The function f(x) = |x - 5| is translated 4 units left. Write the new function.

Step-by-step solution

1. 1

   Recall the rule for a horizontal shift and why it looks backwards. To shift a graph $h$ units left, replace $x$ with $x+h$; to shift it right, replace $x$ with $x-h$. The reason is that the new function must reach a given output sooner, i.e. at a smaller $x$, so the input has to be nudged forward.

2. 2

   Locate the current vertex. $\left|x-5\right|$ is zero when $x=5$, so the V has its vertex at $(5,0)$. Tracking this single point is the fastest way to check any translation.

3. 3

   Apply the substitution $x\mapsto x+4$.

   $f(x+4)=\left|(x+4)-5\right|=\left|x-1\right|$

   The two constants combine inside the bars: $+4-5=-1$.

4. 4

   Check the vertex moved the right way. The new function $\left|x-1\right|$ has its vertex at $(1,0)$, and $5-4=1$ ✓ — four units to the left, as required. Had you wrongly used $x-4$, the result would be $\left|x-9\right|$ with vertex at $9$, four units to the right.

5. 5

   Confirm nothing else changed. The shape stays a V with slopes $\pm1$ and the graph never dips below the $x$-axis, since only the horizontal position was altered. For instance $f(0)$ changes from $5$ to $1$, and the new value equals the old function evaluated at $0+4=4$, which is $\left|4-5\right|=1$ ✓.