Graphing Combined Vertical and Horizontal Shifts

The function $f$ is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of $h$ has transformed $f$ in two ways: $f(x+1)$ is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in $f(x+1)-3$ is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in Figure 9.

Let us follow one point of the graph of $f(x)=\left|x\right|.$

• The point $(0,0)$ is transformed first by shifting left 1 unit: $(0,0)\to (\mathrm{-1},0)$
• The point $(\mathrm{-1},0)$ is transformed next by shifting down 3 units: $(\mathrm{-1},0)\to (\mathrm{-1},\mathrm{-3})$

Figure 10 shows the graph of $h.$