Question

The graph of y = is reflected over the y-axis and then translated down 2 units to form f(x). Which is the graph of f(x)?

Answer 1

It looks like the original formula for y was left out. In general, reflecting a graph over the y-axis and then translating down 2 units corresponds to the transformation

- Reflect over the y-axis: replace x by −x → y = g(−x)
- Translate down 2: subtract 2 → f(x) = g(−x) − 2

So whatever the original function g(x) was, f(x) = g(−x) − 2.

If you give the original expression (or the multiple-choice graph options), I can write f(x) explicitly and pick the matching graph. Here are a few examples:
- If g(x)=x^2, then f(x)= (−x)^2 − 2 = x^2 − 2 (parabola shifted down 2).
- If g(x)=√x, then f(x)=√(−x) − 2 (domain x ≤ 0, square-root curve on the left, shifted down 2).
- If g(x)=1/x, then f(x)=1/(−x) − 2 = −1/x − 2 (hyperbola reflected and shifted down 2).
- If g(x)=|x|, then f(x)=|−x| − 2 = |x| − 2 (V-shape shifted down 2).

Send the original function or the answer choices and I’ll identify the correct graph.