Step 1:
When by either
f(x) or x is multiplied by a number, functions can “stretch” or “shrink” vertically or horizontally, respectively, when graphed.
In general, a vertical stretch is given by the equation
y=bf(x). If b>1, the graph stretches with respect to the y-axis, or vertically. If b<1, the graph shrinks with respect to the y-axis. In general, a horizontal stretch is given by the equation y=f(cx) If c>1, the graph shrinks with respect to the x-axis, or horizontally. If c<1, the graph stretches with respect to the x-axis.
Step 2:
The function is vertically stretched by a factor of 2.
![\begin{gathered} Parent\text{ function} \\ y\text{ = }\sqrt[]{4x-x^2} \\ \text{When a function is stretched by a factor of 2} \\ \text{The new function becomes } \\ y\text{ = 2}\sqrt[]{4x-x^2} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/rfluh6o1kr8osq2a2huxjcde2z42pgy59h.png)
Step 3:
A horizontal translation is generally given by the equation
y=f(x−a). These translations shift the whole function side to side on the x-axis.
Hence, the function is translated 6 units to the right
![y\text{ = 2}\sqrt[]{4(x-6)-(x-6)^2}](https://img.qammunity.org/2023/formulas/mathematics/college/1xhij7tq4vo8svadqyecnakup39c0zoqi9.png)
Final answer
![\begin{gathered} \text{The function is} \\ \text{y = 2}\sqrt[]{4(x-6)-(x-6)^2} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/m45eawd1klkk436wk2hr1uzo5fm04it30x.png)