DHS Graphs & Transformations 1 (9758) (Revision Solutions)

Y5 Topical Revision Package 1 1. Graphs and Transformations (Solutions) 1 1.1 Graphs and Transformations 1 (Suggested Solutions) Curve Sketching 1(i) Asymptotes : y = 3, x = 4 (ii) (iii) 32 4 xkx x += − 2 4 3 2 0kx kx x− − − = 2 ( 4 3) ( 2) 0kx k x+ − − + − = ---(*) For 0,k = (*) is a linear equatio and not possible to have two roots. For 0,k  (*) is a quadratic equation and we wand discriminant > 0 2 2 2 ( 4 3) 4( )( 2) 0 16 24 9 8 0 16 32 9 0 kk k k k kk − − − −  + + +  + +  1.66 0.339 (3 s.f.)k or k −  −  set of values = : 1.66 or 0.339, 0k k k k   −  −  2(i) Asymptotes : 7yx=+ and 2x =− (ii) 2 2 2 2 2 2 Discriminant 0 (9 ) 8)( 5 8) 0 5 8 5 9 16 2 ( 2) 9 16 (9 ) (16 2 ) 0 4(16 2 ) 0 10 17 0 ( 8 5) 8 0 (5 xxk x k x x x x k x k k k k k k k k k ++= + + = + + + − + − = − −  −+  − − +  −  − −  −  − + y = 3 y x O x = 4 -2/3 -0.5

Y5 Topical Revision Package 1 1. Graphs and Transformations (Solutions) 2 (iii) 3(i) Asymptotes: and 12 xyx== (ii) ( ) 2 d1 0d2 1 yA x x = − = − ( ) 2 12xA−= Therefore, for C not to have stationary points, A < 0. 4(i) 1c = When 0, 5, 1x y c= = = , 5 1 b= 5b = 227 2 ( 2) 1 x ax b ay x a x c x + + −= = + − +++ When 1, 1xy= − = , 22 1 2( 1) 2 5 5, 5 y x a a a ab = + − = − + − =  = = (ii) 2225 x ax bx xc +++= + One root. 5(i) y (0,4) (–2,0) O (2,0) x –1 y = –1 y x O

Y5 Topical Revision Package 1 1. Graphs and Transformations (Solutions) 3 (ii) ( ) 222 2 2 2 22 2 2 4 1 4 11x x x x hx h h x    −− = +  + =   ++    −      So insert the graph ( ) 22 2 2 1xy h+= − which is an Ellipse with centre O. For only one real root, the ellipse must meet the graph in (ii) at exactly one point. Hence h = 4 and the corresponding root is x = 0. 6(i) (ii) 2x  (iii) From GC : 2 48 5 3 or 62 xx xx −+ =  =−

Y5 Topical Revision Package 1 1. Graphs and Transformations (Solutions) 4 the range of values of x for which 2 48 52 xx x −+ − is 6x  or 23 x . (iv) 1m  7 For no real solutions to the equation 22 1 ( 1)cos( ), 1x a x bx c x+ = − +  range of values for a is 0.899 0.899a−   8(i) Note that 22 6 16 0x yy− + + = ( )( ) ( ) ( ) 22 2 2 2 2 22 9 16 2 30 3 3 5 ( 0) 155 x x x y y y −+ −= − −= − − − = − The equations of the asymptotes are 5 53 3 or 3y x y x y x− =   = + = − + . O ( 0.225, 0.899)−−

Y5 Topical Revision Package 1 1. Graphs and Transformations (Solutions) 5 (ii) From the graph, it follows that 1m − or 1m  . (iii) The other hyperbola with the same asymptotes are ( ) ( ) ( ) 22 22 22 25 0 3 155 3xy yx = −− −− − = Therefore, 0, 3pq== . (iv) (v) The