# Vertical Translation

The following is the graph of \(f\left( x \right) = \left| x \right|\), as we have seen earlier:

Using this graph, can we plot the graph of \(g\left( x \right) = \left| x \right|- 1\)?

Think carefully: what will be the relation between the graphs of *f* and *g*? For any value of the input *x*, the output produced by *g* will be 1 less than the output produced by *f*, because.

\[g\left( x \right) = \left| x \right| - 1 = f\left(x \right) - 1\]

This means that we can obtain the graph of *g* by simply down-shifting the graph of *f* by 1 unit:

Similarly, the graph of \(h\left( x \right) = \left| x \right| + 1 = f\left( x\right) + 1\) can be obtained by up-shifting the graph of *f* by 1 unit:

In general, the graph of \(y = f\left( x \right) + k\) can be obtained by vertically shifting the graph of \(y = f\left( x\right)\) by \(\left| k \right|\)units . The direction of the shift will depend upon the sign of *k*. If *k *is positive, the shift will be upward, else it will be downward.

The following figure shows the graph of a function \(y = f\left( x \right)\) (dotted)and the graph of \(y = f\left( x \right) - 1\) obtained by down-shifting the graph of \(y =f\left( x \right)\) by 1 unit: