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H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

JC2 Revision Package 1 H2 Mathematics (9740) Graphing Techniques

y l2 l1

x

Page 1 of 15

H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

3. 2011 CJC Prelim P2/Q1(ii) A curve C has parametric equations x = 2 sin t and y = 3 cos t where 0 t 2 . (i) Find the Cartesian equation of C and sketch C. 4. 2011 DHS Prelim/P1/7 (a) The diagram shows the graph of y = f ( x), which has turning points at A(2, 4) and B(2, 3). The horizontal and vertical asymptotes are y = 2 and x = 1 respectively. y

[3]

y=2

O

x

x= 1

Sketch, on separate diagrams, the graphs of (i) y = f ( x ), (ii) y 2 = f ( x), showing clearly all relevant asymptotes, intercepts and turning point(s), where possible (b)x= 1 y

[2] [3]

x

The graph of y = g( x) above intersects the x-axis at ( , 0) and ( , 0), where > 1 and > 1. It has a turning point (0, 1) and a vertical asymptote x = 1.

Page 2 of 15

H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

y = g( 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 x-axis. The resulting graph is y = h( x). 1 Sketch, on separate diagrams, the graphs of y = h( x) and y = , showing clearly [5] g( x ) all relevant asymptotes, intercepts and turning point(s), where possible.5. 2011 DHS Prelim/P2/3(iii)

(i)

1 1 A curve C has parametric equations x = at , y = bt + , t > 0. t t For a = 1 and b = 1, the curve C has two oblique asymptotes y = x and y = x . By considering the curve of C, sketch the graph of y = f '( x). [3]

6. 2011 HCI Prelim/P1/10 (a) The curve C has the equation y 2 = 3 ( x 1)2 + 1 . (i) (ii) Draw a sketch of C , indicating clearly the axial intercepts, the equations of the asymptotes and the coordinates of the stationary points. It is given that the curve y = 3 ( x 1) + 1 intersects another curve2

[4]

y 2 + ( x 1) = 1 at exactly 2 points. Find the range of values of h . h(b) The diagram below shows a sketch of the curve of y = f ( x ) with a maximum point at (1, 0 ) . The lines y = 2 , y = 2 and x = 0 are asymptotes of the curve. y

2

O

1

x

Sketch on separate diagrams, the curves of 1 (i) y= , f ( x) (ii) y 2 = f ( x) , stating the equations of any asymptotes and the coordinates of any intersections with the axes.

Page 3 of 15

H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

7. 2011 JJC Prelim/P1/12 (a) The curve D has the equation y = and x 1 . (i) (ii) (iii) Find the equations of the asymptotes of D. Show that the stationary points of D are (a, 0) and (a 2, 4a 4) . Draw a sketch of D, which should include the asymptotes, turning points and points of intersection with the axes. Hence state the set of values of k for which the line y = k does not intersect D. [2] [2] [3]

( x + a)2 where a is a constant such that 1 < a 3 x +1

(iv)

[1]

(b) The curve G given below has equation y = f(x). Sketch, on separate diagrams, the graphs of [3] (i) y = f (3 x) 1 y= (ii) [3] f (x) y

y = f(x)

2

2

0y = 1 (3, 2)

x

8. 2011 MJC Prelim/P1/10 (a) State a sequence of transformations which transform the graph of x 2 + y 2 = 1 to the graph of ( x 1) + y 2 = 4 . (b) It is given that f ( x ) =2

[3]

2 x 2 + ax 7 , where a is a constant, a 5 . Find the range of x 1 [4] values of a such that y = f ( x ) has no turning points.Using a = 1 , in separate diagrams, sketch (i) y = f ( x ) ,1 (ii) y = . f ( x)

[3] [3]

Page 4 of 15

H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

9. 2011 NJC Prelim/P2/4(ii) Sketch the curve C, with equation equations of any asymptotes. 10. 2011 NYJC Prelim/P2/4(a)(b) (a) The diagrams below show the graphs of y = | f ( x) | and y = f '( x) . y y = |f(x)| y

( y + 2)4

2

( x 3) = 1 , indicating clearly the

2

[2]

3 y=2

y = f (x)

1

2

x

y=0

x=3

x=3

Sketch the graph of y = f ( 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 ( x) k = 0 .

[3] [2]

(b) The diagram below shows the graph of y = f (2 x 1) . The curve passes through the point A( 1, 0) and B(1, 1 ). The asymptotes are x = 0 and y = 0 and y = 3 .

