# How do you graph y=2sin(x/2)?

May 14, 2018

Vertical and horizontal stretches.

#### Explanation:

Starting from the standard sine function $y = \sin \left(x\right)$, you have two transformations:

• $\sin \left(x\right) \setminus \to \sin \left(\frac{x}{2}\right)$. Multipling the input variable means to horizontally stretch/compress the graph of the function. So, in general, $f \left(x\right) \setminus \to f \left(k x\right)$ means to compress the graph if $| k | > 1$, and stretch it otherwise. Since in your case $k = \frac{1}{2}$, the graph will be stretched by a factor of $2$. This means, for example, that the sinusoidal waves will take twice the time to complete their oscillations.
• Then, you have $\sin \left(\frac{x}{2}\right) \setminus \to 2 \sin \left(\frac{x}{2}\right)$. This kind of transformations $f \left(x\right) \setminus \to k f \left(x\right)$ result in a vertical stretch if $| k | > 1$, or a vertical compression otherwise. Since in your case $k = 2$, the graph will be vertically stretched, again by a factor of $2$. This affects the amplitude of the waves, which will no longer range between $- 1$ and $1$, but between $- 2$ and $2$.

Here you can see the two graphs drawn together.