To obtain the function that repreents the final transformation of the solid black function, the following steps are recommended:
Step 1: Express the solid black function as follows:
[tex]\begin{gathered} \text{the solif black function is given as:} \\ y=\lvert x\rvert \end{gathered}[/tex]Step 2: Observe the changes that have occurred from the solid black function to the dotted function, as shown in the graph, and as explained below:
We can see that, to obtain the dotted function, the solid black function was:
1) Translated (moved) 5 units to the right along the x axis
2) Moved one unit downwards along the y axis
Step 3: Interpret the two obervations in mathematical terms, as follows:
[tex]\begin{gathered} After\text{:} \\ -\text{ translation of 5 units to the right, along the x axis, we now have:} \\ y=\text{ }\lvert x-5\rvert \\ -\text{ translation of one unit downwards, along the y axis, we now have:} \\ y=\text{ }\lvert x-5\rvert-1 \end{gathered}[/tex]Therefore, the dotted function can be mathematically represented as:
[tex]f(x)=\lvert x-5\rvert-1[/tex]