DHS Graphs & Transformations 2 (9758) (Revision Solutions)

Y5 Topical Revision Package 1 1.2 Graphs and Transformations 2 (solutions) 1 1.2 Graphs and Transformations 2 (Suggested Solutions) Transformation of Curves 1 (i) Range of values of x for which the graph of f ( )x (a) is strictly increasing : or x a x b (b) concaves upwards: xh (ii) Maximum point at xa= Minimum point at xb= 2 (i) Maximum point at x = a Minimum point at x = c Stationary point of inflexion at x = e (ii) 3(i) f ( )y x a=+ 3(ii) 1 f( )y x= x 0 a c e y = f(x) x y B A (-a,2a) y x

Y5 Topical Revision Package 1 1.2 Graphs and Transformations 2 (solutions) 2 3(iii) f '( )yx= 4(i) y = f(a-x) 4(ii) 1 f( )y x= 5ai) Method 1: ( ) 2 2 2 1x y k+ + = (1): Translation by 2k units in the negative y- direction; (2): Scale parallel to the y-axis by factor 1/2. OR Method 2: ( ) 2 21+ + =x y k (1): Scale parallel to the y-axis by factor ½ (2): Translation by k units in the negative y- direction. 5aii 6(ai) y = f(x+1) 6(aii) f (1 | |)yx=− A B y x X ( ) 2 4 3 1+ + =xy

Y5 Topical Revision Package 1 1.2 Graphs and Transformations 2 (solutions) 3 6(bi) Equation of ellipse: 22 22 ( ) ( ) 1 (2 ) x p y p pp −− += 6(bii) A : Scale the resulting graph parallel to the y-axis by a scale factor of 2. B : Translate the resulting graph in the negative y- direction by p units (positive y-direction by p− units) Alternatively, A’ : Translate the resulting graph in the negative y- direction by 2 p units (positive y-direction by 2 p− units) B’: Scale the resulting graph parallel to the y-axis by a scale factor of 2. 7(a) 1 f( )y x= 8(i) 1 f( )y x= 8(ii) ( )d fdyx x=  9(i) 9(ii)

Y5 Topical Revision Package 1 1.2 Graphs and Transformations 2 (solutions) 4 9(iii) 9(iv) Translation parallel to the 𝑦-axis by 2 units 9(v) ( ) 224 2 4xy+ − = 10a Method 1: ( ) 2 221 149 y x− −= (1): Translate 1 unit in the positive y-direction; (2): Scale parallel to the y-axis by factor 1/2. Method 2: 2 2 12 2 149 y x  − −= (1): Scale parallel to the y-axis by factor 1/2; (2): Translate 1/2 unit in the positive y-direction. 10(aii) 10(bi) 10(bii) 11a (1) Translate 3− units in the direction of x-axis, followed by (2) Scale parallel to the x-axis with a scale factor of 1 2 .

Y5 Topical Revision Package 1 1.2 Graphs and Transformations 2 (solutions) 5 11(bii) 1 f( )y x= 11(biii) ( )y f x=− 12 2 7 2 12 5 2 5 xy xx += = +++ Step 1 Step 2 Step 3 Step 4 1 1 1 5 2 5 22 12 5 2 5 x x x xx ⎯⎯⎯ → ⎯⎯⎯ →++ ⎯⎯⎯ → ⎯⎯⎯ → +++ Steps: 1. Translate 5 units in the negative x-direction. 2. Scale parallel to the x-axis by factor 1 2 . 3. Scale parallel to the y-axis by factor 2. 4. Translate 1 unit in the positive y-direction. 12 cont’d two real roots

Y5 Topical Revision Package 1 1.2 Graphs and Transformations 2 (solutions) 6 13(i) 1 f( )y x= 13(ii) y = f ’(x) 14(i) y = f(−2x) 14(ii) y = f(2−2x) 14(iii) ( )1 f22=yx D(3, -1.5) 1 2y=− B(0, 1) C(1, 0) A(–1, 0) y x ( )1 f22yx= B(1, 2) A(2, 0) C(0, 0) D(–2, –3) y x y = –1 ( )f 2 2yx=− y x y = –1 B(0, 2) A(1, 0) C(–1, 0) D(–3, –3) ( )f2yx=− 7xa= 1y=− 2.5 7a 1 ( )2.5, 2− ( ) 1 fy x= 3x= 1.5x=