How does the graph of [tex]y=\sqrt{x}+2[/tex] compare to the graph of the parent square root function?
A. The graph is a horizontal shift of the parent function 2 units right.
B. The graph is a horizontal shift of the parent function 2 units left.
C. The graph is a vertical shift of the parent function 2 units up.
D. The graph is a vertical shift of the parent function 2 units down.
Answer (2)
The parent function is y = x .
The transformed function is y = x + 2 .
The '+2' represents a vertical shift.
The graph is shifted upwards by 2 units: $\boxed{The graph is a vertical shift of the parent function 2 units up.}
Explanation
Understanding the Problem We are asked to compare the graph of y = x + 2 to the graph of the parent square root function, which is y = x . We need to identify the transformation that maps the parent function to the given function.
Identifying the Transformation The given function is y = x + 2 . Comparing this to the parent function y = x , we see that the only difference is the addition of 2. This addition is outside the square root function, which means it represents a vertical shift.
Determining the Direction of the Shift Since we are adding 2, the graph is shifted upwards by 2 units. Therefore, the graph of y = x + 2 is a vertical shift of the parent function 2 units up.
Conclusion The graph of y = x + 2 is the graph of y = x shifted vertically upwards by 2 units.
Examples
Imagine you're building a staircase where each step's height is determined by the square root of its position. The parent function y = x represents the basic staircase. Now, if you decide to raise the entire staircase by 2 units, that's exactly what the function y = x + 2 represents. This vertical shift is a common transformation in various fields, such as engineering, where adjusting baseline levels is crucial.
The graph of y = x + 2 is a vertical shift of the parent function y = x upwards by 2 units. This is because of the addition of 2 outside the square root function. Therefore, the correct option is C.
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