To determine which graph corresponds to the equation [tex]\( y = \sqrt{-x-3} \)[/tex], we will analyze the characteristics and behavior of the function step by step.
1. Domain Determination:
- The expression inside the square root must be non-negative: [tex]\( -x - 3 \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get:
[tex]\[ -x - 3 \geq 0 \implies -x \geq 3 \implies x \leq -3. \][/tex]
- Therefore, the domain of [tex]\( y = \sqrt{-x-3} \)[/tex] is [tex]\( x \leq -3 \)[/tex].
2. Range Determination:
- Since the square root function produces non-negative results, the range of [tex]\( y \)[/tex] is [tex]\( y \geq 0 \)[/tex].
3. Critical Points:
- When [tex]\( x = -3 \)[/tex]:
[tex]\[ y = \sqrt{-(-3)-3} = \sqrt{0} = 0. \][/tex]
- As [tex]\( x \)[/tex] decreases to negative infinity, [tex]\( -x-3 \)[/tex] increases, and thus [tex]\( y \)[/tex] increases without bound.
4. Behavior Analysis:
- For [tex]\( x \leq -3 \)[/tex], as [tex]\( x \)[/tex] gets smaller (moving left on the x-axis), [tex]\( -x-3 \)[/tex] increases and hence [tex]\( \sqrt{-x-3} \)[/tex] increases.
- This indicates that the graph will start at the point (-3, 0) and move upwards as [tex]\( x \)[/tex] decreases.
5. Plot and Shape:
- The graph will be situated to the left of [tex]\( x = -3 \)[/tex], and will appear to be rising as [tex]\( x \)[/tex] decreases past -3.
- The function [tex]\( y = \sqrt{-x-3} \)[/tex] produces values that make the graph concave down.
Using these properties, we compare our findings with the provided graph options. The correct graph should start at (-3, 0), be defined for [tex]\( x \leq -3 \)[/tex], and rise as it moves to the left. A concave down shape is also a key characteristic.
By comparing these key identifiers, you can successfully determine the graph representing [tex]\( y = \sqrt{-x-3} \)[/tex].
