There are three types of transformations:
[ul]liTranslations[/li]liStretches[/li]liReflections[/li]/ul
In the examples below, the solid line is the original graph, and the dashed line is the transformed graph.
uTranslations/u
[row][col]p(f(x)+a): Translation by vector (\begin{pmatrix} 0 \ a\end{pmatrix}) (Translate (↑))[/p]mtaimg/images/topics/2/2-22-1.png[/mtaimg]/col
[col]p(f(x)-a): Translation by vector (\begin{pmatrix} 0 \ -a\end{pmatrix}) (Translate (↓))[/p]mtaimg/images/topics/2/2-22-2.png[/mtaimg][/col]/row
[row][col]p(f(x+a)): Translation by vector (\begin{pmatrix} -a \ 0\end{pmatrix}) (Translate (←))[/p]mtaimg/images/topics/2/2-22-3.png[/mtaimg]/col
[col]p(f(x-a)): Translation by vector (\begin{pmatrix} a \ 0\end{pmatrix}) (Translate (→))[/p]mtaimg/images/topics/2/2-22-4.png[/mtaimg][/col]/row
uStretches/u
[row][col]p(af(x)): Stretch parallel to the (y)-axis by scale factor (a) (Stretch (\begin{matrix} ↑ \ ↓\end{matrix}))[/p]mtaimg/images/topics/2/2-22-5.png[/mtaimg]/col
[col]p(\dfrac{1}{a}f(x)): Stretch parallel to the (y)-axis by scale factor (\dfrac{1}{a}) (Stretch (\begin{matrix} ↓ \ ↑\end{matrix}))[/p]mtaimg/images/topics/2/2-22-6.png[/mtaimg][/col]/row
[row][col]p(f(ax)): Stretch parallel to the (x)-axis by scale factor (\dfrac{1}{a}) (Stretch (→←))[/p]mtaimg/images/topics/2/2-22-7.png[/mtaimg]/col
[col]p(f(\dfrac{1}{a}x)): Stretch parallel to the (x)-axis by scale factor (a) (Stretch (←→))[/p]mtaimg/images/topics/2/2-22-8.png[/mtaimg][/col]/row
uReflections/u
[row][col]p(-f(x)): Reflection in the (x)-axis[/p]mtaimg/images/topics/2/2-22-9.png[/mtaimg]/col
[col]p(f(-x)): Reflection in the (y)-axis[/p]mtaimg/images/topics/2/2-22-10.png[/mtaimg][/col]/row
[b]uMultiple transformations[/u]/b
Tip: Transformations which involve a combination of translations, stretches and/or reflections, such as (af(bx+c)+d) should be done in the following order:
ol
liTranslate along the (x)-axis (move (c) units to the left)[/li]liStretch parallel to the (x)-axis and reflect in (y)-axis if necessary (stretch by scale factor (\dfrac{1}{b}) parallel to the (x)-axis)[/li]liStretch parallel to the (y)-axis and reflect in (x)-axis if necessary (stretch by scale factor (a) parallel to the (y)-axis)[/li]liTranslate along the (y)-axis (move (d) units up)[/li]/ol
You should be able to apply these transformations to any of the functions covered in the syllabus:
ul
liquadratics, cubics, quartics[/li]lireciprocals[/li]limodulus[/li]li(\sin{x}), (\cos{x}), (\tan{x})[/li]li(ex), (a^x)[/li]/ul
Summary of transformations:
Translations:
(f(x)+a): Translate (↑)
(f(x)-a): Translate (↓)
(f(x+a)): Translate (←)
(f(x-a)): Translate (→)
Stretches:
(af(x)): Stretch (\begin{matrix} ↑ \ ↓\end{matrix})
(\dfrac{1}{a}f(x)): Stretch (\begin{matrix} ↓ \ ↑\end{matrix})
(f(ax)): Stretch (→←)
(f(\dfrac{1}{a}x)): Stretch (←→)
Reflections:
(-f(x)): Reflection in the (x)-axis
(f(-x)): Reflection in the (y)-axis
Multiple transformations:
Translate in (x) direction, stretch/reflect in (x) direction, stretch/reflect in (y) direction, translate in (y) diretcion