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

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

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Answer 1 · 2017-12-16T01:24:39

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

Explanation:

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

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

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

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

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

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

Step3. Leading $- \to$ $- ->$ Reflect about the line $y = 2$ $y=2$

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 $f (x)$ $f(x)$ is $[1, + \infty)$ $[1,+oo)$ and the range of $y$ $y$ is $[2, - \infty)$ $[2,-oo)$

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

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Explanation:

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