Translations

To shift a function vertically up we add a number to the output function.

Click the circle next to the equation to see the transformed curve.

If you look at the table of values (you might need to drag the scrollbar to see it all) you can see how the function output has been increased by 3 for each point: 3 units higher on the graph.

Select the points, or the line to see how the tick moves up by 3 units.

Change the number '3' to translate the curve by different amounts.

A negative value will mean that the tick will move in which direction?

This time think about what is happening to our maths:

Our function is:

f(x) = 2x -1

(which is a straight line of gradient 2, y intercept -1)

So f(x) +3 = 2x -1 + 3 = 2x + 2

(which is also gradient 2, y intercept 2, in other words the same line shifted UP 3 - as expected)

Horizontal shifts are more difficult to think about. We need to 'move the x' before we do the function. Sometimes people think that the direction of the shift is the opposite way round to the way you'd expect. A transform of f(x-3) moves the graph to the RIGHT. If you look at the table of values this might help you understand why it works this way round.

Now let's look at what is happening to the maths with our straight line.

f(x) = 2x - 1

To calculate f(x+2), we use x+2 wherever we see an x in the original function, making

f(x+2) = 2(x+ 2) -1 = 2x + 4 - 1 = 2x + 3

This is again, a curve with a gradient 2. This time it has been shifted horizontally to the left.