Question

question
the graph of $y = \sqrt{x}$ is the solid black graph below. which function represents the dotted graph?
answer
$\circ$ $y = -\sqrt{x - 1}$
$\circ$ $y = -\sqrt{x + 1}$
$\circ$ $y = \sqrt{-x + 1}$
$\circ$ $y = \sqrt{-x - 1}$

Explanation:

Step1: Analyze the original function

The original function is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and starts at the origin \((0,0)\), increasing in the first quadrant.

Step2: Analyze the dotted graph's transformation

Looking at the options, we need to check the transformations (reflection, horizontal shift, etc.). Let's analyze each option:

- Option \( y = -\sqrt{x - 1} \): Horizontal shift right by 1 and reflection over x - axis. But the original starts at (0,0), this would start at (1,0), not matching.
- Option \( y = -\sqrt{x + 1} \): Horizontal shift left by 1 and reflection over x - axis. The domain would be \( x+1\geq0\) or \( x\geq - 1\), and when \( x=-1\), \( y = 0\). But the dotted graph's position (from the visual, though not fully clear, but let's check other options)
- Option \( y=\sqrt{-x + 1}=\sqrt{-(x - 1)} \): This is a reflection over y - axis and horizontal shift right by 1. The domain is \( -x + 1\geq0\) or \( x\leq1\).
- Option \( y=\sqrt{-x}-1 \): Not matching the form of transformation we see. Wait, maybe I misread. Wait, the dotted graph (from the visual, assuming the solid is \( y = \sqrt{x}\), the dotted seems to be a reflection over y - axis and maybe shift? Wait, no, let's re - evaluate. Wait, the solid is \( y=\sqrt{x}\) (first quadrant, starts at (0,0)). The dotted graph, if we look at the options, let's check the domain and the starting point.

Wait, another approach: Let's find a point on the dotted graph. Suppose the solid is \( y = \sqrt{x}\), for the dotted graph, let's assume a point. If we take the option \( y=\sqrt{-x + 1}\), when \( x = 1\), \( y=\sqrt{-1 + 1}=0\); when \( x = 0\), \( y=\sqrt{1}=1\). But the solid at \( x = 0\) is \( y = 0\). Wait, maybe I made a mistake. Wait, the original solid is \( y=\sqrt{x}\) (black), the dotted (let's see the options again). Wait, the option \( y=\sqrt{-x + 1}\) can be written as \( y=\sqrt{-(x - 1)}\), which is a reflection over the y - axis of \( y=\sqrt{-(x)}\) (which is \( y=\sqrt{-x}\), domain \( x\leq0\)) and then shifted right by 1. The graph of \( y=\sqrt{-x}\) is a reflection of \( y = \sqrt{x}\) over the y - axis (domain \( x\leq0\), starts at (0,0) and goes into the second quadrant). Then shifting right by 1 (replacing x with x - 1) gives \( y=\sqrt{-(x - 1)}=\sqrt{-x + 1}\), which has domain \( x\leq1\) and starts at (1,0), and for \( x = 0\), \( y=\sqrt{1}=1\). But maybe the correct option is \( y=\sqrt{-x + 1}\)? Wait, no, wait the options: Wait, the user's options: Let me re - list the options:

Wait the options are:

- \( y = -\sqrt{x - 1}\)
- \( y = -\sqrt{x + 1}\)
- \( y=\sqrt{-x}+1\)? No, the user's options (from the image):

Wait the options are:

Top left: \( y = -\sqrt{x - 1}\)

Top right: \( y = -\sqrt{x + 1}\)

Bottom left: \( y=\sqrt{-x}-1\)? No, bottom left: \( y=\sqrt{-x}-1\)? Wait no, the bottom left is \( y=\sqrt{-x}-1\)? No, the bottom left (from the image text) is \( y=\sqrt{-x}-1\)? Wait, no, the user's options (as per the image):

Wait the options are:

1. \( y = -\sqrt{x - 1}\)

2. \( y = -\sqrt{x + 1}\)

3. \( y=\sqrt{-x}+1\)? No, bottom right: \( y=\sqrt{-x + 1}\) (maybe, I think the correct option is \( y=\sqrt{-x + 1}\) but wait, no, let's check the reflection. Wait, the solid is \( y=\sqrt{x}\) (first quadrant). The dotted graph, if it's a reflection over the y - axis and a shift, but maybe the correct option is \( y=\sqrt{-x + 1}\). Wait, no, maybe I made a mistake. Wait, let's check the domain and the starting point.

Wait, the original function \( y = \sqrt{x}\) has domain \( x\geq0\). The dotted graph, looking at the options, the function \( y=\sqrt{-x + 1}\) has domain \( -x + 1\geq0\Rightarrow x\leq1\). When \( x = 1\), \( y = 0\), and when \( x = 0\), \( y=\sqrt{1}=1\). The solid graph starts at (0,0), the dotted graph (assuming) starts at (1,0) and is in the second quadrant (since it's a reflection over y - axis - like) but shifted. Wait, maybe the correct option is \( y=\sqrt{-x + 1}\) (the option which is \( y=\sqrt{-x + 1}\), maybe labeled as one of the bottom options). Wait, maybe I misread the options. Wait, the user's options (from the image):

Top row:

- \( y = -\sqrt{x - 1}\)

- \( y = -\sqrt{x + 1}\)

Bottom row:

- \( y=\sqrt{-x}-1\)

- \( y=\sqrt{-x + 1}\) (maybe, the last option).

Wait, let's re - analyze. The solid is \( y=\sqrt{x}\) (first quadrant, increasing). The dotted graph, from the visual (even though the graph is a bit unclear, but from the options), the function \( y=\sqrt{-x + 1}\) is a reflection over the y - axis of \( y=\sqrt{x - 1}\)? No, \( y=\sqrt{-x + 1}=\sqrt{-(x - 1)}\), which is a reflection over the y - axis of \( y=\sqrt{x - 1}\). The graph of \( y=\sqrt{x - 1}\) is a shift right of \( y=\sqrt{x}\) by 1, and reflecting over y - axis gives \( y=\sqrt{-(x - 1)}\), which has domain \( x\leq1\) and starts at (1,0), and for \( x = 0\), \( y = 1\). This matches the transformation from \( y=\sqrt{x}\) (starts at (0,0)) to a graph that starts at (1,0) and is in the second - like quadrant (left of x = 1).

But wait, maybe the correct option is \( y=\sqrt{-x + 1}\) (the bottom right option). Wait, but let's check the other options:

- \( y = -\sqrt{x - 1}\): Reflection over x - axis and shift right, starts at (1,0), goes downward, not matching.

- \( y = -\sqrt{x + 1}\): Reflection over x - axis and shift left, starts at (- 1,0), goes downward, not matching.

- \( y=\sqrt{-x}-1\): Shift down, not matching.

- \( y=\sqrt{-x + 1}\): This is the only one that has a domain \( x\leq1\) and a shape that is a reflection and shift, matching the transformation from \( y=\sqrt{x}\).

Answer:

\( y=\sqrt{-x + 1} \) (assuming the bottom right option is \( y=\sqrt{-x + 1}\))