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 . Therefore the oblique asymptote is yx ii) x2 4 = sec2 , y2 = tan2  tan 2  + 1 = sec2  y2 + 1 = x2 4 Intercepts : (–2, 0) , (2, 0)   1 y = x 2 –2 y = y = –

2020 SAJC JC1 H2 Maths Tutorial 4: Graphing Techniques Page 16 of 18 4. [N2016/H2 Maths/2/3 (part)] A curve D has parametric equations cos , 1 cos , for π 20xt t y t t   . Sketch the graph of D. Give in exact form the coordinates of the points where D meets the x- axis, and also give in exact form the coordinates of the maximum point on the curve. 4 ttx cos , ty cos1 (i) On the x-axis, 0cos1  t  1cos t  0t or 2t When ,0t 110 x When ,2t 12  x Hence the coordinates of points on the x-axis are )0,1( and )0,12(  t t dt dx dt dy dx dy sin1 sin  = 0 for maximum or minimum  0sin t   2,,0 ort Since 0t and 2t correspond to the two points on the x-axis, the maximum point occurs when t . When t , 1)1(cos  x , 2)1(1cos1  y Hence the coordinates of the maximum point are  2,1 . The following diagram shows the graph of D:

2020 SAJC JC1 H2 Maths Tutorial 4: Graphing Techniques Page 17 of 18 5. A sketch of the curve dx cbxaxy   2 , where cba ,, and d are constants, is shown, not to scale in the diagram. The equations of the asymptotes, also shown in the diagram, are 2x and 32  xy . (i) Write down the value of d . [1] (ii) Find the value of a and show that 7b . [3] (iii) Given that the curve has a stationary point where 1x , find the value of c and the x-coordinates of the other stationary point. [4] (iv) Copy the above sketch and, by drawing a sketch of another suitable curve in the same diagram, find the number of real roots for the equation 432278 2xxx x  . [3] (i) 2x is the vertical asymptote implies the denominator is  2x . Therefore, 2d . (ii) 2 2 ax bx cy x   By long division   dx abcabaxy   222 As x ,  abaxy 2 .  22 3ax b a x  Comparing coefficients: 2a  7 322 32    b b ab (iii)  2 632 2 227232     x cx x cxy   2 6d 2d 2 cy x x   When 1x , d 0d y x  Hence     22 86d2 22d 22 y x xx    When d 0d y x    2 2 202 2 21 21 o r 1 1 or 3 x x x x        The other stationary point is at 3x . 32  xy 2x

2020 SAJC JC1 H2 Maths Tutorial 4: Graphing Techniques Page 18 of 18     8 2 1 6 21 620 2 2     c c c (iv)      432 22 22 2 2 278 2 27 8 2 27 8 12 27 8 1 2 xxx x xx x x xx x x xx x x         By sketching the curve 2 1 xy  on the same diagram as dx cbxaxy   2 , we can find the number of real roots by counting the number of intersection points. From the graph, we see two intersection points. Hence there are 2 solutions. 32  xy 2x x