1 transformations

1.1 translations

the graph is moved (translated) but the shape stays the same

1.2 dilations

the graph is stretch or squashed, if the factor is negative the graph is reflected over the axis

1.3 transformation formula

\[\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}a&0\\0&b\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}+\begin{bmatrix}h\\k\end{bmatrix}\] \[x'=ax+h\qquad y'=by+k\] \[x=\frac{x'-h}{a}\qquad y=\frac{y'-k}{b}\] \[y=f(x)\] \[\frac{y-k}{b}=f(\frac{x-h}{a})\] \[y=bf(\frac{x-h}{a})+k\]

• dilation of factor \(a\) from the \(y\) axis
• dilation of factor \(b\) from the \(x\) axis
• translation of \(h\) units in the positive \(x\) direction
• translation of \(k\) units in the positive \(y\) direction

1.4 combinations of transformations

using translation and dilation together. order matters!