We start with the function f(x)=x^1/2 and end witht the function g(x)=-2(x-2)^1/2-3.
We have to find the transformations.
The first transformation is a scale with a factor of 2:
[tex]\begin{gathered} f(x)=x^{\frac{1}{2}} \\ f_1(x)=2f(x)=2x^{\frac{1}{2}} \end{gathered}[/tex]
The second transformation is a reflection over the horizontal axis:
[tex]f_2=-f_1=-2x^{\frac{1}{2}}[/tex]
The third transformation is a translation in the horizontal axis, 2 units to the right. Then, we have:
[tex]f_3(x)=f_2(x-2)=-2(x-2)^{\frac{1}{2}}[/tex]
The fourth transformation is a translation 3 units down:
[tex]f_4=f_3-3=-2(x-2)^{\frac{1}{2}}-3=g(x)[/tex]
Answer: The transformation are:
1) Dilation with a scale factor of 2.
2) Reflection over the x-axis.
3) Translation 2 units to the right.
4) Translation 3 units down.
We can sketch the graph of all the transformations as:
