Question

How do you find the period and the amplitude for `y = (1/2)sin(x - pi)`?

Answer 1

`2pi,1/2`

Explanation:

The standard form of the `color(blue)"sine wave function"` is

`color(red)(|bar(ul(color(white)(a/a)color(black)(y=asin(bx+c)+d)color(white)(a/a)|)))`

where amplitude = |a| , period`=(2pi)/b`

c is the horizontal shift and d , the vertical shift.

here a `=1/2,b=1,c=-pi" and "d=0`

hence period`=(2pi)/1=2pi" and amplitude"=|1/2|=1/2`

Answer 2

`color(blue)("Assumption: period is equivalent to 'pitch'")`

Amplitude: `" "-1/2" to "+1/2`

The period (pitch) is `2pi`

Explanation:

`color(blue)("Amplitude")`

The given equation starts with the basis of sine. This has a maximum and minimum values of -1 to +1.

Making this into a product of `1/2xx"sine"` changes the amplitude to values `-1/2" to "+1/2`
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Period")`

I am interpreting this to be the equivalent of pitch. Like that of a thread. That is the distance between maximum values or any other such repeat.

This standard 'untampered with' period is `2pi` radians or `360^0` if you wish to think in degrees. The way to change this is to apply multiplication on the `x` value in `sin(x)`, say for example `sin(3x)" or "sin(1/2x)`.

As we do not have this structure the pitch remains at the standard `2pi` radians.

Suppose we had `sin(2x)` and suppose we were considering `x=3`.

Imagine that we have two graphs. The one we are drawing and a reference one of `sin(x)`

As we are considering `x=3` we turn to the reference graph. We look at `y =sin(2x) " which is "y=sin(6)` and record that `y` value.

We then turn to the graph we are drawing and plot this recorded `y` value against `x=3`. So in effect we have moved that point to the left from `x=6` to `x=3`. In other words we have reduced the pitch (frequency increase)

So multiplying `x` by `n>1` squashes the graph horizontally and likewise multiplying `x` by `n<1` stretches the graph horizontally.

`color(brown)("We do not have a change in period hear!")`

`color(green)("The "sin(x-pi)" moves the whole plot to the right by "pi" radians")`