Question

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

Answer 1

Calculate the period from `b` (the `2` in `2x`), 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(x) = a sin(bx - c) + d`. List down what is known (when it comes to sines and cosines, a full cycle is `2π`.):

`a` (amplitude) `= 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(0) = 1/4 sin(2(0)) = 1/4 sin(0) = 1/4 (0) = 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!