Question

the graph of ( y = sqrt{x} ) is transformed as shown in the graph. which equation represents the transformed function?
options:
( y = sqrt{(-x)} + 2 )
( y = -sqrt{x} - 2 )
( y = sqrt{(-x)} - 2 )
( y = -sqrt{x} + 2 )

Explanation:

Step1: Recall parent function properties

The parent function is $y=\sqrt{x}$, which has a domain $x\geq0$, range $y\geq0$, and starts at $(0,0)$, increasing rightward.

Step2: Analyze transformed graph features

The given graph starts at $(0,2)$ and extends rightward downward. This means:

1. Vertical shift: The starting $y$-value is 2, so there is a vertical shift up by 2 units? No, wait, the direction is downward from the start. Wait, the parent function increases, this one decreases, so there is a reflection over the x-axis: $y=-\sqrt{x}$.
2. Vertical shift: The starting point of $y=-\sqrt{x}$ is $(0,0)$, but the transformed graph starts at $(0,2)$, so we add 2: $y=-\sqrt{x}+2$? No, wait, check the point: when $x=4$, parent $y=\sqrt{4}=2$, transformed at $x=4$, $y=0$. Let's test:

For $y=-\sqrt{x}+2$, when $x=0$, $y=2$ (matches start point). When $x=4$, $y=-\sqrt{4}+2=-2+2=0$ (matches the graph's point at $x=4,y=0$). When $x=9$, $y=-\sqrt{9}+2=-3+2=-1$, which follows the downward trend.
Now check other options:

• $y=\sqrt{(-x)}+2$: domain $x\leq0$, graph leftward, does not match.
• $y=-\sqrt{x-2}$: starts at $(2,0)$, does not match.
• $y=\sqrt{(-x)}-2$: starts at $(0,-2)$, leftward, does not match.

Step3: Confirm correct transformation

The transformation is reflection over x-axis ($y=-\sqrt{x}$) and vertical shift up 2 units, giving $y=-\sqrt{x}+2$.

Answer:

$y = -\sqrt{x} + 2$