Let's analyze the function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex] step by step to understand how to graph it:
1. Identify and Shift:
- The basic form of the radical function is [tex]\( \sqrt{x} \)[/tex].
- Notice the term [tex]\( \sqrt{x+3} \)[/tex]. This indicates a horizontal shift to the left by 3 units.
2. Vertical Scaling and Reflection:
- The function has a multiplier of [tex]\(-\frac{1}{2}\)[/tex] outside the square root.
- The [tex]\(-1/2\)[/tex] factor means we scale the square root function by [tex]\(-1/2\)[/tex], which includes two transformations:
- Reflection: The negative sign reflects the function across the x-axis.
- Vertical Compression: The 1/2 factor compresses the function vertically by half.
3. Vertical Shift Upward:
- There is an additional [tex]\( +2 \)[/tex] outside the square root, which shifts the graph up by 2 units.
To summarize the steps:
- Start from the basic function [tex]\( y = \sqrt{x} \)[/tex].
- Shift it left by 3 to get [tex]\( y = \sqrt{x+3} \)[/tex].
- Reflect and compress vertically by 1/2: [tex]\( y = -\frac{1}{2} \sqrt{x+3} \)[/tex].
- Shift the graph up by 2 units to get the final function: [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex].
### Key Points:
To graph this function, it's helpful to calculate and plot specific points by substituting values for [tex]\( x \)[/tex] and solving for [tex]\( y \)[/tex].
1. Determine the domain:
- The expression under the square root (inside √) must be non-negative.
- [tex]\( x + 3 \geq 0 \)[/tex]
- [tex]\( x \geq -3 \)[/tex]
- Thus, the domain is [tex]\( x \geq -3 \)[/tex].
2. Choose key points:
- When [tex]\( x = -3 \)[/tex]:
[tex]\[
y = -\frac{1}{2} \sqrt{-3 + 3} + 2 = -\frac{1}{2} \sqrt{0} + 2 = 2
\][/tex]
- When [tex]\( x = 1 \)[/tex] (a simple value choice):
[tex]\[
y = -\frac{1}{2} \sqrt{1 + 3} + 2 = -\frac{1}{2} \sqrt{4} + 2 = -\frac{1}{2} \cdot 2 + 2 = -1 + 2 = 1
\][/tex]
3. Plot the points and shape:
- Point (-3, 2)
- Point (1, 1)
- Reflecting the downward trend caused by the negative coefficient.
### Graph:
1. Start from point [tex]\((-3, 2)\)[/tex].
2. Draw the curve reflecting the pattern and using additional intermediate values if required.
3. The graph starts from [tex]\((-3, 2)\)[/tex] and decreases as [tex]\( x \)[/tex] increases, smoothly resembling a vertically compressed, reflected half-parabola.
To conclude, by plotting several points and smoothly connecting them, we can draw the correct graph of the function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex]. The graph starts at [tex]\((-3, 2)\)[/tex] and stretches to the right, moving downward, flattened one-half compared to [tex]\( y = -\sqrt{x} \)[/tex].
