Graph transformations

𝑓(𝑥) 𝑔 𝑥 = 𝑓 𝑥 − 1 2

Reflections – I : reflection in x-axis

Graph transformations 2

𝑓 𝑥 = 𝑥3 + 2𝑥2 − 5𝑥 − 4 𝑔 𝑥 = −𝑓 𝑥 = −𝑥3 − 2𝑥2 + 5𝑥 + 4

Symmetry axis

𝑓

𝑔

Reflections – II : reflection in y-axis

Graph transformations 3

𝑓 𝑥 = 𝑥3 + 2𝑥2 − 5𝑥 − 4 𝑔 𝑥 = 𝑓 −𝑥 = −𝑥3 + 2𝑥2 + 5𝑥 − 4

Symmetry axis

𝑓 𝑔

Reciprocal

Graph transformations 4

𝑓 𝑥 = 𝑥3 + 2𝑥2 − 5𝑥 − 4𝑔 𝑥 =

1

𝑓 𝑥=

1

𝑥3 + 2𝑥2 − 5𝑥 − 4

Zero of 𝑓 becomes asymptote of 1 𝑓

Translations – I : Horizontal

Graph transformations 5

𝑓 𝑥 = 𝑥3 + 2𝑥2 − 5𝑥 − 4

𝑔 𝑥 = 𝑓 𝑥 − 6 = 𝑥 − 6 3 + 2 𝑥 − 6 2 − 5 𝑥 − 6 − 4

= 𝑥3 − 18𝑥2 + 108𝑥 − 216 + 2 𝑥2 − 12𝑥 + 36 − 5𝑥 + 26

= 𝑥3 − 16𝑥2 + 79𝑥 − 118

ℎ 𝑥 = 𝑓 𝑥 + 6 = 𝑥 + 6 3 + 2 𝑥 + 6 2 − 5 𝑥 + 6 − 4

= 𝑥3 + 18𝑥2 + 108𝑥 + 216 + 2 𝑥2 + 12𝑥 + 36 − 5𝑥 − 34

= 𝑥3 − 42𝑥2 + 79𝑥 + 146

𝑓 𝑔ℎ

60

−60

Translations – II : Vertical

Graph transformations 6

𝑓 𝑥 = 𝑥3 + 2𝑥2 − 5𝑥 − 4

𝑔 𝑥 = 𝑓 𝑥 + 6 = 𝑥3 + 2𝑥2 − 5𝑥 + 2𝑓

𝑔

ℎℎ 𝑥 = 𝑓 𝑥 − 6 = 𝑥3 + 2𝑥2 − 5𝑥 − 10

03

0−3

Dilations – I : Horizontal

Graph transformations 7

𝑓 𝑥 = 𝑥3 + 2𝑥2 − 5𝑥 − 4

𝑔 𝑥 = 𝑓𝑥

2=1

8𝑥3 +

1

2𝑥2 −

5

2𝑥 − 4

𝑓𝑔

2x

Dilations – II : Vertical

Graph transformations 8

𝑓 𝑥 = 𝑥3 + 2𝑥2 − 5𝑥 − 4

𝑔 𝑥 =3

2𝑓 𝑥 = 1

1

2𝑥3 + 3𝑥2 − 7

1

2𝑥 − 6

𝑓

𝑔

3

2×

Summary

Graph transformations 9

Transformation (type) Transformation (value) Function

Reflection in x-axis𝑔 𝑥 = −𝑓 𝑥

Reflection in y-axis𝑔 𝑥 = 𝑓 −𝑥

Translation in x-direction𝑟 to the right (𝑟 < 0 = translation to the left)

𝑔 𝑥 = 𝑓 𝑥 − 𝑟

Translation in y-direction𝑟 upwards (𝑟 < 0 = translation downwards)

𝑔 𝑥 = 𝑓 𝑥 + 𝑟

Dilation (stretch) in x-direction 𝑟 ×𝑔 𝑥 = 𝑓

𝑥

𝑟

Dilation (stretch) in y-direction 𝑟 ×𝑔 𝑥 = 𝑟 ∙ 𝑓 𝑥

END

Graph transformations 10

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