To understand how the graph of the function [tex]\( y = \sqrt{x} + 2 \)[/tex] compares to the graph of the parent square root function [tex]\( y = \sqrt{x} \)[/tex], let's analyze the components of the function:
1. The parent function is [tex]\( y = \sqrt{x} \)[/tex]. This is the basic square root function.
2. The given function is [tex]\( y = \sqrt{x} + 2 \)[/tex].
The key difference between the parent function and the given function is the [tex]\( +2 \)[/tex] term. This term is outside the square root, which affects the vertical position of the graph.
Here's the step-by-step reasoning:
- The function [tex]\( y = \sqrt{x} \)[/tex] represents the square root function, whose graph starts at the origin (0,0) and increases gradually as [tex]\( x \)[/tex] increases.
- When we add 2 to [tex]\( \sqrt{x} \)[/tex], the entire graph of [tex]\( y = \sqrt{x} \)[/tex] gets shifted vertically upwards by 2 units. This is because adding a constant outside the function [tex]\( \sqrt{x} \)[/tex] translates the whole graph up by that constant value.
Therefore, the correct comparison is:
The graph of [tex]\( y = \sqrt{x} + 2 \)[/tex] is a vertical shift of the parent function [tex]\( y = \sqrt{x} \)[/tex] by 2 units up.
Hence, the answer to the question is:
The graph is a vertical shift of the parent function 2 units up.
