# How do you graph  y=2+sin(1/2x)?

Aug 14, 2017

Draw a sine wave with twice the wavelength. Then shift it UP two units.

#### Explanation:

You know what the graph of $\sin \left(x\right)$ is.

• when $x$ is $0 , y = 0$
• when $x$ is $\frac{\pi}{2} , y = 1$
• when $x$ is $\pi , y = 0$
• when $x$ is $\frac{3 \cdot \pi}{2} , y = - 1$
• when $x$ is $2 \cdot \pi , y = 0$

graph{sin(x) [-10, 10, -5, 5]}

So now, for $y = \sin \left(\frac{x}{2}\right)$:

• when $x$ is $0 , \frac{x}{2}$ is $0 , y = 0$
• when $x$ is $\pi , \frac{x}{2}$ is $\frac{\pi}{2} , y = 1$
• when $x$ is $2 \cdot \pi , \frac{x}{2}$ is $\pi , y = 0$
• when $x$ is $3 \cdot \pi , \frac{x}{2}$ is $\frac{3 \cdot \pi}{2} , y = - 1$
• when $x$ is $4 \cdot \pi , \frac{x}{2}$ is $2 \cdot \pi , y = 0$

So, it's the same sine wave, but stretched out.

Now, since your function is $2 + \sin \left(\frac{x}{2}\right)$, simply add $2$ to the value for $y$ in the table above:

• when $x$ is $0 , y = 2$
• when $x$ is $\pi , y = 3$
• when $x$ is $2 \cdot \pi , y = 2$
• when $x$ is $3 \cdot \pi , y = 1$
• when $x$ is $4 \cdot \pi , y = 2$

graph{2 + sin(x/2) [-10, 10, -5, 5]}