Answer :
To compare the graph of [tex]\( y = \sqrt{x} + 2 \)[/tex] to the graph of the parent square root function [tex]\( y = \sqrt{x} \)[/tex], let's analyze the effect of the "+ 2" in the equation.
1. Start with the parent function: The parent function is [tex]\( y = \sqrt{x} \)[/tex]. This is a basic square root function where the graph starts at the origin (0,0) and increases slowly.
2. Determine the transformation: The given function is [tex]\( y = \sqrt{x} + 2 \)[/tex]. Notice that the "+ 2" is outside of the square root.
- When a constant is added outside of a function, it represents a vertical shift. Specifically, [tex]\( y = f(x) + k \)[/tex] shifts the graph vertically by [tex]\( k \)[/tex] units.
- In our case, [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( k = 2 \)[/tex]. Thus, the "+ 2" means that every point on the graph of [tex]\( y = \sqrt{x} \)[/tex] will be moved 2 units up.
3. Effect on the graph:
- The original graph of [tex]\( y = \sqrt{x} \)[/tex] at [tex]\( x = 0 \)[/tex] is [tex]\( y = \sqrt{0} = 0 \)[/tex]. After the vertical shift, this point becomes [tex]\( (0, 2) \)[/tex].
- At [tex]\( x = 1 \)[/tex], the original [tex]\( y \)[/tex]-value is [tex]\( \sqrt{1} = 1 \)[/tex]. After the vertical shift, this point becomes [tex]\( (1, 3) \)[/tex].
By analyzing these transformations, the conclusion is:
- 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.
Thus, the correct answer is:
The graph is a vertical shift of the parent function 2 units up.
1. Start with the parent function: The parent function is [tex]\( y = \sqrt{x} \)[/tex]. This is a basic square root function where the graph starts at the origin (0,0) and increases slowly.
2. Determine the transformation: The given function is [tex]\( y = \sqrt{x} + 2 \)[/tex]. Notice that the "+ 2" is outside of the square root.
- When a constant is added outside of a function, it represents a vertical shift. Specifically, [tex]\( y = f(x) + k \)[/tex] shifts the graph vertically by [tex]\( k \)[/tex] units.
- In our case, [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( k = 2 \)[/tex]. Thus, the "+ 2" means that every point on the graph of [tex]\( y = \sqrt{x} \)[/tex] will be moved 2 units up.
3. Effect on the graph:
- The original graph of [tex]\( y = \sqrt{x} \)[/tex] at [tex]\( x = 0 \)[/tex] is [tex]\( y = \sqrt{0} = 0 \)[/tex]. After the vertical shift, this point becomes [tex]\( (0, 2) \)[/tex].
- At [tex]\( x = 1 \)[/tex], the original [tex]\( y \)[/tex]-value is [tex]\( \sqrt{1} = 1 \)[/tex]. After the vertical shift, this point becomes [tex]\( (1, 3) \)[/tex].
By analyzing these transformations, the conclusion is:
- 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.
Thus, the correct answer is:
The graph is a vertical shift of the parent function 2 units up.
