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The radicand of a square root is always greater than or equal to 0.
Graph:
Range: {y|y≤ 1}
Comparison With the Graph of f(x)=sqrt(x): The graph of r is a reflection in the x-axis and a translation 2 units right and 1 unit up of the graph of f.
Let's first graph the given square root function and find its range. Then, we will compare the graph to the graph of its parent function.
| x | -sqrt(x-2)+1 | r(x)=-sqrt(x-2)+1 |
|---|---|---|
| 2 | -sqrt(2-2)+1 | 1 |
| 3 | -sqrt(3-2)+1 | 0 |
| 6 | -sqrt(6-2)+1 | -1 |
| 11 | -sqrt(11-2)+1 | -2 |
Let's plot and connect the obtained points. Remember, the domain is all real numbers greater than or equal to 2, so we do not want to extend the function any farther to the left.
We can see that the function takes values of y that are less than or equal to 1. This tells us the range. {y|y≤ 1}
To compare the graph of our function with the graph of the parent function f(x)=sqrt(x), we will consider some possible transformations.
| Transformations of f(x) | |
|---|---|
| Vertical Translations | Translation up k units, k>0 y=f(x)+ k |
| Translation down k units, k>0 y=f(x)- k | |
| Horizontal Translations | Translation right h units, h>0 y=f(x- h) |
| Translation left h units, h>0 y=f(x+ h) | |
| Reflections | In the x-axis y=- f(x) |
| In the y-axis y=f(- x) | |
Let's now identify the transformations in our function. r(x)=-sqrt(x- 2)+ 1 The graph of the given function is a reflection in the x-axis followed by a translation 2 units right and 1 unit up of the graph of f(x)=sqrt(x).
