Al’s* Box Method for Graphing Sinusoidal Functions
This method is a simple way to graph sinusoidal functions such as
𝜋
𝑦 = 3 cos (2 (𝑥 − )) + 2
2
or
𝜋
𝑦 = 5 sin (3 (𝑥 + )) − 1
2
1) To graph a sinusoidal curve, first put the curve in one of the forms below:
y = A cos (B (x – D) ) + C
or y = A sin (B (x – D)) + C
(or, in Dr. Leisinger’s class:)
y – C = A cos (B (x – D) )
or y – C = A sin (B (x – D))
2) Next, make a box, as shown below.
The height of the box is 2𝐴, twice the amplitude
The width of the box is 2𝜋/𝐵 units, which is one cycle (period) of the cosine or sine curve.
3) A full cycle (period) of the sine or cosine curve is then fitted into the box, so that it looks
like the one below.
𝑦 = 𝐴 sin(𝐵(𝑥 − 𝐷 )) + 𝐶
A=Amplitude
A=Amplitude
2𝜋
1 period = 𝐵
4) Next we place the box (stretched, shifted as needed) in the coordinate plane.
The box is centered vertically a distance 𝐶 units above the 𝑥-axis, and displaced horizontally a
distance 𝐷 units to the right of the 𝑦-axis.
The finished graph is shown below.
𝑦 = 𝐴 sin(𝐵(𝑥 − 𝐷 )) + 𝐶
𝑦
A=Amplitude
𝐶
𝐶 = 𝐶𝑒𝑛𝑡𝑒𝑟
A=Amplitude
2𝜋
1 period = 𝐵
𝐶
𝐷 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑜𝑟 𝑝ℎ𝑎𝑠𝑒 𝑠ℎ𝑖𝑓𝑡
𝑥
𝜋
Example: Graph 𝑓(𝑥) = 3 sin (2 (𝑥 − )) + 3.5
2
1) The curve is in the proper form. It is in the form 𝑓(𝑥) = 𝐴 sin(𝐵(𝑥 − 𝐷)) + 𝐶
2) We make a box with height 2𝐴 = 2(2) = 4 units. It rises 2 units above the center to 5.5,
and descends 2 units below the center to 0.5 units.
The width of the box is
2𝜋
𝐵
=
2𝜋
2
𝜋
𝜋
𝜋
2𝜋
2
2
2
2
= 𝜋. Since it starts at it extends to + 𝜋 = +
=
3𝜋
2
.
3) A full cycle of the sine curve is fit into this box.
4) The box is then positioned on the coordinate plane. It is centered vertically at 𝐶 = 3.5 units
𝜋
above the 𝑥-axis, and 𝐷 = units to the right of the 𝑦-axis.
2
The finished graph is shown below. [You will often want to use multiple boxes to draw several
cycles of the curve].
Note: Given a graph of a sinusoidal function, you may write its formula by using the method in
reverse. This means that, if you can find the coordinates of two adjacent crests and one point
on the center line, you can get the coordinates of the “box” containing one period or cycle.
From these coordinates, get A,C,D, and p. Find B = 2π/p. Then write the equation.
* This excellent teaching method was developed by Al Leisinger. The write-up was done by
Stan Dick.