Question

Question #eb249

Answer 1

Domain: `x in RR`
Range: `-1/2 <= y <= 1/2`
Amplitude: `1/2`
Period: `2pi`

Explanation:

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.

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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`.