© The Visual Classroom 3.7 Transformation of Functions Given y = f(x), we will investigate the function y = af [k(x – p)] + q for different values of a, k, p and q.

© The Visual Classroom 1) Investigating y = f(x – p) (4, 2) (0, 2) (7, 2) (left 4 units) (right 3 units)

© The Visual Classroom 2) Investigating y = f(x) + q (4, 2) (4, 5) (4, –3) (up 3 units) (down 5 units)

© The Visual Classroom 3) Investigating y = af(x) (4, 2) (4, 4) (4, 1) (vertical stretch by 2) (vertical stretch by one-half)

© The Visual Classroom 3) Investigating y = af(x) (4, 2) (4, –2) (4, – 4) (inverted) (inverted stretch)

© The Visual Classroom 4) Investigating y = af(x – p) + q (4, 2) (–1, 1) (–1, 2) (–1, 4) left 5 × 2 down 3

© The Visual Classroom 5) Investigating y = f(kx) (4, 2) (2, 2) horizontal stretch by a factor of :

© The Visual Classroom y = af [k(x – p)] + q Inverse transformations - Shift right p units - Horizontal stretch by a factor of y = af [k(x – p)] + q vertical transformations - Vertical stretch by a factor of a - Vertical shift by a factor of q

© The Visual Classroom If the graph of f is given as y = x 3, describe the transformations that you would apply to obtain the following: a) y = (x + 1) 3 b) y = x 3 – 2 c) y = – 2x 3 d) y = (– 3x) 3 e) y = 2(x – 5) 3 left 1 down 2 vertical stretch ×2 and reflection in the x-axis horizontal stretch by a factor of 1/3 and a reflection in the y-axis, right 5 and a vertical stretch by a factor of 2

© The Visual Classroom Example: Given the ordered pair (3, 5) belongs to g. List the ordered pairs that correspond to: a) y = 2g(x) b) y = g(x) + 4 c) y = g(x – 3) d) y = – g(2x) (3, 10) (3, 9) (6, 5) (1.5, – 5) Vertical stretch × 2 Up 4 Right 3 Horizontal stretch 1/2 Reflect in the x-axis

© The Visual Classroom If the graph of f is given as, describe the transformations that you would apply to obtain the following: right 3 up 4 horizontal stretch ×1/2 reflection in the x-axis

© The Visual Classroom Describe the transformations that you would apply to f(x) to obtain the following: a) f(2x + 6) f [2(x + 3)] b) f(– 3x + 12) + 5 f[–3(x – 4)] + 5 Left 3 Horizontal stretch by a factor of 1/2. Horizontal stretch by a factor of 1/3. Right 4 Reflection in y-axis Up 5

© The Visual Classroom Describe the transformation on f (x): f(x)f(x) x3x3 2x32x3 x4x4 (x – 3) 4 x2x2 (x + 1) Vertical stretch Left 2, reflection in the x-axis Right 3 Horizontal stretch, inverted Left 1, up 4 Horizontal stretch (2), vertical stretch (0.5).

© The Visual Classroom The graph of y = f(x) is given Sketch the graph of y = –f (2x+6) + 1 Input output diagram × 2× –1 f add 3 x +1 y left 3 ÷ 2 reflect in x-axis up 1 y = –f [2(x+3)] + 1

© The Visual Classroom y = –f [2(x+ 3)] + 1 × 2× –1 f add 3 x +1 y left 3 ÷ 2 reflect in x-axis up 1 P(–3, –2) P Q R S (–6, –2) (–3, –2) (–3, 2) P´(–3, 3) left 3 ÷ 2 reflect in x-axis up 1 P´

© The Visual Classroom y = –f [2(x+ 3)] + 1 × 2× –1 f add 3 x +1 y left 3 ÷ 2 reflect in x-axis up 1 Q(0, 2 ) P Q R S (–3, 2 ) (–1.5, 2) (–1.5, –2) Q´(–1.5, –1) left 3 ÷ 2 reflect in x-axis up 1 P´ Q´

© The Visual Classroom y = –f [2(x+ 3)] + 1 × 2× –1 f add 3 x +1 y left 3 ÷ 2 reflect in x-axis up 1 R(3, 2) P Q R S (0, 2) (0, –2) R´(0, –1) left 3 ÷ 2 reflect in x-axis up 1 P´ Q´ R´

© The Visual Classroom y = –f [2(x+ 3)] + 1 × 2× –1 f add 3 x +1 y left 3 ÷ 2 reflect in x-axis up 1 S(4, 0) P Q R S (1, 0) (0.5, 0) S´(0.5, 1) left 3 ÷ 2 reflect in x-axis up 1 P´ Q´ R´ S´