4. Choose the correct graph of the function \( y=-\frac{1}{2} \sqrt{x-2}-3 \)

To help you choose the correct graph of the function \( y=-\frac{1}{2} \sqrt{x-2}-3 \), let's analyze the function step by step: 1. **Domain**: The function contains a square root, which means the expression inside the square root must be non-negative. Therefore, \( x-2 \geq 0 \), which implies \( x \geq 2 \). The domain of the function is \( [2, \infty) \). 2. **Asymptote**: The function has a vertical asymptote at \( x=2 \) because the square root term becomes undefined when \( x \) is less than 2. 3. **Y-Intercept**: To find the y-intercept, set \( x=0 \). However, since the domain is \( [2, \infty) \), there is no y-intercept. 4. **X-Intercept**: To find the x-intercept, set \( y=0 \) and solve for \( x \): \[ 0 = -\frac{1}{2} \sqrt{x-2} - 3 \] \[ \frac{1}{2} \sqrt{x-2} = -3 \] \[ \sqrt{x-2} = -6 \] Since the square root of a real number cannot be negative, there is no x-intercept. 5. **Behavior**: The function is decreasing because the coefficient of the square root is negative, and the function is shifted down by 3 units. 6. **Graph**: Based on the above analysis, the graph should have the following characteristics: - A vertical asymptote at \( x=2 \) - No y-intercept - No x-intercept - Decreasing as \( x \) increases - Shifted down by 3 units Now, let's look at the provided graphs and choose the one that matches these characteristics. Since I cannot see the graphs, I cannot directly tell you which one is correct. However, you can use the above analysis to identify the correct graph.