Question

How do you translate `y=2sin (4x-pi/3)` from the parent function?

Answer 1

Start from `sin(x)`. We did the following transformations:

1. `sin(x)\to sin(4x)`
2. `sin(4x)\to sin(4x-pi/3)`
3. `sin(4x-pi/3)\to 2sin(4x-pi/3)`

Let's see how these changes affect the graph:

1. When we change `f(x)\to f(kx)`, we change the "speed" with which the `x` variable runs. This means that, if `k` is positive, the `x` values arrive earlier. For istance, if `k=4`, we have `f(4)` when `x=4`, of course. But when computing `f(4x)`, we have `f(4)` for `x=1`. This means that `sin(4x)` is a horizontally compressed version of `sin(x)`. Here's the graphs .

2. When we change from `f(x)` to `f(x+k)`, we are translating horizontally the function, and the reasons are similar to those in the first point. Is `k` is positive, the function is shifted to the left, if `k` is negative to the right. So, in this case, the function is shifted to the right by `pi/3` units. Here's the graphs

3. When we change from `f(x)` to `k*f(x)`, we simply multiply every point in the graph by `k`, resulting in a vertical stretch (expanding if `k>0` or contracting if `k<0`). Here's the graphs.