2x = − and 1y x= − are asymptotes of the graph of f ( )y x= , find the

values of a, b and d.

By using differentiation, find the range of values of c such that the graph of

f ( )y x= has no turning points.

[3]

[3]

f ( )y x=

2 and 2x y= =

( )f 2y x= −

( )fy x′=

y

x

(5, 4)

0

y = f (x)


Page 15 of 15

(iii) Given that c = 1, draw a sketch of the graph of f ( )y x= , showing the coordinates of

the axial intercept(s), turning point(s) and the equations of asymptotes.

By drawing a suitable graph, write down the number of roots of the equation

4 3 2 2 0x x x x+ + + + = .

[2]

[2]

5. 2011 PJC Prelim/P1/11

(i)

(ii)

(iii)

(iv)

The curve C has equation 2 24

, ,x x k

y x kx k

− += ≠

− and k is a constant such that

0k ≠ and 2k ≠ .

Find the equations of the asymptotes of C.

Show that if C has 2 stationary points, then 0k < or 2k > .

Given that y x= is an asymptote of C, find the value of k. With this value of k,

sketch C, showing clearly the asymptotes and the stationary points.

By adding a suitable graph which passes through the point ( )4,4 on the sketch of C,

find the range of values of p for which the equation

( ) ( )( )2 24 4 4 0x x k x k px p− + − − + − = has exactly 2 real roots for the value of k

found I.

[2]

[3]

[3]

[2]