Question

how does the graph of $y = \sqrt{x} + 2$ compare to the graph of the parent square root function?
\bigcirc\\ the graph is a horizontal shift of the parent function 2 units right.
\bigcirc\\ the graph is a horizontal shift of the parent function 2 units left.
\bigcirc\\ the graph is a vertical shift of the parent function 2 units up.
\bigcirc\\ the graph is a vertical shift of the parent function 2 units down.

Explanation:

The parent square root function is \( y = \sqrt{x} \). For a function of the form \( y = f(x)+k \), if \( k>0 \), the graph of \( f(x) \) is shifted vertically up by \( k \) units. Here, the function is \( y=\sqrt{x} + 2 \), which is in the form \( y = f(x)+2 \) (where \( f(x)=\sqrt{x} \)) with \( k = 2>0 \). So it's a vertical shift up by 2 units. The other options about horizontal shifts are incorrect because horizontal shifts are of the form \( y = f(x - h) \), not adding a constant outside the function.

Answer:

C. The graph is a vertical shift of the parent function 2 units up.