The graph of a linear equation g(x)=-1/3x +2 can be obtained from the graph f(x)=1/3x by using infinite sets of elementary translation (i.e reflection and shifting). List out five of those sets

We assume you are not interested in five infinite sets of translations. Rather, we assume you want to pick 5 translations from the infinite set of possibilities.

The attached graph shows f(x), g(x), and 5 lines (dashed or dotted) that represent possible reflections and shifts of the function f(x).

The function f1 represents a reflection of f(x) about the x-axis, followed by a left-shift of 6 units. To make it match g(x), we need to shift it upward 4 units. Then the set if translations is ...

g(x) = f(x) ... {reflected over the x-axis, shifted left 6, shifted up 4}

Along the same lines, other possibilities are ...

g(x) = f(x) ... {reflected over the x-axis, shifted left 3, shifted up 3}

g(x) = f(x) ... {reflected over the x-axis, shifted left 0, shifted up 2}

g(x) = f(x) ... {reflected over the x-axis, shifted right 3, shifted up 1}

g(x) = f(x) ... {reflected over the x-axis, shifted right 9, shifted down 1}

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Additional comment

All of the transformations listed above use reflection in the x-axis. Reflection could use the y-axis, as well. Reflection of the basic function f(x) in the y-axis will have the identical effect as reflection in the x-axis. The listed translations would apply unchanged.