Answer
As there is no sign of any symmetry across neither the $x$-axis nor the $y$-axis, the graph shown should have undergone a combination of the transformations of $g(x) = \sqrt{x}$.
The equation of the graph is $y = 2\sqrt{x+4} - 4.$
Work Step by Step
As there is no sign of any symmetry across neither the $x$-axis nor the $y$-axis, the graph shown should have undergone a combination of the transformations of $g(x) = \sqrt{x}$.
Since horizontal translation to the left by 4 units is found, $x$ should be replaced by '$x+4$' and vertical translation down by 4 units is shown, a value of '4' should be deducted.
Apparently, the equation of the graph should be $y = \sqrt{x+4} - 4.$
But, since the graph passes through points $(0, 0)$ and $(5, 2)$, we can see that
when
$x = 0, y = -2$, which $y \neq 0$, and
$x = 5, y = -1$ which $y \neq 2$
Therefore, the transformed graph should have undergone a vertical stretching or shrinking.
Let "m" be the factor, the equation will become $y = m\sqrt{x+4} - 4$
Substitute $(0, 0)$ into the equation, we have
$0 = m\sqrt{0 + 4} - 4$
$4 = m\sqrt{4}$
$4 = 2m$
$m = 2$
Hence, the final equation of the graph is $y = 2\sqrt{x+4} - 4.$
In addition to the horizontal translation to the left by 4 units and a vertical translation down by 4 units, the graph has undergone a vertical stretching by a factor of 2 also.
