The document discusses various types of graph transformations including reflections, translations, and dilations. Reflections involve reflecting the graph across the x or y-axis. Translations involve shifting the graph horizontally or vertically by some value. Dilations involve stretching or compressing the graph by some scale factor along the x or y-axis. For each transformation type, the document provides the formula used to transform the original function f(x) to the transformed function g(x).

1. Graph transformations
푓(푥) 푔 푥 = 푓 푥 − 1
2

2. Reflections – I : reflection in x-axis
Symmetry axis
푔
푓 푥 = 푥3 + 2푥2 − 5푥 − 4 푔 푥 = −푓 푥 = −푥3 − 2푥2 + 5푥 + 4
Graph transformations 2
푓

3. Reflections – II : reflection in y-axis
푓 푥 = 푥3 + 2푥2 − 5푥 − 4 푔 푥 = 푓 −푥 = −푥3 + 2푥2 + 5푥 − 4
푓 푔
Symmetry axis
Graph transformations 3

4. Reciprocal
Zero of 푓 becomes
asymptote of 1 푓
Graph transformations 4
푓 푥 = 푥3 + 2푥2 − 5푥 − 4
푔 푥 =
1
푓 푥
=
1
푥3 + 2푥2 − 5푥 − 4

5. Translations – I : Horizontal
푓 푥 = 푥3 + 2푥2 − 5푥 − 4
푔 푥 = 푓 푥 − 6 = 푥 − 6 3 + 2 푥 − 6 2 − 5 푥 − 6 − 4
= 푥3 − 18푥2 + 108푥 − 216 + 2 푥2 − 12푥 + 36 − 5푥 + 26
= 푥3 − 16푥2 + 79푥 − 118
ℎ 푥 = 푓 푥 + 6 = 푥 + 6 3 + 2 푥 + 6 2 − 5 푥 + 6 − 4
= 푥3 + 18푥2 + 108푥 + 216 + 2 푥2 + 12푥 + 36 − 5푥 − 34
= 푥3 − 42푥2 + 79푥 + 146
Graph transformations 5
ℎ 푓 푔
6
0
−6
0

6. Translations – II : Vertical
푓 푥 = 푥3 + 2푥2 − 5푥 − 4
푓 푔 푥 = 푓 푥 + 6 = 푥3 + 2푥2 − 5푥 + 2
Graph transformations 6
푔
ℎ
ℎ 푥 = 푓 푥 − 6 = 푥3 + 2푥2 − 5푥 − 10
0
3
0
−3

7. Dilations – I : Horizontal
푓 푥 = 푥3 + 2푥2 − 5푥 − 4
푔 푥 = 푓
푥
2
=
1
8
푥3 +
1
2
푥2 −
5
2
푥 − 4
Graph transformations 7
푔 푓
2x

8. Dilations – II : Vertical
푓 푥 = 푥3 + 2푥2 − 5푥 − 4
푔 푥 =
3
2
푓 푥 = 1
1
2
푥3 + 3푥2 − 7
1
2
푥 − 6
Graph transformations 8
푔
푓
3
2
×

9. Summary
Transformation (type) Transformation (value) Function
Graph transformations 9
Reflection in x-axis
푔 푥 = −푓 푥
Reflection in y-axis
푔 푥 = 푓 −푥
Translation in x-direction
푟 to the right (푟 < 0 =
translation to the left)
푔 푥 = 푓 푥 − 푟
Translation in y-direction
푟 upwards (푟 < 0 =
translation downwards)
푔 푥 = 푓 푥 + 푟
Dilation (stretch) in x-direction 푟 ×
푔 푥 = 푓
푥
푟
Dilation (stretch) in y-direction 푟 ×
푔 푥 = 푟 ∙ 푓 푥

10. END
Disclaimer
This document is meant to be apprehended through professional teacher mediation (‘live in class’)
together with a mathematics text book, preferably on IB level.
Graph transformations 10