Question

How do you graph `g(x) = -2sin(pix+pi/3)`?

Answer 1

There are several steps.

Explanation:

`y = Asin(Bx+C)`

(or, in some treatments `y = Asin(Bx-C)` for the method advocated in this answer, it doesn't matter.)

We know that sine takes on values between `-1` and `1`, so when we multiply after finding sine that will change the range of values.

The amplitude reflects this fact and is equal to `absA`

For the basic sine graph, `y = sinx`, we often think of this as starting the first cycle when we take the sine of 0 and finishing when we take the sine of `2pi`. The period of the basic function is `2 pi`.

The period of `y = Asin(Bx+C)` is `(2pi)/B" "` (Multiplying by `B` changes the scale on the `x` axis.)

Phase shift (or Horizontal Shift) tells us where we "start the first period" it tells us when we take the sine of 0.

To find the shift, solve `Bx+C = 0`
(Or `Bx-C=0` depandng on your textbook.)

So here we go:

For `g(x) = -2sin(pix+pi/3)`,

We have Amplitude = `2" "`
(The minus sign will flip the graph across the `x` axis.)

Period is `(2pi)/pi = 2`

Phase Shift is the solution to `pix+pi/3 = 0`,

it is `-1/3`

The graph starts on the `x` axis at `x = -1/3`, then decreases to a minimum of `-2`, back up to and through the `x` axis, up to a maximum of `2`, before returning to the `x` axis. Then repeat the pattern as required.

Here is one period:

graph{y = -2sin(pix+pi/3) *(sqrt(1-(x-2/3)^2))/(sqrt(1-(x-2/3)^2)) [-4.177, 5.69, -2.59, 2.343]

Note

Another way of finding the period is to find the beginning of the first cycle (phase shift) by solving `Bx+C = 0` as before, and the find the end of the cycle by solving `Bx+C = 2pi`

The period is the difference between the end and the beginning. If you go through the algebra, you'll see that the difference is `(2pi)/B`