Which equation best represents the following graph? (A) $y = \sqrt{x}$ (B) $|y| = \sqrt{x}$ (C) $y = \sqrt{|x|}$ (D) $|y| = \sqrt{|x|}$ - HSC - SSCE Mathematics Extension 2 - Question 1 - 2016 - Paper 1

Let's analyze each option:

• (A) $y = \sqrt{x}$: This function is only defined for $x \geq 0$ and would not be symmetrical.
• (B) $|y| = \sqrt{x}$: This could give a partial graph but again would not be fully symmetrical.
• (C) $y = \sqrt{|x|}$: This function is defined for all $x$ and produces a curve that is symmetrical about the y-axis, fitting the graph well.
• (D) $|y| = \sqrt{|x|}$: While this provides symmetry, it defines two separate branches above and below the x-axis, which does not match the graph shape.

Thus, option C emerges as the best match.

Based on the analysis, the equation that best represents the graph is (C) $y = \sqrt{|x|}$.