To determine which graph represents [tex]\(S(x) = \frac{1}{2} \sqrt{x+3}\)[/tex], we can analyze the function step by step. We will identify key features and behaviors of the function.
### Step 1: Identify the Domain
The expression inside the square root, [tex]\(x + 3\)[/tex], must be non-negative for the function to be real-valued. Therefore:
[tex]\[ x + 3 \geq 0 \][/tex]
[tex]\[ x \geq -3 \][/tex]
So the domain of the function is [tex]\(x \geq -3\)[/tex].
### Step 2: Identify Key Points
To plot the function, let's calculate the values of [tex]\(S(x)\)[/tex] for some specific values of [tex]\(x\)[/tex].
1. At [tex]\(x = -3\)[/tex]:
[tex]\[ S(-3) = \frac{1}{2} \sqrt{-3 + 3} = \frac{1}{2} \sqrt{0} = 0 \][/tex]
2. At [tex]\(x = 0\)[/tex]:
[tex]\[ S(0) = \frac{1}{2} \sqrt{0 + 3} = \frac{1}{2} \sqrt{3} = \frac{\sqrt{3}}{2} \][/tex]
3. At [tex]\(x = 1\)[/tex]:
[tex]\[ S(1) = \frac{1}{2} \sqrt{1 + 3} = \frac{1}{2} \sqrt{4} = \frac{1}{2} \cdot 2 = 1 \][/tex]
4. At [tex]\(x = 4\)[/tex]:
[tex]\[ S(4) = \frac{1}{2} \sqrt{4 + 3} = \frac{1}{2} \sqrt{7} \approx \frac{2.64575}{2} = 1.322875 \][/tex]
### Step 3: Analyze the Behavior of the Function
- Increasing Behavior: As [tex]\(x\)[/tex] increases, the value of [tex]\(\sqrt{x+3}\)[/tex] increases, and since it is multiplied by [tex]\(\frac{1}{2}\)[/tex], [tex]\(S(x)\)[/tex] also increases.
- Shape: The function represents a square root function, scaled vertically by 0.5, and shifted left by 3 units. This results in a graph that starts at [tex]\(x = -3\)[/tex] and increases gradually.
### Step 4: Sketch the Graph
1. Start at [tex]\( (-3, 0) \)[/tex] since [tex]\( S(-3) = 0 \)[/tex].
2. Pass through [tex]\( (0, \frac{\sqrt{3}}{2}) \approx (0, 0.866) \)[/tex].
3. Continue through [tex]\( (1, 1) \)[/tex].
4. The function will continue to increase as [tex]\(x\)[/tex] increases.
### Choose the Correct Graph
The correct graph should have the following features:
1. The domain should start at [tex]\(x = -3\)[/tex].
2. The value at [tex]\(x = -3\)[/tex] should be 0.
3. The graph should be an increasing curve, resembling a square root function.
By analyzing these steps and features, you should be able to identify the correct graph based on the function [tex]\(S(x)\)[/tex].
