# How do you graph y=1/4sin(2x+2pi)?

Dec 9, 2017

Calculate the period from $b$ (the $2$ in $2 x$), divide the period into 4 angles, solve the equation for the 4 angles, plot the solutions, then estimate the rest of the graph.

#### Explanation:

The general form is $f \left(x\right) = a \sin \left(b x - c\right) + d$. List down what is known (when it comes to sines and cosines, a full cycle is 2π.):

$a$ (amplitude) $= \frac{1}{4}$

$b$ ("cycles per full cycle") $= 2$

Period (radians in a cycle) = ("full cycle")/b = (2π)/2 = π radians

$c$ (phase shift) = -2π radians

$d$ (vertical shift) $= 0$

What we'll need here is the period. Since the period is π radians, we obtain several angles to solve by dividing that into four (plus $0$):
0, π/4, π/2, (3π)/4, π

The reason we divided by four is so that we can reach all "four corners": the midline, the highest point, the midline again, and then the lowest point. Values in between them could later be estimated.

Shifting by a full cycle, regardless of the direction, does not affect the graph, so the phase shift could also just be $0$ radians and that it does nothing. Vertical shift is supposed to shift the "midline" of the graph vertically, but here it is $0$.

Now input each angle into the function and obtain the results:

$f \left(0\right) = \frac{1}{4} \sin \left(2 \left(0\right)\right) = \frac{1}{4} \sin \left(0\right) = \frac{1}{4} \left(0\right) = 0$

f(π/4) = 1/4 sin(2(π/4)) = 1/4 sin(π/2) = 1/4 (1) = 1/4

f(π/2) = 1/4 sin(2(π/2)) = 1/4 sin(π) = 1/4 (0) = 0

f((3π)/4) = 1/4 sin(2(3π/4)) = 1/4 sin(3π/2) = 1/4 (-1) = -1/4

f(π) = 1/4 sin(2(π)) = 1/4 sin(2π) = 1/4 (0) = 0

Now that we have these, let's plot them! It should look something like this:

graph{((x - 0)^2 + (y - 0)^2 - (0.05)^2)((x - pi/4)^2 + (y - 1/4)^2 - (0.05)^2)((x - pi/2)^2 + (y - 0)^2 - 0.05^2)((x - (3pi)/4)^2 + (y + 1/4)^2 - 0.05^2)((x - pi)^2 + (y - 0)^2 - 0.05^2) = 0 [-1.421, 3.882, -1.302, 1.35]}

Now estimate the graph:

graph{((x - 0)^2 + (y - 0)^2 - 0.05^2)((x - pi/4)^2 + (y - 1/4)^2 - 0.05^2)((x - pi/2)^2 + (y - 0)^2 - 0.05^2)((x - 3pi/4)^2 + (y + 1/4)^2 - 0.05^2)((x - pi)^2 + (y - 0)^2 - 0.05^2)(y - 1/4 sin(2x)) = 0 [-0.305, 4.764, -1.144, 1.39]}

And continue the cycle, both forwards and backwards!

graph{1/4 sin(2x)}

There you have it!