Sure! Let's examine each option to determine which one correctly describes the transformation applied to the square root parent function [tex]\( f(x) = \sqrt{x} \)[/tex].
1. Option A: [tex]\( h(x) = \sqrt{x} + 6 \)[/tex]
- This represents a vertical shift. The graph of [tex]\( f(x) = \sqrt{x} \)[/tex] is shifted upwards by 6 units.
2. Option B: [tex]\( h(x) = \sqrt{x + 6} \)[/tex]
- This represents a horizontal shift. The graph of [tex]\( f(x) = \sqrt{x} \)[/tex] is shifted to the left by 6 units.
3. Option C: [tex]\( h(x) = \sqrt{x} - 6 \)[/tex]
- This represents a vertical shift. The graph of [tex]\( f(x) = \sqrt{x} \)[/tex] is shifted downwards by 6 units.
4. Option D: [tex]\( h(x) = \sqrt{x - 6} \)[/tex]
- This represents a horizontal shift. The graph of [tex]\( f(x) = \sqrt{x} \)[/tex] is shifted to the right by 6 units.
Now let's consider the function transformations one by one:
### Vertical Shifts
- Vertical Shift Up: If a function [tex]\( f(x) \)[/tex] is transformed to [tex]\( f(x) + c \)[/tex], it means the graph is shifted upwards by [tex]\( c \)[/tex] units. For option A ([tex]\( h(x) = \sqrt{x} + 6 \)[/tex]), the graph shifts up by 6 units.
- Vertical Shift Down: If a function [tex]\( f(x) \)[/tex] is transformed to [tex]\( f(x) - c \)[/tex], it means the graph is shifted downwards by [tex]\( c \)[/tex] units. For option C ([tex]\( h(x) = \sqrt{x} - 6 \)[/tex]), the graph shifts down by 6 units.
### Horizontal Shifts
- Horizontal Shift Left: If a function [tex]\( f(x) \)[/tex] is transformed to [tex]\( f(x + c) \)[/tex], it means the graph is shifted to the left by [tex]\( c \)[/tex] units. For option B ([tex]\( h(x) = \sqrt{x + 6} \)[/tex]), the graph shifts left by 6 units.
- Horizontal Shift Right: If a function [tex]\( f(x) \)[/tex] is transformed to [tex]\( f(x - c) \)[/tex], it means the graph is shifted to the right by [tex]\( c \)[/tex] units. For option D ([tex]\( h(x) = \sqrt{x - 6} \)[/tex]), the graph shifts right by 6 units.
### Final Choice
Given the transformations:
- Option A represents a vertical shift upwards by 6 units.
- Option B represents a horizontal shift to the left by 6 units.
- Option C represents a vertical shift downwards by 6 units.
- Option D represents a horizontal shift to the right by 6 units.
So, the appropriate transformations given the parent function [tex]\( f(x) = \sqrt{x} \)[/tex] are accurately described by each of these options based on the types of shifts they perform.
