# 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 $f \left(x\right)$ is $\left[1 , + \infty\right)$ and the range of $f \left(x\right)$ is $\left[2 , - \infty\right)$

#### Explanation:

$f \left(x\right) = - \sqrt{x - 1} + 2$

Consider the "parent" graph $y = \sqrt{x}$ below.

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

The graph of $f \left(x\right)$ above can be produced using the following three transformations of the parent graph.

Step1. $\left(x - 1\right) \to$Shift 1 unit positive ("right") on the $x -$axis

Step2. $+ 2 \to$Shift 2 units positive ("up") on the $y -$axis

Step3. Leading $- \to$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 $f \left(x\right)$ is $\left[1 , + \infty\right)$ and the range of $y$ is $\left[2 , - \infty\right)$