Question

transform the solid black function to match the dotted function.
$y = \sqrt { x }$
$f ( x + \square ) + \square$

Explanation:

Step1: Analyze the horizontal shift

For a function \(y = f(x + h)\), if \(h>0\), the graph shifts left by \(h\) units; if \(h < 0\), the graph shifts right by \(|h|\) units. The original function \(y=\sqrt{x}\) has its vertex at \((0,0)\). The dotted function has its vertex at \((- 4,1)\). To move from \(x = 0\) (for \(y=\sqrt{x}\)) to \(x=-4\) (for the dotted function), we use the transformation \(x\to x + 4\) (since \(x+4 = 0\) when \(x=-4\)).

Step2: Analyze the vertical shift

For a function \(y=f(x)+k\), if \(k>0\), the graph shifts up by \(k\) units; if \(k < 0\), the graph shifts down by \(|k|\) units. The \(y\) - coordinate of the vertex changes from \(y = 0\) (for \(y=\sqrt{x}\)) to \(y = 1\) (for the dotted function). So we use the transformation \(y=f(x)+1\).

Answer:

\(f(x + 4)+1\)