Start by considering the root trigonometric function, sin(x). Its domain is all real x, but let's restrict the domain between 0 and 2pi. What's so interesting about the sine and cosine curve is that it oscillates between two points (or doesn't converge to a single point), thus, we have a period of 2pi for both, the cosine and sine waves.
Now, consider what happens with x/2 when we graph the curve. This indicates that the sine wave will have a period of 4pi, because it has a much more free space. So, it will take longer to complete a period. We know this because a period is calculated as:
[tex]\text{Period of wave: } \frac{2\pi}{x} = \frac{2\pi}{\frac{x}{2}} = 4\pi[/tex]
The range remains the same.
Now, multiplying by 4 (4sin(x/2)) means that the range has been affected. Instead of the regular -1 <= y <= 1, we oscillated it between the points -4 <= y <= 4, so it is stretched by a factor of 4.
Below is the comparison between the three waves. Red represents the normal: y = sin(x) Blue represents a change in the domain: y = sin(x/2) Green is the final function: y = 4sin(x/2)
