1. **Problem statement:** Given the function $f(x) = \sqrt{x}$ and its transformations in Aufgabe (A), describe how multiplication by a factor $a$ affects the graph. 2. **Formula and rules:** Multiplying a function by a factor $a$ scales the graph vertically by $a$. The transformed function is $f_a(x) = a \cdot f(x) = a \sqrt{x}$. 3. **Explanation:** - If $a > 1$, the graph stretches vertically, making it steeper (e.g., $f_1(x) = 2\sqrt{x}$). - If $0 < a < 1$, the graph compresses vertically, making it flatter (e.g., $f_2(x) = \frac{1}{2} \sqrt{x}$). - The domain remains $x \geq 0$ since $\sqrt{x}$ is defined for $x \geq 0$. 4. **Intermediate work:** - For $f_1(x) = 2\sqrt{x}$, each $y$-value of $f(x)$ is doubled. - For $f_2(x) = \frac{1}{2} \sqrt{x}$, each $y$-value of $f(x)$ is halved. 5. **Final answer:** Multiplying $\sqrt{x}$ by a factor $a$ scales the graph vertically by $a$, stretching it if $a>1$ and compressing it if $0