4. For each \( g(x) \), describe, in the appropriate order, the combination of transformations that must be applied to the base function \( f(x)=\sqrt{x} \). (hint: make sure the function is in the form \( y=a f[b(x-h)]+k \) a) \( g(x)=-\frac{1}{2} \sqrt{5-x}+1 \) b) \( g(x)=2 \sqrt{x-3}-4 \)

To describe the transformations applied to the base function \( f(x) = \sqrt{x} \) for each given function \( g(x) \), we will rewrite each function in the form \( y = a f[b(x-h)] + k \) and identify the transformations step by step. ### a) \( g(x) = -\frac{1}{2} \sqrt{5-x} + 1 \) 1. **Start with the base function**: \( f(x) = \sqrt{x} \) 2. **Horizontal transformation**: The expression inside the square root is \( 5 - x \). We can rewrite this as \( -(x - 5) \), which indicates a reflection and a horizontal shift. Specifically, we can express it as: \[ g(x) = -\frac{1}{2} f[-1(x - 5)] \] This indicates a reflection across the y-axis (due to the negative sign) and a shift to the right by 5 units. 3. **Vertical transformation**: The coefficient \( -\frac{1}{2} \) indicates a vertical reflection (due to the negative sign) and a vertical compression by a factor of \( \frac{1}{2} \). 4. **Vertical shift**: The \( +1 \) at the end indicates a vertical shift upwards by 1 unit. **Summary of transformations for part (a)**: - Reflect across the y-axis (due to the negative sign). - Shift right by 5 units. - Compress vertically by a factor of \( \frac{1}{2} \). - Reflect across the x-axis (due to the negative sign). - Shift up by 1 unit. ### b) \( g(x) = 2 \sqrt{x-3} - 4 \) 1. **Start with the base function**: \( f(x) = \sqrt{x} \) 2. **Horizontal transformation**: The expression inside the square root is \( x - 3 \), which indicates a horizontal shift to the right by 3 units. 3. **Vertical transformation**: The coefficient \( 2 \) indicates a vertical stretch by a factor of 2. 4. **Vertical shift**: The \( -4 \) at the end indicates a vertical shift downwards by 4 units. **Summary of transformations for part (b)**: - Shift right by 3 units. - Stretch vertically by a factor of 2. - Shift down by 4 units. In conclusion, the transformations for each function are as follows: - **For \( g(x) = -\frac{1}{2} \sqrt{5-x} + 1 \)**: 1. Reflect across the y-axis. 2. Shift right by 5 units. 3. Compress vertically by a factor of \( \frac{1}{2} \). 4. Reflect across the x-axis. 5. Shift up by 1 unit. - **For \( g(x) = 2 \sqrt{x-3} - 4 \)**: 1. Shift right by 3 units. 2. Stretch vertically by a factor of 2. 3. Shift down by 4 units.