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.

1) Investigating y = f(x – p) (4, 2) (0, 2) Horizontal shift (–4) (7, 2) Horizontal shift (+3)

2) Investigating y = f(x) + q (4, 2) (4, 5) Vertical shift (+ 3) (4, –3) Vertical shift (–3)

3) Investigating y = af(x) (4, 2) (4, 4) (vertical stretch by a factor of 2 or (x2)) (4, 1) vertical compression by a factor of ½ or x

3) Investigating y = af(x) (4, 2) (4, –2) (Reflection in the y-axis) (4, – 4) Reflection in the y-axis and a vertical stretch by a factor of 2

4) Investigating y = af(x – p) + q (4, 2) (–1, 1) Horizontal shift (-5) vertical stretch by a factor of 2 vertical shift (–3)

5) Investigating y = f(kx) (4, 2) (2, 2) horizontal compression by a factor of :

vertical transformations y = af [k(x – p)] + q Horizontal transformations – apply opposite operations!! Add p units to the x-coordinate Multiply the x-coordinate by y = af [k(x – p)] + q vertical transformations - Vertical stretch by a factor of a (if a > 1) - Vertical compression by a factor of a (if 0< a < 1) - Vertical shift of q units

If the graph of f is given as y = x3, describe the transformations that you would apply to obtain the following: a) y = (x + 1)3 Horizontal shift (-1) b) y = x3 – 2 Vertical shift (-2) vertical stretch ×2 and reflection in the x-axis c) y = – 2x3 horizontal compression by a factor of 1/3 and a reflection in the y-axis, d) y = (– 3x)3 e) y = 2(x – 5)3 right 5 and a vertical stretch by a factor of 2

Example: Given the ordered pair (3, 5) belongs to g. List the ordered pairs that correspond to: a) y = 2g(x) (3, 10) Vertical stretch × 2 b) y = g(x) + 4 (3, 9) Vertical shift (+ 4) c) y = g(x – 3) (6, 5) Horizontal shift (+ 3) Horizontal compression by a factor of ½ and a d) y = – g(2x) (1.5, – 5) reflection in the x-axis

If the graph of f is given as , describe the transformations that you would apply to obtain the following: Horizontal shift (+ 3) Vertical shift (+4) horizontal compression (× 0.5)) reflection in the x-axis

Describe the transformations that you would apply to f(x) to obtain the following: a) f(2x + 6) Horizontal shift (- 3) f [2(x + 3)] Horizontal compression by a factor of 0.5 b) f(– 3x + 12) + 5 Horizontal shift (- 3) Horizontal compression by a factor of 1/3. f[–3(x – 4)] + 5 Reflection in y-axis Vertical shift (+5)

Describe the transformation on f (x): Vertical stretch (x 2) Horizonal shift (-2), reflection in the x-axis x4 (x – 3)4 Horizonal shift (+3) Horizontal compression (x 0.5) reflection in the x-axis Horizontal shift (-1) Vertical shift (+4) x2 (x + 1)2 + 4 horizontal stretch (x2), vertical compression (x0.5).

(x,y) (0.5x – 3,– y+1) y = –f (2x+6) + 1 y = –f [2(x+3)] + 1 The graph of y = f(x) is given Sketch the graph of y = –f (2x+6) + 1 y = –f [2(x+3)] + 1 (x,y) (0.5x – 3,– y+1)

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

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

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

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