Question

how is this graph obtained from the graph $f(x)=\frac{1}{x}$ using transformations?
a reflected across the y - axis and shifted up 2 units.
b reflected across the x - axis and shifted left 2 units.
c reflected across the y - axis and shifted down 2 units.
d reflected across the x - axis and shifted right 2 units.
e reflected across the x - axis and shifted right $\frac{1}{2}$ units.

Explanation:

1. Identify the parent function $f(x)=\frac{1}{x}$ has a vertical asymptote at $x=0$, horizontal asymptote at $y=0$, with the right branch in the first quadrant and left branch in the third quadrant.
2. The given graph has vertical asymptote at $x=2$, so it is shifted right 2 units: $f(x-2)=\frac{1}{x-2}$.
3. The branches are flipped across the x-axis: $-f(x-2)=-\frac{1}{x-2}$, matching the graph's quadrant placement (right branch in fourth quadrant, left branch in first quadrant).

Answer:

D. Reflected across the x-axis and shifted right 2 units.