Horizontal Translations of Trig Functions

This is our last type of change to a base trig function! We see horizontal translations when there are parentheses in the function, such as in f(x) = sin(x + 2/3pi). You will usually see the term inside the parentheses in radians, and since we don't like radians, you'd want to change it to degrees first then re-write the function using degrees: f(x) = sin(x + 120).

Horizontal translations change the start and end point of the function on the x-axis. We know that sine/cosine normally start at 0 and end at 360. So we form an inequality: 0 <= x + 120 <= 360 and solve to get x by itself in the middle by subtracting 120 from everything. This gives us -120 <= x <= 240. So, now the function begins at -120 and ends at 240! From there, find your new midpoints and graph your function.

Note that horizontal translations ONLY change the start and end points. They do not change the amplitude, period, direction, or starting point on the y-axis. There may be other numbers in the function that do that, but anything inside the parentheses is a horizontal translation only.