Question

How do you graph `y=sqrt(x-1)` and how does it compare to the parent function?

Answer 1

Translate the parent graph by `1` to the right.

Explanation:

The parent function is `f(x)=sqrt(x)`, so the child function is obtained by computing `f(x-1)` instead of `f(x)`

This trasformation belong to the family of the horizontal translations, which happens everytime you change from `f(x)` to `f(x-k)`.

In particular, you translate `k` units to the left if `k>0`, or `k` units to the right if `k<0`.

In this case, `k=-1`, so this function is drawn by shifting the parent function one unit right: see below.

`f(x)=sqrt(x)`
graph{sqrt(x) [-1, 20, -1, 5]}

`f(x)=sqrt(x-1)`
graph{sqrt(x-1) [-1, 20, -1, 5]}

As you can see, the two graphs are identical, except for that `1` unit right translation.