Vertical translation horizontal translation

reflection over the x-axis

reflection over the y-axis

vertical translation

Step-by-step explanation:

A function is even if:

f(x) = f(-x)

and a function is odd if:

f(-x) = -f(x).

Now, we want to find a translation that only affects the symmetry of an odd function (but not that one of the even function).

The correct answer will be a vertical translation, now let's prove it.

when we have a function f(x) and we want to transform it into g(x), such that g(x) is a vertical translation of (for example) A units up, we can write the relation as:

g(x) = f(x) + A.

Now, let's see if this preserves the symmetry of the original function.

1) if f(x) is even, then:

g(x) = f(x) + A

g(-x) = f(-x) + A

But we know that f(-x) = f(x)

then:

g(-x) = f(-x) + A = f(x) + A = g(x)

this means that g(x) is even, the symmetry of an even function is not affected by this transformation.

Now suppose that f(x) is odd.

Then:

g(x) = f(x) + A

g(-x) = f(-x) + A

and f(-x) = -f(x)

then:

g(-x) = f(-x) + A = -f(x) + A

and this is clearly different than g(x) = f(x) + A.

Then the odd symmetry is broken.

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step-by-step explanation:

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