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 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. **Identify the transformations**: - Start with \( f(x) = \sqrt{x} \). - The expression inside the square root is \( 5 - x \). We can rewrite this as \( -(x - 5) \), which indicates a horizontal reflection and a horizontal shift. - The coefficient \( -\frac{1}{2} \) indicates a vertical reflection and a vertical compression. - The \( +1 \) at the end indicates a vertical shift upwards. 2. **Transformations in order**: - **Horizontal shift**: Shift to the right by 5 units (from \( x \) to \( x - 5 \)). - **Horizontal reflection**: Reflect across the y-axis (due to the negative sign). - **Vertical compression**: Compress vertically by a factor of \( \frac{1}{2} \). - **Vertical reflection**: Reflect across the x-axis (due to the negative sign). - **Vertical shift**: Shift up by 1 unit. ### Summary of transformations for \( g(x) = -\frac{1}{2} \sqrt{5-x} + 1 \): 1. Shift right by 5. 2. Reflect across the y-axis. 3. Reflect across the x-axis (due to the negative sign). 4. Compress vertically by a factor of \( \frac{1}{2} \). 5. Shift up by 1. --- ### b) \( g(x) = 2 \sqrt{x-3} - 4 \) 1. **Identify the transformations**: - Start with \( f(x) = \sqrt{x} \). - The expression inside the square root is \( x - 3 \), indicating a horizontal shift. - The coefficient \( 2 \) indicates a vertical stretch. - The \( -4 \) at the end indicates a vertical shift downwards. 2. **Transformations in order**: - **Horizontal shift**: Shift to the right by 3 units (from \( x \) to \( x - 3 \)). - **Vertical stretch**: Stretch vertically by a factor of 2. - **Vertical shift**: Shift down by 4 units. ### Summary of transformations for \( g(x) = 2 \sqrt{x-3} - 4 \): 1. Shift right by 3. 2. Stretch vertically by a factor of 2. 3. Shift down by 4. These transformations provide a clear understanding of how the base function \( f(x) = \sqrt{x} \) is modified to obtain each \( g(x) \).