Question

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).