1. **Problem Statement:** Investigate the function $f(x) = \sqrt{x}$ in the form $y = \sqrt{kx}$ and analyze the transformations for $y = \sqrt{x}$, $y = \sqrt{2x}$, $y = \sqrt{\frac{1}{2}x}$, and $y = \sqrt{-x}$.\n\n2. **Formula and Rules:** The general form is $y = \sqrt{kx}$. Key transformation rules are:\n- If $|k| > 1$, the graph compresses horizontally (it gets narrower).\n- If $0 < |k| < 1$, the graph stretches horizontally (it gets wider).\n- If $k$ is negative, the graph reflects across the y-axis.\n- The shape is congruent only if there is no reflection or horizontal stretch/compression that changes shape proportions.\n\n3. **Function Analysis:**\n- $y = \sqrt{x}$: Domain $[0, \infty)$, Range $[0, \infty)$, Shape congruent: YES, Transformation: None (base function).\n- $y = \sqrt{2x}$: Domain $[0, \infty)$, Range $[0, \infty)$, Shape congruent: YES, Transformation: Horizontal compression by factor $\frac{1}{\sqrt{2}}$.\n- $y = \sqrt{\frac{1}{2}x}$: Domain $[0, \infty)$, Range $[0, \infty)$, Shape congruent: YES, Transformation: Horizontal stretch by factor $\sqrt{2}$.\n- $y = \sqrt{-x}$: Domain $(-\infty, 0]$, Range $[0, \infty)$, Shape congruent: YES, Transformation: Reflection across the y-axis.\n\n4. **Summary Table:**\n| Function | Domain | Range | Congruent to $y=\sqrt{x}$? | Transformation |\n|---|---|---|---|---|\n| $y=\sqrt{x}$ | $[0, \infty)$ | $[0, \infty)$ | YES | None |\n| $y=\sqrt{2x}$ | $[0, \infty)$ | $[0, \infty)$ | YES | Horizontal compression |\n| $y=\sqrt{\frac{1}{2}x}$ | $[0, \infty)$ | $[0, \infty)$ | YES | Horizontal stretch |\n| $y=\sqrt{-x}$ | $(-\infty, 0]$ | $[0, \infty)$ | YES | Reflection across y-axis |\n\n5. **Final Notes:**\n- For $|k| > 1$, the graph compresses horizontally.\n- For $0 < |k| < 1$, the graph stretches horizontally.\n- For negative $k$, the graph reflects across the y-axis.\n- The shape remains congruent in all cases since these are rigid transformations (reflection, stretch, compression) without altering the fundamental shape.