Lesson 6 - 3: Dilatations of Sinusoidal Functions
Part 1: Vertical Dilatations – Graphs of y = asinx & y = acosx
Graph the following functions on the grid below. Use a domain of 0  x  360. State the period
and amplitude for each.
Function
Period
Amplitude
y = sinx
y = 2 sinx
y = 1/2 sinx
y = -1/2 sinx
Part 2: Horizontal Dilatations – Graphs of y = sin kx & y = coskx
Graph the following functions on the grid below. Use a domain of 0  x  360. State the period
and amplitude for each.
Function
Period
Amplitude
y = cosx
y = cos(2x)
y = cos(1/2x)
y = cos (-1/2x)
Lesson 6 - 3: Dilatations of Sinusoidal Functions
y = asin(kx) and y=acos(kx)
Dilatations are applied to sinusoidal functions in the same way they are applied to
other functions. Consider the effects of the parameters “a” and “k” on
y=sinx and y=cosx .
Part 1: Vertical Dilatations – Graphs of y = asinx & y = acosx
Graph the following functions on the grid below. Use a domain of 0  x  360.
Describe the transformation and then state the period and amplitude for each.
𝟏
𝟏
y = sinx, y = 2sinx, y = 𝟐 sinx, y = - 𝟐 sinx
Function
y = sinx
y = 2 sinx
𝟏
𝟏
Transformation None
Vertical stretch
by factor of 2
y = 𝟐 sinx
Vertical
compression by
factor of 0.5
y = - 𝟐 sinx
Vertical stretch by
factor of 2 and
reflection in x-axis
Period
360°
360°
360°
360°
Amplitude
1
2
0.5
0.5
Note: The vertical dilatation factor “a” gives the amplitude of the function:
│ a│ = amplitude
Part 2: Horizontal Dilatations – Graphs of y = sin kx & y = coskx
Graph the following functions on the grid below. Use a domain of 0  x  360.
Describe the transformation and then state the period and amplitude for each.
𝟏
𝟏
y = cosx, y = cos(2x), y = cos (𝟐 𝒙), y = cos (− 𝟐 𝒙)
Function
y = cosx
y = cos(2x)
𝟏
y = cos(𝟐 𝒙)
𝟏
y = cos (− 𝟐 𝒙)
Transformation None
Horizontal
Horizontal
horizontal stretch
compression by stretch by factor by factor of 2 and
factor of 0.5
of 2
reflection in y-axis
Period
360°
180°
720°
720°
Amplitude
1
1
1
1
Note: The horizontal dilatation factor “k” is used to calculate the period of the
function:
Period = 360°
│k│
Lesson 6 - 3: Dilatations of Sinusoidal Functions
y = asin(kx) and y=acos(kx)
Dilatations are applied to sinusoidal functions in the same way they are applied to other
functions. Consider the effect of the parameters “a” and “k” on y=sinx and y=cosx .
Recall:
Function
Transformation
y = asin x
y = acos x
If
a > 1  graph is
If 0 <
If
y = sin kx
y = cos kx
If
a
k
a < 1  graph is
by a factor of a
< 0  graph is
in the x-axis
>1  graph is
If 0<
If
by a factor of a
Effect on
key points
Multiply
y-values by
“
”
by a factor of 1/k
k <1  graph is
by a factor of 1/k
k < 0  graph is
Divide
x-values by
“
”
in the y-axis

The amplitude of each function is

The period of each function is P =
.
.
Graphing Using the Five Point Method

To sketch sine and cosine functions it is convenient to use the five key points of one cycle.
These five points are:
1. Zero
2. Maximum
3. Zero
4. Minimum
5. Zero
Recall: The 5 key points for the sine and cosine function are:
(  , sin  )
(00,
)
0
(90 ,
)
0
(180 ,
)
0
(270 ,
)
(3600,
)
(  , cos  )
(00,
)
0
(90 ,
)
0
(180 ,
)
0
(270 ,
)
(3600,
)
Each of the five key points are evenly spaced along the x-axis and divide the function into
4 quarters or INTERVALS. The length of one interval, I =
.
Ex. 1a): Graph one cycle of sine function that has amplitude of 2 and a period of 720°, θ≥0°.
b) Write the equation of the sine function.
c) State the domain and range of the function.
Ex. 2: Sketch one cycle of y=
1
cos 2 x , starting at (0, 0), x  0. State the domain and range of one
3
cycle.
Ex. 3: Write the equation for the cosine function with amplitude of 7 and a period of 1080°.
Ex. 4: Determine the equation of the sine function below:
1
9
0
o
1
8
0o
2
7
0o
-1
Homefun: Handout
360°
Exit Ticket
Sketch the graph of
f(x) = 3sin[2x]
6 - 4 Using Transformations to Sketch Graphs of Sinusoidal Functions
Unit 6: Sinusoidal Functions
-vertical translation;
-affects eq. of axis, max. & min. values, range
-no effect on period, amplitude or domain
_____________________
-a maximum occurs at 90o
_____________________
-horizontal translation
-no effect on period, amplitude eq. of axis,
domain or range
_____________________
-vertical stretch or compression
-affects max & min, amplitude, and range
-no effect on period or domain
_____________________
-a minimum occurs at 180o
_____________________
-horizontal stretch or compression
-affects period
_____________________
-no effect on amplitude, eq. of axis, max. & min., domain, and range
Combinations of Transformations:
Graphing
y = a sin [k(x – d)] + c
&
y = a cos[k(x – d)] + c
Perform transformations of trigonometric functions in the following order:
1.
2.
3.
State:
1.
2.
3.
4.
5.
Pg, 379 #1-3; Pg 383 #1-2, (3calculators), 4ace, 5, (6-7)ace