Question

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

Answer 1

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

Explanation:

You know what the graph of `sin(x)` is.

• when `x` is `0, y = 0`
• when `x` is `pi/2, y=1`
• when `x` is `pi, y=0`
• when `x` is `(3*pi)/2, y=-1`
• when `x` is `2*pi, y=0`

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

So now, for `y = sin(x/2)`:

• when `x` is `0, x/2` is `0 , y= 0`
• when `x` is `pi, x/2` is `pi/2, y= 1`
• when `x` is `2 * pi, x/2` is `pi , y= 0`
• when `x` is `3 * pi, x/2` is `(3*pi)/2 , y= -1`
• when `x` is `4 * pi, x/2` is `2 * pi , y= 0`

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

Now, since your function is `2 + sin(x/2)`, 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 * pi, y= 2`
• when `x` is `3 * pi, y= 1`
• when `x` is `4 * pi , y= 2`

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