To determine the graph of the function [tex]\( y = -\sqrt{x} + 1 \)[/tex], we will analyze the function step-by-step.
### Step 1: Understanding the Basic Function [tex]\( y = \sqrt{x} \)[/tex]
The function [tex]\( y = \sqrt{x} \)[/tex] represents the square root of [tex]\( x \)[/tex] and looks like this:
- It starts from the point [tex]\( (0, 0) \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases, but at a decreasing rate.
- The function only exists for [tex]\( x \geq 0 \)[/tex] because the square root of a negative number is not real.
Graphically, it looks like a curve starting from the origin and gradually rising upward to the right.
### Step 2: Reflecting the Function
Next, we consider [tex]\( y = -\sqrt{x} \)[/tex], which reflects [tex]\( \sqrt{x} \)[/tex] about the x-axis:
- For every [tex]\( x \ge 0 \)[/tex], the value of [tex]\( y \)[/tex] will be the negative of the square root function.
- This curve starts at [tex]\( (0, 0) \)[/tex] and moves downward as [tex]\( x \)[/tex] increases.
### Step 3: Vertical Translation
Finally, we translate the function vertically by adding 1 to it: [tex]\( y = -\sqrt{x} + 1 \)[/tex].
- This means we take the [tex]\( y \)[/tex]-values from [tex]\( y = -\sqrt{x} \)[/tex] and shift them up by 1 unit.
- The new starting point will be [tex]\( (0, 1) \)[/tex] because [tex]\( y \)[/tex] value at [tex]\( x = 0 \)[/tex] is [tex]\( 1 \)[/tex].
### Generating Key Points
To visualize the function better, let’s calculate some key points:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = -\sqrt{0} + 1 = 1 \)[/tex]
- When [tex]\( x = 1 \)[/tex]: [tex]\( y = -\sqrt{1} + 1 = 0 \)[/tex]
- When [tex]\( x = 4 \)[/tex]: [tex]\( y = -\sqrt{4} + 1 = -1 \)[/tex]
- When [tex]\( x = 9 \)[/tex]: [tex]\( y = -\sqrt{9} + 1 = -2 \)[/tex]
### Overall Shape and Behavior
- The graph starts at [tex]\( (0, 1) \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases.
- The graph is curved and concave down.
- The function only exists for [tex]\( x \ge 0 \)[/tex].
### Sketching the Graph
Considering the observations above, the graph of [tex]\( y = -\sqrt{x} + 1 \)[/tex] can be sketched as follows:
1. Start at the point [tex]\( (0, 1) \)[/tex].
2. Draw a downward opening curve, moving slowly rightward and downward as [tex]\( x \)[/tex] increases.
This sketch of the function clearly captures the behavior and key characteristics.
In a multiple-choice context, the graph that fits this description will be the correct one, showing a curve starting at [tex]\( (0, 1) \)[/tex] and moving downward as [tex]\( x \)[/tex] increases.
