Chp 3 Tutorial 3A APQ Soln (RVHS)

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2022 Chp 3 Graphing Techniques (Part 1) (Solutions to Tutorial and Assignment) 29 Additional Practice Questions 1. Given the graph of y = x2 + a x + b , where a > 0 and b > 0, i) State the coordinates of the intersection(s) of the graph with the axes. When ݔ= 0,ݕ= (଴)మା௔ (଴)ା௕ = ௔ ௕, so ቀ0, ௔ ௕ቁ is the y-intercept. When ݕ= 0, 0 = ௫మା௔ ௫ା௕ ,ݔଶ = −ܽ< 0 so there is no x-intercept. ii) Find the equations of the asymptote(s). x = –b ; y = x – b iii) Draw a sketch of the curve, labelling the equations of its asymptotes and coordinates of any intersection with the axes. 2. [MI PU2 Promo 9758/2019/02/Q2] The curve C has equation 2 4 9 3 x xy x    . (i) Find algebraically the set of values that y can take, leaving your answer in exact form. [4] (ii) Sketch C, stating the coordinates of the axial intercept, turning points and the equations of any asymptotes. [3] x = –b y = x – b

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2022 Chp 3 Graphing Techniques (Part 1) (Solutions to Tutorial and Assignment) 30 M2 JC1 Promo 9758/2019/02/Q2 (Solutions) 2(i)       2 2 2 2 2 2 2 2 4 9 3 ( 3) 4 9 4 3 9 0 (4 ) 3 9 0 ..... (*) For values that can take, the equation (*) has real roots: 4 0 (4 ) 4(1)(3 9) 0 (4 ) 12 36 0 4 20 0 2 1 6 or 2 1 6 x xy x y x x x x x xy y x y x y y b ac y y y y y y y y                                        : 2 1 6 or 2 1 6 y y y     2(ii) 2 4 9 6 13 3 Asymptotes are 1 and 3. x xy x x x y x x           2 1 6  2 1 6

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2022 Chp 3 Graphing Techniques (Part 1) (Solutions to Tutorial and Assignment) 31 3. The curve C has equation y = 2x 2  x  1 x  1 . (i) Find the equations of the asymptotes of C. Vertical Asymptote: ݔ= −1 By long division: ݕ= 2ݔ− 3 + ଶ ௫ାଵ Therefore, oblique asymptote: ݕ= 2ݔ− 3

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2022 Chp 3 Graphing Techniques (Part 1) (Solutions to Tutorial and Assignment) 32 (ii) Draw a sketch of the curve C, making clear the main relevant features. (iii) Find the range of possible values of k such that the line y = k (x  1)  5 and the curve C do not intersect. ݕ= ݇(ݔ+ 1) − 5 will pass through the point of intersection of the two asymptotes: (-1, 5) It has gradient k. For the curve and the line to not intersect, ݇≤ 2

H2 Mathematics (9758) JC1 River Valley High School, Mathematics Department, 2022 Chp 3 Graphing Techniques (Part 1) (Solutions to Tutorial and Assignment) 33 (iv) Find the set of values of x for wh