methods / transformations.mdon commit tidy up polynomials notes & add more to transformations notes (b195e8d)
   1# Transformation
   2
   3## $f(x) = x^n$ to $f(x)=a(x-h)^n+K$##
   4
   5- $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
   6- if $a<0$, graph is reflected over $x$-axis
   7- $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
   8- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
   9
  10## Translations
  11
  12For $y = f(x)$, these processes are equivalent:
  13
  14- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f$(x)$
  15- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
  16
  17## Dilations
  18
  19For the graph of $y = f(x)$, there are two pairs of equivalent processes:
  20
  211. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
  22   - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
  23
  242. - Dilating from $y$-axis: $(x, y) \rightarrow (ax, y)$
  25   - Replacing $x$ with $x \over a$ to obtain $y = f({x \over a})$
  26
  27For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.