1. 1.6-1.8 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)
(left 4 units)
(7, 2)
(right 3 units)

3. 2) Investigating y = f(x) + q
(4, 2)
(4, 5)
(up 3 units)
(4, –3)
(down 5 units)

4. 3) Investigating y = af(x)
(4, 2)
(4, 4)
(vertical stretch by 2)
(4, 1)
(vertical compression by one-half)

5. 3) Investigating y = af(x)
(4, 2)
(4, –2)
(reflection in x-axis)
(4, – 4)
(reflection in x-axis and vertical stretch of 2)

6. 4) Investigating y = af(x – p) + q
(4, 2)
left 5
(–1, 2)
× 2
(–1, 4)
down 3
(–1, 1)

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

8. vertical transformations
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

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
left 1
b) y = x3 – 2
down 2
vertical stretch ×2 and reflection in the x-axis
c) y = – 2x3
horizontal stretch 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)
Up 4
c) y = g(x – 3)
(6, 5)
Right 3
d) y = – g(2x)
(1.5, – 5)
Horizontal stretch 1/2
Reflect in the x-axis

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

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