Question

How do I horizontally translate a trigonometric graph?

Answer 1

By changing the "c" in your basic trigonometric equation.

The standard trig equation for sine is `y=a*sin[b(x-cpi)]+d`. In this, the variable `a` represents the amplitude. The variable `b` represents the period (`(2pi)/b` = period). Now, the variable `c` represents what is known as the phase shift - more commonly known as a horizontal translation. You shift the graph `cpi` units from the original parent function, which in this case is `y=sinx`. If `c` is positive, shift the graph to the right `cpi` unites. If `c` is negative, shift the graph to the left `cpi` units.

If you're wondering, `d` represents the vertical translation.

I hope this helps, and I'f strongly suggest going to google and typing in functions like `y=sin(x-2pi)` and comparing them to the parent function, `y=sinx`.

Answer 2

Generally speaking, if you have a function `y=f(x)` and know its graph, the function `y=f(x-c)` has a graph that is similar to the one of `y=f(x)` but shifted by `c` to the right for `c>0` or to the left for `c<0`.

Continuing the graph transformation, the graph of `y=f(x)+d` is similar to the graph of `y=f(x)` but shifted by `d` up for `d>0` or down for `d<0`.

Next transformation is related to a graph of a function `y=a*f(x)`. The graph of this function can be obtained from a graph of `y=f(x)` by stretching (if `|a|>1`) or squeezing (if `|a|<1`) it by a factor `a` vertically. That is, point `(x,y)` on a graph of `y=f(x)` is transferred into `(x,a*y)` on a graph of `y=a*f(x)`. This includes reflection relative to the X-axis for `a<0`.

Finally, the graph of a function `y=f(b*x)` can be obtained from the graph of `y=f(x)` by horizontal squeezing (if `|b|>1`) or stretching (if `|b|<1`) it by a factor of `b`. That is, point `(x,y)` on a graph of `y=f(x)` is transferred into `(x/b,y)` on a graph of `y=f(b*x)`. This includes reflection relative to the Y-axis for `b<0`.

You can find more detailed explanation of these manipulations with graphs in a lecture on Unizor following the menu items Algebra - Graphs - Transformation.