Page 5 of 15

H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

y

y = f(2x-1)

y=3

x A ( 1,0) B (1, 1) x=0Sketch, on separate clearly labelled diagrams, the graphs of

(i) (ii)

y = f (2( | x |) 1) , y = f ( x) , describing the sequence of transformations involved.

[2] [5]

Your sketch should show clearly the equations of any asymptotes and the coordinates of the points corresponding to A and B.

11. 2011 PJC Prelim/P1/10(a)(b) (a) The diagram below shows the graph of y = f ( x) . The graph crosses the x-axis at x = 0 , x = 2 and has a turning point at ( 3,3) . The asymptotes of the graph are x = 1and y = 2 .

y

(3,3) 2

0

1

2

x

Page 6 of 15

H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

Sketch, on separate clearly labelled diagrams, the graphs of 1 (i) , y= f ( x)

[3] [3]

(ii)

y = f ( x + 1) .

(b) A graph with the equation y = f ( x ) undergoes, in succession, the following transformations: A: A translation of 1 unit in the direction of the x-axis. 1 B: A stretch parallel to the x-axis by a scale factor . 2 C: A reflection in the y-axis.The equation of the resulting curve is y =4 . Determine the equation of 4x + 4x +1 the graph y = f ( x ) , giving your answer in the simplest form. [4]2

12. 2011 RJC Prelim/P1/10(i),(ii),(iii) The curves C1 and C2 have equations

y=

1 x( x 2 39 x + 399) 10

and

y2 =

1 x( x 2 39 x + 399) 10

respectively.

(i) Find, by differentiation, the coordinates of the turning points of C1 and determine [3] their nature. (ii) Sketch the curve C1 , indicating clearly any relevant features. (iii) Hence sketch, on a separate diagram, the curve C2 . 13. 2011 RVHS Prelim/P1/10[2] [2]

ax 2 + bx 5 where a, b, c are constants and x c . x+c (i) Given that x = 1 is an asymptote of C and C has a turning point on the y-axis, [3] determine the values of b and c.The curve C has equation y = 5 [2] (ii) Given also that C has no x-intercept, show that a < . 4 5 5 (iii) Sketch the curve C for < a < , stating clearly the coordinates of any 2 4 stationary point, point of intersection with the axes, and the equations of any [3] asymptotes.

Page 7 of 15

H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

(iv) By adding an additional line on the same diagram, determine in terms of a, the set of ax 2 + bx 5 5 5 values of x which satisfies the inequality > ax + 1 for < a < . [3] x+c 2 4 ax 2 + bx 5 (v) Sketch on a separate diagram, the graph of y = f (x ) , where f ( x ) = , for x+c . 5 5 0 for C to have two stationary points. (iii) Given that a = 1, b = 2, c = 3 , 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 k ( x + 3) = x 2 +3x + 2 has exactly two real roots. [6]

16. 2011 SRJC Prelim/P1/6Find the equations of the asymptotes of the hyperbola 4 x 2 24 x 9 y 2 + 36 y = 36 . Hence sketch the hyperbola, stating clearly the asymptotes. Hence2

[2]

find

the

range2

of

values

of

k,

such

that

the

equation [3]

4 x 24 x 9 ( kx + 4 ) + 36 ( kx + 4 ) 36 = 0 has no real solutions.

17. 2011 SRJC Prelim/P1/7 (a)A graph with equation y = g( x) 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 x 12 The equation of the resulting curve is given by y = . Find the equation 2x 9 [3] y = g( x) .

(b) Given the curves of y = f ( x) and y = f ( x ) below, sketch the graph of y = f ( x)stating clearly the turning points, asymptotes and axial intercepts (if any). [3]

y

(1, 3 ) y=1 0 X

x=2

Page 9 of 15

H2 Revision Package 1: Graphing Techniques

2012 Meridian Junior College

y

x=2

0 x

y = 1

18. 2011 TJC Prelim/P1/11

1 (a) The graph of y = f(x) is shown below. The curve cuts the y-axis at 0, and the 2 1 x-axis at (1, 0) and (7, 0). Sketch the graph of y = , showing clearly the main f ( x) [3] relevant features of the curve.y 2

y = f(x)

O

1

3

5

7

x

(b) The curve C has equation y = (i) (ii) (iii)

x+ p , where p is a non-zero constant. x( x + 3)

[2] State the equations of the asymptotes. [4] Show that if C has 2 stationary points, then p < 0 or p > 3. Given p = 4, sketch the curve C, showing clearly the equations of the [2] asymptotes and the coordinates of the axial intercepts and stationary points.

Page 10 of 15

H2 Revision Package 1: Graphing Techni