DHS Graphs & Transformations 1 (9758) Topical Revision

1. Graphs and Transformations 1 1.1 Graphs and Transformations 1 Curve Sketching 1 MI/2009Prelim/I/5a The curve C has equation 4 23 − += x xy . (i) Write down the equations of all the asymptotes of C. [2] (ii) Sketch the curve C, indicating all turning points, asymptotes and axial intercepts, if any. [3] (iii) Find the set of values of k for which the line y = kx intersects C at 2 points. Leave your answers to 3 significant figures. [4] 2 IJC/2009Prelim/II/1 The curve C has equation 2 9 16 2 xxy x ++= + . (i) Find the equations of the asymptotes of C. [2] (ii) Find, leaving your answers in exact form, the range of values of k for which the equation 2 9 16 2 xxk x ++= + has no real roots. [4] (iii) Sketch C. Show, on your diagram, the equations of the asymptotes, the coordinates of the stationary points, and the coordinates of the points of intersection of C with the x- and y- axes. [3] 3 DHS/2009Prelim/I/3(b)part The curve C has the equation 21 xAy x=+ − , where A is a non-zero constant. (i) State the equations of the asymptotes of C. [2] (ii) Find the range of values of A such that C does not have any stationary points. [2] 4 MJC/2011Promo/2 A sketch of the curve 22x ax by xc ++= + , where a, b and c are constants, is shown, not to scale, in the diagram. Given that the asymptotes of the graph intersect at the point ( 1,1)− and the curve passes through the point ( )0,5 , (i) write down the value of c and find the values of a and b. [4] (ii) hence determine the number of roots of the equation 2225 x ax bx xc +++= + . [1] x y O

1. Graphs and Transformations 2 5 NYJC Promo 9758/2018/Q4 Sketch the graph of 2 2 4 xy x −= +   in each of the following two cases: (i) 0 , [2] (ii) 0 . [3] Your sketches should include clear labelling of asymptotes, the exact coordinates of any points where the curve crosses the x- and y- axes intercepts and stationary point(s). By using your graph in (ii) and considering a suitable graph whose cartesian equation is to be stated, find the positive value of h such that the equation 222 2 4 1xx hx h  − =+ +   where 0 , has only one real root. State the value of the real root for this value of h. [3] 6 NJC/2015Promo/5 A curve C has equation (i) Sketch C, labelling the equations of the asymptotes and the coordinates of the turning points. [3] (ii) State the range of values of x for which C is concave upwards. [1] (iii) Find the range of values of x for which [2] (iv) State the range of values of m such that has exactly two real roots. [1] 7 RI/2010Promo/3 Sketch the graph of 221 1 xy x += − , showing clearly the coordinates of any turning point(s) and axial intercept(s), and the equation(s) of any asymptote(s). Hence, state the range of values for a such that there are no real solutions to the equation ( ) ( ) 22 1 1 cosx a x bx c+ = − + , 1x , for all values of b and c. [6] 8 JJC/2014Promo/6 The curve 1C has equation 22 6 16 0x y y− + + = The curve 2C has equation 22 15 ,5 x p y q−−    =    −  where p and q are constants. (i) Sketch 1C , stating the coordinates of any points of intersection with the x- and y-axes and the equations of any asymptotes. [3] (ii) The line y mx c=+ intersects 1C twice. State the range of values of m. [2] (iii) Given that 1C and 2C have the same asymptotes, determine the values of p and q .[2] (iv) On the same diagram in part (i), sketch 2C . [1] (v) Let n be the number of points of intersection of the curve ( ) 222 3 rx y+ − = where 0r  , with the curve 2C . Determine the possible values of n. [1] 2 48 .2 xxy x −+= − 2 48 5.2 xx x −+ − 2 48 ( 2)2 xx mxx −+ =−−

