Question #eb249

The domain is all the #x# values for which the problem is defined.
The range is all the #y# values for which the problem is defined.
The amplitude is how big the wave is (i.e. how far it goes from the x-axis).
The period is how long it takes for the wave to complete one cycle.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Here is the graph of #y = 1/2sinx# :
graph{y=0.5sinx [-6.28, 6.28, -1, 1]}

Domain:

This graph goes on and on forever along the x axis, so you could pretty much plug in any value of #x# and it would be defined.

In mathematics, we would say that #x# could be all real numbers, or for short, #x in RR#.

Range:

This isn't the same as domain. Notice that the curve only goes from #-1/2# to #+1/2#. If you tried to find a point on the curve where the #y#-value was 2, for example, you couldn't do it, because 2 is outside the function's range.

Therefore, we would say that #y# could be any number from #-1/2# to #1/2#. In mathematics we would write this as #-1/2 <= y <= 1/2#.

Amplitude:

This is actually answered by the last part; amplitude is just how far the wave travels from the x-axis at its furthest point. So, the amplitude for this particular wave is #1/2#.

Period:

Let's say that one wave starts at #0#. We need to find the point where the next wave starts.

The wave goes above the x-axis, and crosses it again at #x = pi#. Then, the wave goes below the x-axis, and finally comes back to its starting point at #x = 2pi#. Therefore, the period of the wave is #2pi#.