RVHS H2 Math Transformations Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Transformations 1 Transformations 1. AJC/I/11 The graph of      −= 23 f xy has 2 stationary points at (8, -2) and (14, 2) and intersects the x- axis at 18 and 10 ,6 === xxx as shown in the diagram below. Sketch, on separate clearly labeled diagrams, the graphs of (a)      − = 23 f 1 xy ; [3] (b) )f( xy= . [3] Show clearly the asymptotes, stationary points and points of intersection with the axes in your diagrams. 2. VJC/I/6 A curve C1 has equation 2 2 2 2x a y a−= where a >1. Sketch C1, indicating the axial intercepts, asymptotes and stationary points, if any. [2] C1 undergoes a single transformation to become C2. Given that C2 has a line of symmetry 4y= and the point (4, 3) lies on C2, find a. [3] -4 (8, -2) x y 6 10 18 (14, 2)

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Transformations 2 3. EJC Prelim/2020/02/Q1 The diagram above shows the curve ( )f.yx= The curve cuts the x-axis at the point ( )1.5,0− and has a maximum point at ( )0,6 . The curve also has vertical asymptote 2x=− and horizontal asymptote 0.y= Stating the equations of the asymptotes and coordinates of any turning points and points where the curves cross the axes, if it is possible to do so, sketch on separate diagrams, the curves (i) ( )1 f12yx=− , [3] (ii) ( ) 1 fy x= . [3] 4. ACJC/2011/II/5 The curves 1C and 2C have equations 229 ( ) 9y x k= + − and 2 2 2 1x y k += respectively, where k is a real constant such that 3k  . (a) Describe a sequence of transformations which transforms the graph of 22 1xy−= to the graph of 1C . (b) (i) On the same diagram, sketch the graphs of 1C and 2C , stating clearly the coordinates of any points of intersection with the axes and the equations of any asymptotes. (ii) Find the range of values of the positive constant a, where ak , such that the equation ( ) 22 2 9 19 xkx a +−+= has two real roots. y x O ( )0,6

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Transformations 3 5. TJC/2013/I/7 (a) The diagram below shows the graph of y = f(x). The graph has a minimum point at ( )1,3 and intersects the axes at x = −2 and y = 4. The equations of the asymptotes are y = 2x and x = −1. On separate diagrams, sketch the graphs of (i) y = f (x), [2] (ii) 1 f ( )y x= , [2] In each case, give if possible, the equations of the asymptotes, the coordinates of the turning points and the coordinates of the points where the graph crosses the x- and y- axes. (b) A curve undergoes the transformations A, B and C in succession: A: a translation of 2 units in the negative direction of the x-axis. B: a scaling parallel to the x-axis by a factor 2. C: a translation of 1 unit in the positive direction of the y-axis. The equation of the resulting curve is 229 4yx x= + − + . Find the equation of the curve before the three transformations were effected. [3] y x 0 (1, 3) 4 x= −1 −2 y = 2x

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Transformations 4 6. AJC/2011/I/10 The sketch below shows the graph of y = f(x). The curve passes through the point A (1,0) and has a minimum point at B(0,2). The equation of the asymptotes are y = -2x+ 1 and 1 2x= . On separate diagrams, sketch the following graphs indicating the points corresponding to A, B and asymptotes where necessary. (i) 1f 2yx =+ [3] (ii) ( ) 1 fy x= [3] (iii) ( )f ' 2yx= [3] 7. NYJC/2011/II/4 (a) The diagrams below show the graphs of | f ( ) |yx= and f '( )yx= . Sketch the graph of f ( )yx= , stating the equations of any asymptotes and the coordinates of any axial intercepts and turning points. [3] Hence, find the range of values of k if there is exactly 1 real root to the equation f ( ) 0xk−= . [2] (b) The diagram below shows the graph of f (2 1)yx=− . The curve passes through the point A( − 1, 0) and B(1, 1− ). The asymptotes are x = 0 and y = 0 and 3y= . x = 3 y = 2 3 1 2 y = |f(x)| y = f (x) x y y x = 3 y = 0 x B(0,2) A(1,0) x y y = –2x+1

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Transformations 5 Sketch, on separate clearly labelled diagrams, the graphs of (i) f ( | 2 1|)yx= − − , [6*] (ii) f ( )yx= , describing the sequence of transformations involved. [5] Your sketch should show clearly the equations of any asymptotes and the coordinates of the points corresponding to A and B. 8. AJC/2013/I/7 The diagram shows the graph of . The curve has a minimum point at ( -1, 0) and a stationary point of inflexion at (1, -1). The asymptotes are 4y= , 0y= , 0x= and 2x= . Sketch the graph of f '( )yx= . [3] In each case, indicate clearly the coordinates of any points where the curve crosses the axes, turning points and equations of asymptotes whenever possible. Find the number of points of intersection between the curves ( ) = f 1yx − and ( )f'yx= . [3] ( )fyx= x = 0 A ( 1,0) B (1, 1) y = 3 x y

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Transformations 6 9. DHS/2015/I/5 The diagram shows the graph of f( ).yx= The graph has a minimum point at ( 1, 1)−− and a maximum point at ( 4, 7).−− It intersects the axes at 2, 1xx=− = and 2 3 .y −= The equations of the asymptotes are 2yx=− and 3.x=− (i) Sketch the graph of 1 ,f ( )y x= giving the coordinates of any stationary points, points of intersection with the axes and the equations of any asymptotes. [3] (ii) Solve the inequality ( )1f 0.x  [3] 10. SAJC/2012/II/3 Find the values of the constants A and B such that 𝑥2−4𝑥 (𝑥−2)2 = 𝐴 + 𝐵 (𝑥−2)2 for all values of x except 2x  . Hence state a sequence of transformations by which the graph of 𝑦 = 𝑥2−4𝑥 (𝑥−2)2 may be obtained from the graph of 2 1y x  . [4] x y 1 O

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Transformations 7 11. CJC/2012/I/6 The sketch below shows the graph of )f(xy= . The curve intersects the y-axis at the point      − 2 11,0 , has a maximum point at ( )ba, and a minimum point at ( )dc, . The equation of the asymptotes are 3=y , 1−=x and 2=x . On separate diagrams, sketch the following graphs indicating the points corresponding to the stationary points, axial intercepts and asymptotes where necessary. (i) f(| |)yx= [2] (ii) 1 f( )y x= [3] (iii) )('f xy= [2] 12. JJC/2015/I/8 The diagram below shows the curve with equation ( )fyx= which has vertical asymptotes xa= and xa=− where 0,a horizontal asymptote 0y= and a stationary point at ( )0, 1− . Sketch, on separate diagrams, the graphs of (a) ( ) 1 fy x= , [3] (b) ( )f'y x= , [3] indicating clearly the asymptotes and the axial intercepts if any. The graph of ( )fyx= undergoes in succession, the following transformations: A: A translation of 1 unit in the negative x-direction. B: A reflection in the x-axis. C: A scaling parallel to the y-axis by a factor of 2. The equation of the resulting curve is given by ( ) 2 1 2 1 y x =