The equation of y=2x-1 with a translation of 3 units to the right followed by a translation of 1 unit down.

y = 2x - 1

Lets see something:

y = 2x - 1

y = f (x)

If we want to do translations in vertical we just have to add an constant, where postive moves it upward and negatives, downward

y = f (x) + c

y = f (x) - 1

y = 2x - 1 - 1

y = 2x - 2

Now let's see this translation to the right

If we want to translate it in horizontal we just have to add a constant inside the function, where positive moves it to the left, and negative to the right

y = f (x) - 1

y = f (x - 3) - 1

y = 2. (x - 3) - 1 - 1

y = 2x - 6 - 2

y = 2x - 8