Question

How do you use shifts and reflections to sketch the graph of the function `f(x)=-sqrt(x-1)+2` and state the domain and range of f?

Answer 1

Transformations below.
The domain of `f(x)` is `[1,+oo)` and the range of `f(x)` is `[2,-oo)`

Explanation:

`f(x) =-sqrt(x-1)+2`

Consider the "parent" graph `y=sqrtx` below.

graph{sqrtx [-10, 10, -5, 5]}

The graph of `f(x)` above can be produced using the following three transformations of the parent graph.

Step1. `(x-1) ->`Shift 1 unit positive ("right") on the `x-`axis

Step2. `+2 ->`Shift 2 units positive ("up") on the `y-`axis

Step3. Leading `- ->`Reflect about the line `y=2`

To produce:

graph{-sqrt(x-1)+2 [-2.05, 10.436, -2.995, 3.25]}

As can be deduced from the graph above, the domain of `f(x)` is `[1,+oo)` and the range of `y` is `[2,-oo)`