6. Transformations Solutions 2023 (RVHS)

River Valley High School, Mathematics Department 2023 Transformations T1 Solutions Transformation 1(a) (b) (-1, -2) 0 -2 -6 (-4, 2) -4 3 y = -1/4 (8, -1/2) (14, 1/2) 6 10 18 x y 1 f 3 2 y x  

River Valley High School, Mathematics Department 2023 Transformations T2 2(a) 3

River Valley High School, Mathematics Department 2023 Transformations T3 4. Suggested solution Suggested solution 5(a) 1C : 229 ( ) 9y x k   2 2 2 () 13 xk y   Sequence of transformations: EITHER (1) Scaling parallel to the x-axis by factor 3 (2) Translation in the negative x-axis direction by k units OR (1) Translation in the negative x-axis direction by 3 k units (2) Scaling parallel to the x-axis by factor 3 y x O y x O (1, 3) (2.5, 0) (0, 1/6)

River Valley High School, Mathematics Department 2023 Transformations T4 (b)(i) 1C : 2 2 2 () 13 xk y  2C : 2 2 2 1x yk  Intercepts:  3 ,0k  3 ,0k Intercepts:  ,0k ,  0, 1 2 0, 1 9 k  Asymptotes: () 3 xky  i.e. ,33 x k x kyy    b(ii) 22 2 ( ) 9 19 x x k a  22 2 () 119 x k x a   ------ (*) Sketch the graph of 1C : 2 2 2 () 13 xk y  (i.e. 2 2 () 19 xky  ) on the same axes as graph of 2C : 2 2 2 1x ya  (i.e. 2 2 21 xy a ). For equation (*) to have 4 real roots, the two graphs intersect at 4 points. Since a is a positive constant, 3ak . O x y

River Valley High School, Mathematics Department 2023 Transformations T5 6(a) (i) (ii) (b) 229 4yx x    228 4yx x      212 2 8 4 8 2 4 2y x x xx       114( 2) 8 4 ( 2) 2y x x xx       The equation of the curve before the transformations were effected is 14yx x x= 2 y x 0 1 (1, ) y x 0 1 x= 1 y = 2 y = f (x) replace y by y + 1 replace x by 2x replace x by (x2)

River Valley High School, Mathematics Department 2023 Transformations T6 7 (i) 1 2y f x   (ii)   1y fx (iii)  '2y f x y B(- , 2) x y = 2x + 2 y = -2x O x B’’(0, ) y O B’’’(0,0) y x y = -4 O

River Valley High School, Mathematics Department 2023 Transformations T7 8a range of values of k is 2 or 3kk  . b Original solutoin was incorrect. Question was a lot more challenging than intended. Side working: replace with / 2 replace with 1 Scale 2 units // to Translate 1 unit in dirnf (2 1) f ( 1) f ( )x x x x xxy x y x y x            replace with replace with Reflect abt axis Transformf ( ) f ( ) f ( ) xxxx yxy x y x y x          replace with 1 replace with 2 Translate 1 units in dirn Scale factor o f 0.5 // f ( ) f ( 1) f ( 2 1)x x x x xxy x y x y x             y = 0 y = -2 x = 3 (2,0) (0,3) (-1,0) O x y

River Valley High School, Mathematics Department 2023 Transformations T8 (ii) Scale by factor 2 parallel to the x- axis followed by translation by -1 unit in the direction of x-axis. 9(i) (ii) From the given graph, f ( ) 0 for 3 or 2 1x x x     Replacing x by 1 ,x 1 1 1 13 0 or 2 1 or 132 x x xxx          x y A’ (  3,0) B’ (1,  1) y = 3 x y x = 1 f ( )yx y y = 3 x O y y = 2 x O y =  1

River Valley High School, Mathematics Department 2023 Transformations T9 10a b Let   22 44h 4 4 1 (2 1)x x x x    Before C,       22 44h 1221 yx xx      Let     2 4p 12 x x   Before B,   22 44p 2 112 2 xy xx        Let     2 4g 1 x x   Before A,       2 2 44f =g 1 11 y x x xx      Let   2 4h 4 4 1x xx  . Before C,       2 2 44h 4 4 14 4 1 yx xxxx         Let   2 4p 4 4 1x xx  Before B, 2 2 44p 2 2 1 4 4 122 xy xxxx               Let   2 4g 21x xx  Before A,         2 2 44f =g 1 1 2 1 1 y x x xxx         O (3, 1/3) 1

River Valley High School, Mathematics Department 2023 Transformations T10 11 From graph, there are 2 points of intersection. Thus, there are 2 solutions. 12(i)