# Transformations

Given the curve of a given function y=f(x), you may be required to sketch transformations of the curve. Transformations can shift, stretch and flip the curve of a function.

## 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).

• Upward shift: $f(x)\rightarrow f(x)+4$, this is a shift in y. The x coordinates are unaffected but all the y coordinates go up by 4.
• Downward shift: $f(x)\rightarrow f(x)-3$, this is a shift in y. The x coordinates are unaffected but all the y coordinates go down by 3.
• Vertical stretch: $f(x)\rightarrow 2f(x)$, this is a stretch in y. The x coordinates are unaffected but all the y coordinates are doubled.
• Reflect in x-axis: $f(x)\rightarrow -f(x)$, this is a flip in y. The x coordinates are unaffected 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 a shift in the x direction. The y coordinates are unaffected 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 a shift in the x direction The y coordinates are unaffected 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. The y coordinates are unaffected 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. The y coordinates are unaffected 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.

See Curve Sketching for some more practice questions on Transformations.

JOKE: I’ll do algebra, geometry, trigonometry and probability…. but graphing is where I draw the line.

More graph jokes. 🙂

Also see other Things to Remember.

### Past Transformation Exam Questions

We have collated past exam questions on Transformations so that you may focus your concentration on this particular subject (answers on the back pages). Alternatively, visit our Questions by Topic page to see which topics you can focus on.

### Pure Maths Practice Papers

Are you ready to test your knowledge on all of the Pure Maths topics so far? If so…

### Transformation Exam Question

This exam question involves completing the square and using the transformations to sketch the curve of the quadratic.