1. 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.

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

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

4. 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

5. 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

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

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

8. 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

9. 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

10. 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

11. 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

12. 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)

13. 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).

14. (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)

15. 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)

16. 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)

17. 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)

18. 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)