We see a vertical translation when there is a number behind the x-value in the function. However, there may be multiple numbers in the function that we need to address.
For example:
y = -2sin4x - 3.
All of the numbers in the function do different things to change sinx from its base function position.
-2, since it is in front of the sin, changes the amplitude of the function. Normally sine has an amplitude of 1, but this makes the amplitude -2. Instead of going up first, the negative flips it, and makes it go down first.
4 changes the period since it is in front of the x. We start with 2pi (always), and divide it by 4 to get the new period. Since we would get 1/2pi, convert this from radians to degrees by multiplying by (180/pi). This will give you a new period of 90 degrees.
-3 is the vertical translation since it is at the end of the function, behind the x. This means that instead of starting and ending at 0, as sin usually does, the function starts and ends at -3 on the y-axis.
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