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?

1 Answer
Dec 16, 2017

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)#