which of the following is the graph of $y = -sqrt{x} + 1$?

Explanation:

Step1: Analyze the domain of the function

The function is ( y = -\sqrt{x}+1 ). The square root function ( \sqrt{x} ) is defined for ( x\geq0 ), so the domain of ( y = -\sqrt{x}+1 ) is ( x\geq0 ). This means the graph should only exist for ( x ) values from 0 to positive infinity. So we can eliminate the graphs where ( x ) is negative (the first two graphs on the left since they have parts where ( x < 0 )).

Step2: Find the y - intercept

To find the y - intercept, we set ( x = 0 ) in the function ( y=-\sqrt{x}+1 ). Substituting ( x = 0 ), we get ( y=-\sqrt{0}+1=0 + 1=1 )? Wait, no, wait: ( y=-\sqrt{0}+1=1 )? Wait, no, ( \sqrt{0}=0 ), so ( y=- 0+1 = 1 )? Wait, no, let's recalculate. Wait, ( y=-\sqrt{x}+1 ), when ( x = 0 ), ( y=-(0)+1=1 )? Wait, no, maybe I made a mistake. Wait, no, let's check again. Wait, the function is ( y=-\sqrt{x}+1 ). So when ( x = 0 ), ( y= - 0+1=1 )? But wait, let's check the third and fourth graphs. Wait, maybe I messed up the y - intercept. Wait, no, let's check the value when ( x = 0 ): ( y=-\sqrt{0}+1=1 ). But let's check the third graph (the one on the right - upper) and the fourth graph (the one on the bottom - right). Wait, maybe I made a mistake in the domain. Wait, the square root of ( x ) requires ( x\geq0 ), so the graph starts at ( x = 0 ). Let's check the value when ( x=0 ): ( y = 1 ). Wait, but let's check the fourth graph: when ( x = 0 ), what's ( y )? The fourth graph (bottom - right) at ( x = 0 ) seems to have ( y=- 0 + 1=1 )? Wait, no, the fourth graph's curve at ( x = 0 ) is at ( y=-1 )? Wait, maybe I made a mistake. Wait, let's check the function again. Wait, the function is ( y=-\sqrt{x}+1 ). Let's find the point when ( x = 0 ): ( y=1 ). When ( x = 1 ), ( y=-\sqrt{1}+1=-1 + 1=0 ). So the graph passes through ( (0,1) ) and ( (1,0) ). Let's check the third graph (upper - right): when ( x = 0 ), what's the y - value? The third graph (upper - right) at ( x = 0 ) has a y - value of 1? Wait, no, the third graph (upper - right) at ( x = 0 ) seems to have a y - value of 1? Wait, no, let's look at the coordinates. Wait, the third graph (upper - right) has a curve that starts at ( x = 0 ), y - value around 1? Wait, no, maybe I mixed up the graphs. Wait, the fourth graph (bottom - right) at ( x = 0 ) has ( y=-1 )? No, wait, let's re - evaluate. Wait, the function ( y = -\sqrt{x}+1 ):

• Domain: ( x\geq0 ) (so only ( x\geq0 ) is valid, eliminate graphs with ( x < 0 ), so first two graphs are out).
• When ( x = 0 ), ( y=1 ).
• When ( x = 1 ), ( y=0 ).
• When ( x = 4 ), ( y=-\sqrt{4}+1=-2 + 1=-1 ).

Now, let's check the third graph (upper - right) and the fourth graph (bottom - right). The third graph (upper - right) has a curve that starts at ( x = 0 ), y - value 1, and goes downwards, passing through ( (1,0) ) and then to ( (4,-1) ). The fourth graph (bottom - right) has a curve that starts at ( x = 0 ), y - value - 1, which is incorrect. Wait, maybe I made a mistake in the y - intercept. Wait, no: ( y=-\sqrt{x}+1 ), when ( x = 0 ), ( y = 1 ). So the graph should start at ( (0,1) ) and decrease as ( x ) increases (since the coefficient of ( \sqrt{x} ) is negative). So the third graph (the upper - right graph, the third one from the left) has ( x\geq0 ), starts at ( (0,1) ), and decreases, passing through ( (1,0) ) and then to lower y - values as ( x ) increases. The fourth graph starts at ( (0,-1) ), which is incorrect. Wait, maybe I messed up the function. Wait, the function is ( y=-\sqrt{x}+1 ), so the parent function is ( y = \sqrt{x} ), reflected over the x - axis (because of the negative sign) and shifted up by 1 unit. The parent function ( y=\sqrt{x} ) has a domain ( x\geq0 ), starts at ( (0,0) ), and increases. After reflection over the x - axis, it becomes ( y = -\sqrt{x} ), which starts at ( (0,0) ) and decreases. Then shifting up by 1 unit, it starts at ( (0,1) ) and decreases. So the correct graph should start at ( (0,1) ), be defined for ( x\geq0 ), and decrease as ( x ) increases. So among the remaining two graphs (third and fourth), the third graph (upper - right) starts at ( (0,1) ) (since at ( x = 0 ), y = 1) and decreases, passing through ( (1,0) ) (when ( x = 1 ), y=0), which matches our calculations. The fourth graph starts at ( (0,-1) ), which is incorrect. Also, the first two graphs have ( x < 0 ), which is not in the domain of ( y = -\sqrt{x}+1 ) (since ( \sqrt{x} ) is not defined for ( x < 0 )). So the correct graph is the third one (the upper - right graph, the third graph in the original set, or the one where ( x\geq0 ), starts at ( (0,1) ) and decreases).

Wait, maybe the original graphs are: first graph (top - left), second (top - middle), third (top - right), fourth (bottom - right). So the third graph (top - right) is the one with ( x\geq0 ), starts at ( (0,1) ) (y - intercept 1) and decreases, passing through ( (1,0) ), which is the graph of ( y=-\sqrt{x}+1 ).

Answer:

The third graph (the upper - right graph, or the graph where ( x\geq0 ), starts at ( (0,1) ) and decreases as ( x ) increases).