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Graph transformations
π(π₯) π π₯ = π π₯ β 1 2
Reflections β I : reflection in x-axis
Graph transformations 2
π π₯ = π₯3 + 2π₯2 β 5π₯ β 4 π π₯ = βπ π₯ = βπ₯3 β 2π₯2 + 5π₯ + 4
Symmetry axis
π
π
Reflections β II : reflection in y-axis
Graph transformations 3
π π₯ = π₯3 + 2π₯2 β 5π₯ β 4 π π₯ = π βπ₯ = βπ₯3 + 2π₯2 + 5π₯ β 4
Symmetry axis
π π
Reciprocal
Graph transformations 4
π π₯ = π₯3 + 2π₯2 β 5π₯ β 4π π₯ =
1
π π₯=
1
π₯3 + 2π₯2 β 5π₯ β 4
Zero of π becomes asymptote of 1 π
Translations β I : Horizontal
Graph transformations 5
π π₯ = π₯3 + 2π₯2 β 5π₯ β 4
π π₯ = π π₯ β 6 = π₯ β 6 3 + 2 π₯ β 6 2 β 5 π₯ β 6 β 4
= π₯3 β 18π₯2 + 108π₯ β 216 + 2 π₯2 β 12π₯ + 36 β 5π₯ + 26
= π₯3 β 16π₯2 + 79π₯ β 118
β π₯ = π π₯ + 6 = π₯ + 6 3 + 2 π₯ + 6 2 β 5 π₯ + 6 β 4
= π₯3 + 18π₯2 + 108π₯ + 216 + 2 π₯2 + 12π₯ + 36 β 5π₯ β 34
= π₯3 β 42π₯2 + 79π₯ + 146
π πβ
60
β60
Translations β II : Vertical
Graph transformations 6
π π₯ = π₯3 + 2π₯2 β 5π₯ β 4
π π₯ = π π₯ + 6 = π₯3 + 2π₯2 β 5π₯ + 2π
π
ββ π₯ = π π₯ β 6 = π₯3 + 2π₯2 β 5π₯ β 10
03
0β3
Dilations β I : Horizontal
Graph transformations 7
π π₯ = π₯3 + 2π₯2 β 5π₯ β 4
π π₯ = ππ₯
2=1
8π₯3 +
1
2π₯2 β
5
2π₯ β 4
ππ
2x
Dilations β II : Vertical
Graph transformations 8
π π₯ = π₯3 + 2π₯2 β 5π₯ β 4
π π₯ =3
2π π₯ = 1
1
2π₯3 + 3π₯2 β 7
1
2π₯ β 6
π
π
3
2Γ
Summary
Graph transformations 9
Transformation (type) Transformation (value) Function
Reflection in x-axisπ π₯ = βπ π₯
Reflection in y-axisπ π₯ = π βπ₯
Translation in x-directionπ to the right (π < 0 = translation to the left)
π π₯ = π π₯ β π
Translation in y-directionπ upwards (π < 0 = translation downwards)
π π₯ = π π₯ + π
Dilation (stretch) in x-direction π Γπ π₯ = π
π₯
π
Dilation (stretch) in y-direction π Γπ π₯ = π β π π₯
END
Graph transformations 10
DisclaimerThis document is meant to be apprehended through professional teacher mediation (βlive in classβ) together with a mathematics text book, preferably on IB level.