Combining Transformations

Use transformations to graph the function \(f( x) =\sqrt{1-x}+2\).

Solution We graph \(f\) in steps:

Observe that \(f\) is basically a square root function, so we begin by graphing \(y=\sqrt{x}\). See Figure 47(a).

Now we replace the argument \(x\) with \(x+1\) to obtain \(y=\sqrt{x+1},\) which shifts the graph of \(y=\sqrt{x}\) horizontally to the left \(1\) unit, as shown in Figure 47(b).

Then we replace \(x\) with \(-x\) to obtain \(y=\sqrt{-x+1}=\sqrt{1-x},\) which reflects the graph about the \(y\)-axis. See Figure 47(c).

Finally, we add \(2\) to each \(y\)-coordinate, which shifts the graph vertically up 2 units, and results in the graph of \(f( x) = \sqrt{1-x}+2\) shown in Figure 47(d).