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