# Question #eb249

Mar 30, 2017

Domain: $x \in \mathbb{R}$
Range: $- \frac{1}{2} \le y \le \frac{1}{2}$
Amplitude: $\frac{1}{2}$
Period: $2 \pi$

#### 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 $y = \frac{1}{2} \sin x$ :
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, $x \in \mathbb{R}$.

Range:

This isn't the same as domain. Notice that the curve only goes from $- \frac{1}{2}$ to $+ \frac{1}{2}$. 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 $- \frac{1}{2}$ to $\frac{1}{2}$. In mathematics we would write this as $- \frac{1}{2} \le y \le \frac{1}{2}$.

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 $\frac{1}{2}$.

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 $x = 2 \pi$. Therefore, the period of the wave is $2 \pi$.