Question

How do you graph `f(x)=sin (x/2 + pi/8)`?

Answer 1

Graph it as you would normally graph `sin(x/2)`, but you need to shift the response to the left by `pi/4` radians, or 45 degrees.

Explanation:

Multiplying or dividing x in a sinusoidal function changes the frequency of the oscillation. Normally, `sin(x)` looks like this:
graph{sin(x) [-10, 10, -5, 5]}

If we plot `sin(x/2)`, it looks like this:
graph{sin(x/2) [-10,10,-5,5]}
See how it's half of the oscillations?

finally, I will plot `sin(x/2+pi/8)`, showing the leftward phase-shift
graph{sin(x/2+pi/8) [-10,10,-5,5]}

notice that the equation is 0 at x=`-pi/4`, which is that leftward shift.