To understand how the graph of [tex]\( y = \sqrt{x} + 2 \)[/tex] is transformed in comparison to the parent function [tex]\( y = \sqrt{x} \)[/tex], let's break it down step by step.
1. Parent Function:
The parent function is [tex]\( y = \sqrt{x} \)[/tex]. This function describes a curve that starts at the origin (0,0) and increases gradually as [tex]\( x \)[/tex] increases, forming the familiar square root curve.
2. Transformation Description:
The given function is [tex]\( y = \sqrt{x} + 2 \)[/tex].
- Vertical Shifts: Any addition or subtraction outside the square root function ([tex]\( +2 \)[/tex] in this case) affects the vertical position of the graph. Adding a positive number, such as 2, shifts the graph upwards by that amount.
- Horizontal Shifts: Any addition or subtraction inside the function (under the square root) would signify horizontal movement. However, since there is no such term in this function, the horizontal position remains unchanged.
3. Applying the Transformation:
In the function [tex]\( y = \sqrt{x} + 2 \)[/tex], the term [tex]\( +2 \)[/tex] indicates that every point on the parent graph [tex]\( y = \sqrt{x} \)[/tex] is shifted 2 units vertically upwards.
4. Understanding the Graph:
- The parent function [tex]\( y = \sqrt{x} \)[/tex] for [tex]\( x = 0 \)[/tex] is [tex]\( y = 0 \)[/tex].
- For the transformed function, when [tex]\( x = 0 \)[/tex], [tex]\( y = \sqrt{0} + 2 = 2 \)[/tex].
This confirms that the new graph starts at (0, 2) instead of (0, 0), and every other point on the curve is similarly shifted upwards by 2 units.
Based on these observations, the graph of [tex]\( y = \sqrt{x} + 2 \)[/tex] is a vertical shift of the parent function [tex]\( y = \sqrt{x} \)[/tex] 2 units up.
So, the correct interpretation is:
The graph is a vertical shift of the parent function 2 units up.
