To determine which graph corresponds to the function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex], let's analyze the function step by step:
1. Domain: The function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex] involves a square root. For the square root to be defined, the argument under the square root must be non-negative:
[tex]\[
x - 5 \geq 0 \implies x \geq 5
\][/tex]
Thus, the domain of the function is [tex]\( x \geq 5 \)[/tex].
2. Range: Since [tex]\( \sqrt{x - 5} \)[/tex] is non-negative (as it is the square root of a non-negative number), the smallest value [tex]\( \sqrt{x - 5} \)[/tex] can take is 0 (when [tex]\( x = 5 \)[/tex]). Subtracting 1 from the smallest value (0) gives:
[tex]\[
y = \sqrt{x - 5} - 1 \rightarrow \text{as} \quad \sqrt{x - 5} \geq 0, \quad -1 \text{ is the smallest value of } y.
\][/tex]
Thus, the range of [tex]\( y \)[/tex] is [tex]\( y \geq -1 \)[/tex].
3. Key Points:
* At [tex]\( x = 5 \)[/tex]:
[tex]\[
y = \sqrt{5 - 5} - 1 = 0 - 1 = -1
\][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x - 5} \)[/tex] increases, so [tex]\( y \)[/tex] will also increase.
For larger [tex]\( x \)[/tex]:
[tex]\[
\text{If } x = 6, \quad y = \sqrt{6 - 5} - 1 = \sqrt{1} - 1 = 1 - 1 = 0
\][/tex]
[tex]\[
\text{If } x = 9, \quad y = \sqrt{9 - 5} - 1 = \sqrt{4} - 1 = 2 - 1 = 1
\][/tex]
4. Behavior: The function will start at the point (5, -1) and increase as [tex]\( x \)[/tex] increases. The shape of [tex]\( \sqrt{x - 5} \)[/tex] is characteristic of a square root function shifted to the right by 5 units and down by 1 unit due to the "-1".
Now, let's use this information to identify the correct graph from the options A, B, C, and D:
- The correct graph should:
- Start at [tex]\( (5, -1) \)[/tex]
- Show an increasing trend as [tex]\( x \)[/tex] increases from 5.
- Resemble a square-root function shape shifted and translated accordingly.
Examining the given characteristics, the correct graph is:
Graph A
This graph correctly represents the function [tex]\( y = \sqrt{x - 5} - 1 \)[/tex], starting at the point (5, -1) and increasing as [tex]\( x \)[/tex] increases while maintaining the correct shape of a square-root function.
