Transformations

Given the curve of a given function $y=f(x)$, they may require you to sketch transformations of the curve. Transformations can shift, stretch and flip the curve of a function. Don’t confuse these with the shape transformations in coordinate geometry at GCSE (transformations at GCSE).

y-transformations

A $y$-transformation affects the y coordinates of a curve. You can identify a $y$-transformation as changes are made outside the brackets of $y=f(x)$. We examine $y$-transformations first since they behave as expected.

  • Upward shift: $f(x)\rightarrow f(x)+4$, this is a $y$-shift. This does not affect $x$ coordinates but all the $y$ coordinates go up by 4.
  • Downward shift: $f(x)\rightarrow f(x)-3$, this is also a $y$-shift. This does not affect $x$ coordinates but all the $y$ coordinates go down by 3.
  • Vertical stretch: $f(x)\rightarrow 2f(x)$, this is a $y$-stretch. This does not affect $x$ coordinates but all the $y$ coordinates are doubled.
  • Reflect in x-axis: $f(x)\rightarrow -f(x)$, this is a flip in $y$. This does not affect $x$ coordinates but all the $y$ coordinates are flipped across the $x$-axis.

x-transformations

$x$-transformations always behave in the opposite way to what is expected. They can be identified when changes are made inside the brackets of $y=f(x)$.

  • Left shift: $f(x)\rightarrow f(x+4)$, this is an $x$-shift. This does not affect $y$ coordinates but all the $x$ coordinates go to the left by 4, the opposite direction to what is expected.
  • Right shift: $f(x)\rightarrow f(x-3)$, this is also an $x$-shift. This does not affect $y$ coordinates but all the $x$ coordinates go to the right by 3, the opposite direction to what is expected.
  • Shrink in x: $f(x)\rightarrow f(2x)$, this is a stretch in the $x$ direction. This does not affect $y$ coordinates but all the $x$ coordinates are halved, the opposite to what is expected.
  • Reflect in y-axis: $f(x)\rightarrow f(-x)$, this is a flip in the $x$ direction. This does not affect $y$ coordinates but all the $x$ coordinates are flipped across the $y$-axis.

Note that $y$-transformations usually behave as expected as opposed to $x$-transformations that seem to do the opposite.

Examples

Sketch the graph of $y=\ln(3-2x)$.

Videos

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