Graphing the Sine and Cosine Functions
Definition 1. The generalized sine and cosine families of functions can be described by
f (x) = A sin[ω x − φ] + B and
f (x) = A cos[ω x − φ] + B
where A, ω, φ, and B are any real numbers.
• |A| represents the amplitude.
• The period is Pd =
•
φ
ω
2π
ω
is the phase shift, or the horizontal shift.
• B is the amount of vertical shift.
• QP is the quarter-point width given by QP =
Pd
4
Graphing Algorithm (short version): One way to draw the graph of
y = A sin[ω x − φ] + B or y = A cos[ω x − φ] + B is to carry out the following steps:
1. Graph one period of either y = sin(ω x) or y = cos(ω x) using Pd =
2π
B.
2. Change the amplitude to follow y = A sin(ω x) or y = A cos(ω x). If A < 0,
then a reflection (about the x-axis) must be applied to the graph.
3. Translate the cycle
φ
ω
units right if
φ
ω
> 0. Shift the graph left if
φ
ω
< 0.
4. Shift the cycle B units up or down, depending upon if B > 0 or B < 0.
5. Locate the next four quarter-points using the quarter-point width (QP), then
draw another cycle.
Graphing Algorithm (long version):
y = A sin[ω x − φ] + B or
y = A cos[ω x − φ] + B
1. Recall the graphs of one fundamental cycle of y = sin(x) and y = cos(x), as
well as the five key points associated with each of those graphs.
y
y
y = sin(x)
y = cos(x)
1
π
2
−1
π
x
3π
2
2π
1
π
2
−1
π
3π
2
x
2π
2. Identify and write down the values of A, ω, φ, and B corresponding to the
given function. Then calculate the period (PD) and the quarter point width (QP)
according to:
Pd =
2π
ω
QP =
Pd
4
y
y
x = Pd
3. Locate and label x = Pd
along the x-axis.
x
QP
2QP
3QP P d
4. Locate and label each of the
quarter points (QP’s)
5. Then locate and label y = A and y = −A along the y axis.
6. Identify which of the two graphs
y
you were given: either sine or cosine.
Recall the placement of the five key
|A|
points on the graph of y = sin(x) or y =
cos(x) for x ∈ [0, 2π] (See step 1).
x
Translate (mark) these five key points
QP 2QP 3QP P d
onto your graph, just as you would if
−|A|
we were to place the graph of y =
sin(x) or y = cos(x) underneath your
graph and draw a trace. The x-coord–
inate of the five key points will be located at x = 0, QP, 2QP, 3QP, and PD; the
y-coordinate of the five key points will be located at either y = −A, 0, or A.
7. If A is negative, then apply a reflection of the five key points around the x-axis.
8. Horizontal Shift: now translate horizontally each of the five points on your
graph ωφ units. Set the given argument of the sine or cosine function equal to zero
and solve for x to find x = ωφ . Then if ωφ > 0 shift right, else if ωφ < 0 shift left.
9. Vertical Shift: If B is not zero, vertically shift each of the five points on your
graph B.
10. If you need to graph more than one period, repeat steps 3-6, starting step 3
by locating twice the length of Pd along the x-axis, followed by locating the three
QP’s between x = Pd and x = 2 Pd, using the quarter-point width (QP). You will
then have your next five key points to draw the trace of one cycle of the sine or
cosine curve through.