MCR3U1
U5L4
TRANSFORMATIONS OF THE TRIGONOMETRIC GRAPHS
(Vertical and Horizontal Translations)
PART A ~ VERTICAL TRANSLATIONS
Vertical translations shift the graph either up (+) or down (–).
Ex.
Graph the following functions for one complete cycle. Include a sketch of the parent
function, y = sin  or y = cos , and describe each transformation.
a)
y = sin  + 2
____________________________________________________________
y

b)
y = cos  – 3
____________________________________________________________
y

c
c>0
VERTICAL TRANSLATIONS ( y = sin  + c )
Transformation
c<0
NOTE: The equation of the axis of the transformed function becomes y = c.
MCR3U1
U5L4
PART B ~ HORIZONTAL TRANSLATIONS (PHASE SHIFTS)
A horizontal translation applied to a trigonometric function is called a phase shift.
To describe a phase shift, include the amount and direction of the shift. Phase shifts
are either left or right.
Ex.
Graph the following functions for one complete cycle. Include a sketch of the parent
function, y = sin  or y = cos , and describe each transformation.
a)
y = sin ( – 900)
____________________________________________________________
y

b)
y = cos ( + 600)
____________________________________________________________
y

d
d>0
d<0
PHASE SHIFTS ( y = sin ( – d) )
Transformation
MCR3U1
U5L4
PART C ~ COMBINED EXAMPLES
Write an equation and graph one cycle for the following. Include a sketch of the
parent function and state the range.
sine function
phase shift right 450
vertical translation down 2 units
y

range: _______________________________________________
PART D ~ HOMEWORK
1.
p.379 #3
2.
Determine the vertical translation and the phase shift of each function with
respect to 𝑦 = sin x. Sketch each function.
a)
c)
3.
y = sin x + 3
y = sin (x – 600) + 1
b)
d)
y = sin (x – 450)
y = sin (x + 150) – 4.5
Determine the vertical translation and the phase shift of each function with
respect to 𝑦 = cos x. Sketch each function.
a)
c)
ANSWERS:
y = cos x – 6
y = cos (x – 300) – 2
b)
d)
y = cos (x + 300)
y = cos (x +900) + 1.5
2.a) VT up 3 units
c) VT up 1 unit
phase shift right 600
b) phase shift right 450
d) VT down 4.5 units
phase shift left 150
3.a) VT down 6 units
c) VT down 2 units
phase shift right 300
b) phase shift left 300
d) VT up 1.5 units
phase shift left 900