2. Given the graph of \( f(x)=\sqrt{x} \), what happens to the graph when \( f(x) \) is replaced by \( 2 f(-(x+4))-3 \) ?

To understand the transformation of the graph of \( f(x) = \sqrt{x} \) into \( F(x) = 2 f(-(x+4)) - 3 \), let's break down each step of the transformation: 1. **Horizontal Shift:** - **Inside the function:** \( -(x + 4) \) indicates a horizontal shift. - **Interpretation:** - The \( +4 \) inside the function shifts the graph **4 units to the left**. - The negative sign \( - \) reflects the graph across the **y-axis**. 2. **Vertical Stretch:** - **Outside the function:** The coefficient \( 2 \) multiplies the entire function. - **Interpretation:** This **stretches the graph vertically by a factor of 2**. Every y-value of the original function is doubled. 3. **Vertical Shift:** - **Outside the function:** The \( -3 \) subtracts from the entire function. - **Interpretation:** This **shifts the graph **3 units downward**. ### Summary of Transformations: 1. **Shift 4 units to the left:** \( x \) replaced by \( x + 4 \). 2. **Reflect across the y-axis:** \( x + 4 \) replaced by \( -(x + 4) \). 3. **Vertical stretch by a factor of 2:** \( \sqrt{-(x + 4)} \) multiplied by 2. 4. **Shift downward by 3 units:** Subtract 3 from the entire function. ### Visual Representation: - **Original Function:** Starts at the origin (0,0) and curves upward to the right. - **Transformed Function:** - Starts at (-4, -3) due to the horizontal shift and vertical shift. - Reflected across the y-axis, so it now curves to the left. - Stretched vertically, making it steeper. ### Final Transformed Function: \[ F(x) = 2\sqrt{-(x + 4)} - 3 \] **Graphically, the transformed \( F(x) \) is obtained by shifting \( f(x) = \sqrt{x} \) left 4 units, reflecting it across the y‑axis, stretching it vertically by a factor of two, and then moving it down three units.**