ANSWER:
[tex]\begin{gathered} g(x)=\frac{1}{2}x^2-2 \\ \text{vertex = }(0,-2) \end{gathered}[/tex]
STEP-BY-STEP EXPLANATION:
The first thing is to calculate step by step the transformations applied to the function f (x):
1. A vertical shrink
Is the compression of the graph towards the x-axis.
if 0 2. The reflection
If the function f (x) is reflected in the x-axis, then its image is g (x) = f (x)
3. Vertical translation
If the function f (x) is translated vertically down in n units, then your image would be:
g (x) = f (x) - n
Now, we apply this and the function would be:
[tex]g(x)=\frac{1}{2}x^2-2[/tex]
We can calculate the vertex as follows:
[tex]\begin{gathered} x_v=-\frac{b}{2a} \\ a=\frac{1}{2} \\ b=0 \\ \text{ replacing} \\ x_v=-\frac{0}{2\cdot\frac{1}{2}}=0 \\ \text{ now, for y:} \\ y_v=\frac{1}{2}\cdot0^2-2 \\ y_v=-2 \\ \text{therefore, the vertex is:} \\ \text{vertex = }(0,-2) \end{gathered}[/tex]
