# 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

## What Next?

- See more on Curve Sketching
- Go back to PURE MATHS
- See QUESTIONS BY TOPIC
- Go to PAST and PRACTICE PAPERS