Chp 3 Tutorial 3B APQ Soln (RVHS)

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 23 Additional Practice Questions Solutions 1. NYJC/2006/Promo/Q6 The diagram shows a sketch of  fy x . On separate diagrams, sketch the graphs of i)  fy x , [2]

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 24 ii)  f 1y x  , [3]

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 25 2. RJC/2006/Promo/Q2 The diagram below shows the graph of  fy x . The points A, B and C have coordinates 1( ,0)2 , (1, 2) and (3, 1) respectively. Sketch on separate diagrams, the graphs of (i) f 2 xy      , [1] x y  fy x C(3,1) B(1,2) 1 2( ,0)A  

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 26 (ii) f 12 xy      , [2] State, in each case, the coordinates of the points corresponding to A, B and C, where appropriate.

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 27 3. HCI/2006/Promo/Q6 The diagram below shows the graph of  f 1y x  . The curve has a maximum point at A 31, 2      , and axes intercepts at B(0, 1) and C(–1, 0). The lines 2x and 1y  are the asymptotes. Sketch the graphs of (a)   1 f 1y x  [3] x y B(0, 1) A C(–1, 0) x y (0, 1) ݔ= −1

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 28 (b)  fy x , [3] ݕ= f(1 −ݔ)→ f(1 +ݔ)→ f(ݔ) 4. SAJC/2006/Promo/Q9 (modified) a) The curve C has equation 2 3 x axy  where a is a constant. Given that C has a stationary point at x = 2, find the value of a and determine the nature of the stationary point. Write down the equation of the asymptotes of C. [4] ݕ= ௫యା௔ ௫మ =ݔ+ ܽݔିଶ ௗ௬ ௗ௫ = 1 − 2ܽݔିଷ = 0 when ݔ= 2 1 − ଶ ଼ܽ= 0 ܽ= 4 ௗమ௬ ௗ௫మ = 24ݔିସ > 0, therefore (2, 3) is a minimum point. Asymptotes: ݔ= 0 and ݕ= ݔ x y ൬0, 3 2൰ (1,1) ݔ= 3 (0, 2)

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 29 Hence sketch C, showing clearly the asymptotes and the coordinates of any point(s) of intersection with the coordinate axes. [2] b) The given diagram shows the graph of y = f(x). On separate diagrams, sketch the graphs of (i)  xy f [1] O x y B(4, – 2) C(7, 0) A(2, 0) 2

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 30 (ii)   1 fy x [2] in each case showing clearly the asymptotes and the points corresponding to A, B and C. 5. MI/2006/Promo/Q8 The equations of the curves 1C and 2C are given by 1C : 2 2 3 3x y  , 2C : 2 2 2x y  . (i) Give a geometrical description of the curve 1C and state any axes of symmetry. [4] C1 is an ellipse, centred on (0, 0). Horizontal diameter is 2√3, vertical diameter 2. Axes of symmetry are ݕ= 0 and ݔ= 0. (ii) State the equations of the asymptotes of 2C . [1] ݕ= ±ݔ Sketch 1C and 2C on the same diagram, indicating clearly the asymptotes and state the coordinates of the points of intersection of the curves 1C and 2C . [5]

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 31 6. CJC/2006/Promo/Q3 A graph with equation y = f(x) undergoes 3 successive transformations as follows: I: A scaling parallel to x–axis with scale factor 1 2 II: A translation of 3 units in the direction of the positive x–axis III: A scaling parallel to y–axis with scale factor 2 The resulting equation is y = 2 ln (6 – 2x). Determine the equation of the graph y = f(x), showing your workings clearly. [3] Let: III’: Scaling parallel to y-axis with scale factor ½ II’: Translation of -3 units in the direction of the positive x-axis I’: Scaling parallel to x-axis with scale factor 2. These will undo the original three transformations. Applying them successively to ݕ= 2 ln(6 − 2ݔ):      ' ' ' 12ln(6 2 ) ln(6 2 ) ln 6 2 3 ln( 2 ) ln 2 ln 2 III II Ix x x x x x                        Therefore the original graph was ݕ= ln(−ݔ)

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2021 Chp 3 Graphing Techniques (Part 2) (Solutions to Tutorial and Assignment) 32 7. VJC/2006/Promo/Q12 (modified) The diagram shows the graph of y = f(x) . The curve passes through the origin and has minimum point (1, 2). The asymptotes are 2x , 0y and 2y . Sketch, on separate diagrams, the graphs of (i) y = f (x  1) , [2] y O x y = 2 x = 2 (1, 2) 