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H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
Page 1 of 15
JC2 Revision Package 1
H2 Mathematics (9740)
Graphing Techniques
l1
l2
y
x
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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)
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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)
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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(b)
Sketch, on separate clearly labelled diagrams, the graphs of
(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
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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(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 < .
Sketch, on separate clearly labelled diagrams, the graphs of
(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 ●
●
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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(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 )
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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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)
H2 Revision Package 1: Graphing Techniques 2012 Meridian Junior College
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]
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