To determine the transformed function [tex]\( h(x) \)[/tex] of the square root parent function [tex]\( f(x) = \sqrt{x} \)[/tex], we need to analyze each of the given options based on standard function transformation rules.
The parent function [tex]\( f(x) = \sqrt{x} \)[/tex] can undergo several types of transformations, such as horizontal shifts, vertical shifts, reflections, and stretches/compressions. The transformations involved in this problem appear to be horizontal and vertical shifts.
Let's examine each option in detail:
### Option A: [tex]\( h(x) = \sqrt{x+6} \)[/tex]
- Horizontal Shift: The function [tex]\( h(x) = \sqrt{x+6} \)[/tex] represents a horizontal shift of the parent function.
- Direction and Magnitude: The expression [tex]\( \sqrt{x+6} \)[/tex] results from shifting the parent function [tex]\( \sqrt{x} \)[/tex] horizontally to the left by 6 units.
- Mechanism: To understand why, notice the general form of a horizontal shift: [tex]\( h(x) = \sqrt{x + c} \)[/tex] shifts the function left if [tex]\( c \)[/tex] is positive and right if [tex]\( c \)[/tex] is negative.
### Option B: [tex]\( h(x) = \sqrt{x} + 6 \)[/tex]
- Vertical Shift: The function [tex]\( h(x) = \sqrt{x} + 6 \)[/tex] represents a vertical shift of the parent function.
- Direction and Magnitude: The term [tex]\( 6 \)[/tex] is added directly to the output of the square root function, indicating a vertical shift upwards by 6 units.
- Mechanism: This aligns with the general form of a vertical shift: [tex]\( h(x) = \sqrt{x} + c \)[/tex], where positive [tex]\( c \)[/tex] shifts the graph upward and negative [tex]\( c \)[/tex] shifts it downward.
### Option C: [tex]\( h(x) = \sqrt{x} - 6 \)[/tex]
- Vertical Shift: The function [tex]\( h(x) = \sqrt{x} - 6 \)[/tex] also represents a vertical shift.
- Direction and Magnitude: Subtracting [tex]\( 6 \)[/tex] from the output of the square root function indicates a vertical shift downwards by 6 units.
- Mechanism: Using the general form [tex]\( h(x) = \sqrt{x} - c \)[/tex], a positive [tex]\( c \)[/tex] means a shift downward.
### Option D: [tex]\( h(x) = \sqrt{x 6} \)[/tex]
- Validity: This form is invalid in standard mathematical notation.
- Explanation: The expression [tex]\( \sqrt{x \quad 6} \)[/tex] does not follow a recognized pattern or notation for transforming functions, suggesting it is an error or misprint.
### Conclusion
Among the options, Option D does not make sense as a valid transformation of the square root parent function. The other options are valid transformations, with Option A representing a horizontal shift left by 6 units, Option B representing a vertical shift up by 6 units, and Option C representing a vertical shift down by 6 units.
To identify which specific function [tex]\( h(x) \)[/tex] is correct:
1. If the problem specifies looking for a horizontal shift, the correct answer is:
- Option A: [tex]\( h(x) = \sqrt{x+6} \)[/tex]
If the problem specifies a vertical shift, the correct answer will be:
- Option B: [tex]\( h(x) = \sqrt{x} + 6 \)[/tex]
- Option C: [tex]\( h(x) = \sqrt{x} - 6 \)[/tex]
However, as there is no specific reference to the type of shift in the question, we must exclude the invalid option, which is:
- Option D: [tex]\( h(x) = \sqrt{x \quad 6} \)[/tex]
Thus, the correct function among the options is:
[tex]\[ \boxed{\sqrt{x+6}} \][/tex]
