We are given the following two functions
[tex]\begin{gathered} f(x)=x^2 \\ g(x)=(-x)^2-1 \end{gathered}[/tex]
Recall that the rule for reflection over the y-axis is given by
[tex]f(x)\rightarrow f(-x)[/tex]
As you can see, the graph of g(x) will be a reflection over the y-axis of the graph f(x).
Recall that the rule for vertical translation (upward) is given by
[tex]f(x)\rightarrow f(x)-d[/tex]
The above translation will shift the graph vertically upward by d units.
For the given case, d = 1
As you can see, the graph of g(x) will be a vertical translation of the graph f(x)
Therefore, we can conclude that the graph of g(x) will be a reflection over the y-axis and a vertical translation of the graph f(x).
1st option is the correct answer.
