To determine which statement is true regarding the transformation of the parent square root function [tex]\( f(x) = \sqrt{x} \)[/tex] to the function [tex]\( g(x) = \sqrt{x + 3} - 4 \)[/tex], follow these steps:
1. Identify the Parent Function:
The parent function is given by [tex]\( f(x) = \sqrt{x} \)[/tex].
2. Transformation Analysis:
The function [tex]\( g(x) = \sqrt{x + 3} - 4 \)[/tex] undergoes two transformations from the parent function [tex]\( f(x) \)[/tex]:
- Horizontal Shift: The term [tex]\( x + 3 \)[/tex] inside the square root function indicates a horizontal shift. Specifically,
- Adding 3 to [tex]\( x \)[/tex] translates the graph of the function [tex]\( \sqrt{x} \)[/tex] 3 units to the left. This is because [tex]\(\sqrt{x + 3}\)[/tex] is equivalent to taking the parent function and shifting it left by 3 units.
- Vertical Shift: The term [tex]\( -4 \)[/tex] outside the square root function indicates a vertical shift. Specifically,
- Subtracting 4 from the entire function translates the graph of the function [tex]\( \sqrt{x} \)[/tex] 4 units down. This is a direct vertical translation.
3. Combine the Effects:
Combining these transformations, the graph of the parent function [tex]\( \sqrt{x} \)[/tex] is translated:
- 3 units to the left (due to [tex]\( x + 3 \)[/tex]), and
- 4 units down (due to [tex]\( -4 \)[/tex]).
4. Correct Statement:
Given the analysis, the correct statement that describes these transformations is:
B. The graph of [tex]\( f \)[/tex] is translated 3 units to the left and 4 units down.
So, the correct answer is option B.
