To determine the correct graph of the function [tex]\( y = -\sqrt{x} + 1 \)[/tex], let's analyze and conceptualize the properties of this function step-by-step:
1. Understanding the Square Root Function:
- The function [tex]\( \sqrt{x} \)[/tex] is the square root of [tex]\( x \)[/tex]. It is defined for [tex]\( x \geq 0 \)[/tex] and produces non-negative values.
- The general shape of [tex]\( \sqrt{x} \)[/tex] is an increasing, concave-down curve starting from the point (0, 0).
2. Negating the Square Root:
- The function [tex]\( -\sqrt{x} \)[/tex] involves multiplying [tex]\( \sqrt{x} \)[/tex] by -1, which reflects its graph over the x-axis.
- This changes the curve into a decreasing function that starts at (0, 0) and continues to decrease as [tex]\( x \)[/tex] increases.
3. Shifting the Function Upwards:
- Adding 1 to the function, [tex]\( -\sqrt{x} + 1 \)[/tex], translates the entire graph upwards by 1 unit.
- The new starting point of the function is at (0, 1).
4. Key Points:
- For [tex]\( x = 0 \)[/tex], [tex]\( y = -\sqrt{0} + 1 = 1 \)[/tex]. This gives us the point (0, 1).
- For [tex]\( x = 1 \)[/tex], [tex]\( y = -\sqrt{1} + 1 = 0 \)[/tex]. This gives us the point (1, 0).
- For [tex]\( x = 4 \)[/tex], [tex]\( y = -\sqrt{4} + 1 = -2 + 1 = -1 \)[/tex]. This gives us the point (4, -1).
5. Graph Characteristics:
- The graph of [tex]\( y = -\sqrt{x} + 1 \)[/tex] starts at the point (0, 1).
- As [tex]\( x \)[/tex] increases, the value of [tex]\( y \)[/tex] decreases.
- The curve will be a downward-opening shape (not a full parabola, but a half-parabolic shape).
Based on this analysis, the correct graph is a downward curve that starts at (0, 1) and gradually decreases as [tex]\( x \)[/tex] increases. The key points to plot are (0, 1), (1, 0), and (4, -1), forming the characteristic shape of the downward-opening half-parabola.
