Transformations of the Trig Graphs
(The Five-point Method)
The five-point method is a convenient way to sketch the graphs of the
sine and cosine functions and their transformations.
One cycle of the sine or cosine function is divided into 5 key points:
y = sin 
y
10
-1-


2




3
2
2




3
2
2

y = cos 
y
10
-1-


2

COMBINATIONS of TRANSFORMATIONS
(Sine and Cosine Functions)
y  a sin(k (x  d))  c
amplitude
changes
period
changes
horizontal
translations
a 
(–) reflection
in the x-axis
3600/2
k
(–) reflection
in the y-axis
(+) left
(–) right
a
max  min
2
c
max  min
2
vertical
translations
(+) up
(–) down
k
2
period
SUGGESTED STRATEGIES for GRAPHING:

Perform the transformations on one cycle of the function in the
following order:
1)
2)
3)
4)
5)

period changes
amplitude changes
changes the shape
reflections
horizontal translations
changes the position
vertical translations
Use the five key points and apply the mapping:
x

( x , y)    d, ay  c 
k

Ex 
State the period, amplitude, horizontal translation, and
equation of the axis for each of the following functions:
a)
Ex 
y=
1
sin(2x – )
2
b)
1

y = 5cos  x   – 5
2
6
A=
A=
period =
period =
HT:
HT:
axis:
axis:
Write the equation of each of the following functions:
a)
b)


cosine function
2
period
3

sine function

max at y = 8

axis y = 3
Ex 


Graph the function, y = 3sin  2x   – 1, for one cycle.
2

y
x
A = __________ period = __________ equation of axis: __________
Ex 


Graph the function, y = 3sin  2x   – 1, using mapping.
2

y
x