RI C8A Graphing Techniques Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________ Tutorial 8A: Graphing Techniques Page 1 of 4 Tutorial 8A: Graphing Techniques Section A (Basic Questions) 1 The equation of a curve is 32236 4yx x x  . Find ddyx and hence show that y is strictly increasing for all real values of x. [ 266 6xx ] 2 For each of the curves with the following equations, find ddyx and hence find the exact coordinates of the stationary points. Determin e the nature of each stationary point using the first or second derivative test. (a) y = x(ln x)2 , where x > 0, (b) 2 2 1 1 xxy xx  , [ (a) (1,0) min. (e2, 4e2) max (b) (–1, 3) max, (1, 13 ) min ] 3 Sketch, on separate diagrams, the graphs of (a) 21 xy x  , (b) 621yx x  , giving the equations of any asymptotes and the coordinates of any points of intersection with the x- and y-axes. Section B (Discussion Questions) 1 A function f is defined by  23f 35xx x  . (i) Show that f x can be written in the form 35ba x  , where a and b are constants to be determined. [1] (ii) Sketch the graph of  fyx . [3] (iii) Hence, state the range of f. [2] [ (i) 23a ; 13b ]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8A: Graphing Techniques Page 2 of 4 2 It is given that f( ) ax bx cx d  , for non-zero constants a, b, c and d. (i) Given that 0ad bc , show by differentiation that the graph of =f( )yx has no turning points. (ii) What can be said about the graph of f( )yx when 0ad bc ? (iii) Deduce from part (i) that the graph of 3721xy x  has a positive gradient at all points of the graph. (iv) S ketch the graph of 37 21 xy x  , including the coordinates of the points where the graph cross the axes and the equations of any asymptotes. 3 It is given that 2 1 1 x xy x  , ,1x x . (i) Without using a calculator, find the set of values that y can take. (ii) Sketch the graph of 2 1 1 x xy x  , showing clearly any asymptotes, intercepts and turning points. (iii) The functions f is defined as follows: 2 1f: 1x xx x , ,1x x . Find the range of f. (iv) Explain whether 1f exists. [ ,3 2 3(] [ 3 2 3 , ) .   ] 4 The curve C has equation 2x ax by x c  where a, b and c are constants. The line 1x  is an asymptote to C and the range of values that y can take is given by 0 or 4yy . (i) State the value of c and show that 4 and 4.ab [4] (ii) Sketch C indicating clearly the equations of the asymptotes and coordinates of the turning points and axial intercepts. [3] (iii) State the coordinates of the point of intersection of the asymptotes. Hence state the range of values of k such that the line  12yk x  cuts C at two distinct points. [2] [ (i) c = 1 (iii) (1 , 2 ) , 1k ]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8A: Graphing Techniques Page 3 of 4 Conic Sections 5 Sketch, on separate diagrams, the graphs of (a) 2 1(1 ) 1 2yx (b) 22(3 )(4 )5xy  , (c) 22 22 123xy (d) 22 149yx , (e) 22 42 2 0 0xx yy . 6 The curve C1 has equation 22xy x  . The curve C2 has equation 22 163xy . (i) Sketch C1 and C2 on the same diagram, stating the exact coordinates of any points of intersection with the axes and the equations of any asymptotes. (ii) Show algebraically that the x-coordinates of the points of intersection of C1 and C2 satisfy the equation       22 2 22 2 6x xx   . (iii) Use your calculator to find these x-coordinates. [ (iii) x = –0.515 or 2.45] 7 It is given that 2 21f o r ,f( ) 0 for 2 , x axax a ax a        and that f( 3 ) f( )x ax for all real values of x, where a is a real constant. Sketch the graph of f( )yx for 46ax a  . 8 (i) Sketch the curve with equation 2 21 4 x xy 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 4xxkx k x     has at least one real root. [3] [ (ii) 10k  ]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8A: Graphing Techniques Page 4 of 4 Parametric Equations 9 Sketch, on separate diagrams, the curves de fined by the following pair of parametric equations: (a) 23,f o r 0xt yt t  , (b) 23cos , sin , for 0 2xt y t t    , (c) 22(1 ) , 1xt y t  for t  , (d) 2 , f o r , 111ttx yt ttt    , (e) 11, f o r , 0xt yt t ttt     . Find the equations of the curves in (b), (d) and (e) in Cartesian form. State the equations of any asymptotes on your sketch of the curves for (b), (d) and (e). 10 A curve D has parametric equations cos , 1 cos , for 0 2 .xt 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 give in exact form the coordinates of the maximum point on the curve [4] 11 A curve C has parametric equations sin , 1 cos ,xy   where 02 .  (i) Show that d1 cotd2yx   and find the gradient of C at the point where .  What can be said about the tangents to C as 0 and 2  ? [5] (ii) Sketch C, showing clearly the features of the curve at the points where 0, and 2 .  [3] (iii) A point P on C has parameter p, where 120. p  Show that the normal to C at P crosses the x-axis at the point with coordinates (p, 0). [3]