1. Graphs and Transformations 3 9 EJC Promo 9758/2018/Q9 The curve C has equation 2 4x axy xb +−= + , where a and b are constants, and x b− . It is given that the asymptotes of C are 2yx=− and 1x= . (i) Find the values of a and b. Hence sketch C, stating the coordinates of the point(s) where the curve crosses the axes, and the equations of the asymptotes. [5] (ii) By drawing a suitable curve on the same diagram in part (i), find the number of real distinct roots of the equation 22 2 42 20 21 0 x axxx xb  +−− − + =  + . [3] 10 ACJC/2011MYE JC1/6 (a) The diagram below shows a sketch of the graph of 2 8xxy xk += + , where k is a constant. The graph has stationary points at (−2, 4) and (4, 16). It also passes through the point ( )8,0− and has a vertical asymptote of 1x= . (i) State the value of k and the equation of the oblique asymptote. [2] (ii) Sketch the graph of 2 8 xky xx += + , labelling all asymptotes, axial intercepts and turning points, if any. [4] (b) The curve C is given by the equation ( ) ( ) 22 21 10x y r− + − = , where r is a positive integer. Write down the smallest value of r such that C intersects the graph of 2 8xxy xk += + at four points. [1] y O x

1. Graphs and Transformations 4 11 NYJC/2010Promo/10 The curve C has equation and a is a non-zero constant. (i) Given that the oblique asymptote of C is y = x + 2, find the value of a. [1] (ii) Find the set of values of a if C has two turning points. [3] (iii) Using the value of a found in part (i) and using an algebraic method, show that C cannot exist for where values of are to be determined in exact form. [3] (iv) Hence sketch the graph of C, showing clearly the equations of the asymptotes and coordinates of the stationary points. [3] 12 NJC/2011MYE JC1/10 The equation of a curve C is given by 2()px qy xr += + , where p, q and r are non-negative real constants. It is also given that C has a vertical asymptote 0x= and an oblique asymptote of 9yx =+ , where λ is a positive real constant. (i) State the value of r and show that 6q = . [4] (ii) Determine the coordinates and the corresponding nature of the stationary point(s). Express your answer in terms of  . [4] For 18 = , (iii) sketch C. Label clearly the coordinates of the stationary point(s) and axial intercept(s) (if any). [3] The curve D is defined parametrically by sin 1xa =+ , cos 18ya =+ , where , a  . Find the Cartesian equation of D in terms of a. [2] Deduce the least integer value of a for which C and D intersect each other more than once. [1] 13 NJC Promo 9758/2021/Q6 A curve 1C has equation 22 2 100xy+= and a curve 2C has parametric equation 222e 4e , 3e e .t t t txy −−= − = + (i) On the same diagram, sketch 1C and 2,C labelling the coordinates of the points where both curves cross the x- and y-axes. [5] (ii) Show that 2C has a Cartesian equation of the form ( ) ( ) 2 ax by cx dy k+ + = for some integer constants a, b, c, d and k to be determined. [3]  y  x2  a x  a , x  a y1 y y2 y1 and y2

1. Graphs and Transformations 5 14 NJC/2014Promo/12 (a) The curve 1C has parametric equations 2x t t=+ , 24y t t=− , 11 t−   . (i) Sketch 1C , labelling the coordinates of the end -points and the axial intercepts (if any) of this curve. [2] (ii) Calculate the gradient of the curve 1C at the point where 5 16x= . [3] (iii) The curve 2C is defined parametrically by the equations 2x t t=+ , 24y t t=− , t . Find a Cartesian equation of 2C . [2] (b) The curve 3C has equation 1 1 xy x −= + . The curve 4C has equation 22 120 5 xy−= . Sketch 3C and 4C on the same diagram, stating the exact coordinates of any points of intersection with the axes and the equations of any asymptotes. [4] Hence find the number of solutions to the equation ( ) ( ) 2 2 2 41 20 1 xx x − = + − . [2] 15 NJC/2010Promo/6 Sketch the graph of ( ) 2y xx = − , where μ is a p