For the transformation y=f(x+k), for k>0:

• f(x+k) is f(x) moved k to the left
• f(x-k) is f(x) moved k to the right

In this example, we have f(x) = x^2 - 4 and y=f(x+2)

So, subtract 2 from the x-coordinates of f(x) to get y=f(x+2)

For the transformation y=f(x)+k, for k>0:

• f(x)+k is f(x) moved k upwards
• f(x)-k is f(x) moved k downwards

In this example, we have f(x) = x^2 - 4 and y=f(x)+3

So, add 3 to the y-coordinates of f(x) to get y=f(x)+3

For the transformation y=af(x):

• If |a|>1 then af(x) is f(x) stretched vertically by a factor of a
• If 0<|a|<1 then f(x) is squashed vertically
• If a<0 then is f(x) also reflected in the x-axis

In this example, we have f(x) = x^2 - 4 and y=2f(x)

This is a stretch vertically, so multiply the y-coordinates of f(x) by 2 to get y=2f(x)

For the transformation y=f(ax):

• If |a|>1 then f(ax) is f(x) squashed horizontally by a factor of a
• If 0<|a|<1 then f(x) is stretched horizontally
• If a<0 then is f(x) also reflected in the y-axis

In this example, we have f(x) = x^2 - 4 and y=f(2x)

This is a squash horizontally, so divide the x-coordinates of f(x) by 2 (or multiply by \dfrac{1}{2}) to get y=f(2x)

For combinations of transformations, it is easy to break them up and do them one step at a time (do the bit in the brackets first). You can sketch the graph at each step to help you visualise the whole transformation.

e.g. for f(x) = x^2 - 4 and y=2f(x+2), draw the graph of
y=f(x+2) first, and then use this graph to draw the graph of
y=2f(x+2)

Note: These transformations can also be combined with modulus functions.