Combining Vertical and Horizontal Shifts

Use transformations to graph the function \(f( x) = ( x+3) ^{2}-5\).

Solution Graph \(f\) in steps:

Observe that \(f\) is basically a square function, so begin by graphing \( y=x^{2}\) in Figure 43(a).

Replace the argument \(x\) with \(x+3\) to obtain \(y=( x+3) ^{2}\). This shifts the graph of \(f\) horizontally to the left \(3\) units, as shown in Figure 43(b).

Finally, subtract \(5\) from each \(y\)-coordinate, which shifts the graph in Figure 42(b) vertically down \(5\) units, and results in the graph of \( f( x) =( x+3) ^{2}-5\) shown in Figure 43(c).