Graphing Mastery Solutions

2020 SAJC JC1 H2 Maths Tutorial 4: Graphing Techniques Page 12 of 18 Mastery Questions 1. [CJC/17/Promos/Q3] (i) Sketch the curve with equation 2 21 4 xxy x   , stating the equations of any asymptotes, the coordinates of any turning points and any points of intersection with the axes. [3] (ii) By drawing a suitable graph on the same diagram in part (i), find the range of values of k, where 0k  , such that  22 222 2111 0 4 xxkx k x      has at least one real root. [3] [Ans: (ii) 10k  ] 1(i) y x O (i)

2020 SAJC JC1 H2 Maths Tutorial 4: Graphing Techniques Page 13 of 18 1(ii) Draw    2222 11 0kx y k  or   2 2 2 1011 yx k   on the same diagram, ellipse centre 1, 10 and horizontal axis fixed 1 and vertical axis k variable. For at least one root, 0,0.25 2. (Tutors can highlight this question for students to try as self-practice) A curve is defined parametrically by the equations, t tx  1 ; t ty  1 2 t ℝ, 1t  (i) Find the Cartesian equation of the curve, expressing your answer in the form fyx . (ii) Sketch the curve. Label your graph clearly, indicating any asymptote(s) and stationary point(s). (iii) By sketching another suitable graph on the same diagram as in (ii), determine the number of real roots of the equation 32f( ) 6xx x  . [Ans : (i) Cartesian Equation : x xy  1 2 (iii) Number of real roots = 2] y x O (i) (ii)

2020 SAJC JC1 H2 Maths Tutorial 4: Graphing Techniques Page 14 of 18 (i)   2 2 22 2 2 2 ,11 1 1 1 1111 1 1 ttxy tt xx tt xt x x x x xxy x xx x xy x               (ii) To find the equation of the asymptotes of the curve 2 1 11 1 ,1 xy x yx x x yx            2 2 d1 1d( 1 ) d1 01 0 o r 2d( 1 ) When 0, 0 min . When 2, 4 max . y xx y xxxx xy xy          Equation of the asymptotes: 1 xy and 1x (iii) 32 32f( ) 6 f( ) 6xx x x x x      . We need to find the number of intersections between the curve representing 623  xxy and f( )y x Cartesian Equation of the curve

2020 SAJC JC1 H2 Maths Tutorial 4: Graphing Techniques Page 15 of 18 From the sketch, there are 2 intersection points between the graphs, hence, 2 real roots to the equation. 3. [CJC/06/Promo/Q8] Find the cartesian equations and coordinates of the intersections of the following curves with the x and y–axes (if any): (i) x = t 2, y = t4 + 1 [2] (ii) x = – 2 sec , y = tan  [2] On separate diagrams, sketch the curves in (i) and (ii), indicating clearly the equation(s) of any asymptotes. [4] [Ans : i) y = x2 + 1 ; (0, 1), ii)    2 2 1 , 2, 0 , 2, 04 xy   ] i) y = x2 + 1 ; (0, 1) As x , 2yxx Note: 2xx  ! Since 2xt , it means 0x , hence xx . The