5.2 Transformations of Sinusoidal Functions
Electric power and the light waves it generates
are sinusoidal waveforms.
Math 30-1
1
Graphing a Horizontal Translation
Graph y = sin x
Use 5 key points to plot the graph of y = sin x
x
sin x
y = sin x
0

2

0
1
0
3
2
-1
2
0
Math 30-1
2
Graphing a Horizontal Translation
Vertical Translation - upward or downward shift in the graph of the function.
y  a cos(bx  c )  d
y  a sin(bx  c )  d
The constant “d” determines the magnitude and direction of
the vertical displacement of a periodic function.
y  sin x  3
Math 30-1
3
Graphing y = sinx + d
Sketch the graphs of y = sinx – 3 and y = sinx + 2
y = sin x + 2
y = sin x
y = sin x - 3
Math 30-1
4
Graphing a Horizontal Translation
Horizontal Translation - upward or downward shift in the graph of the function.
y  a cos(bx  c )  d
y  a sin(bx  c )  d
The constant “c” determines the magnitude and direction of
the phase shift of a periodic function.
Math 30-1
5
Graphing y = sin(x – c)
Sketch the graph of y  cos( x 
3π
)
2
y  cos( x 
y
3

2



1

2
2
1

3
2
2
5
2
3π
)
2
x
y = cosx
Math 30-1
6
Graphing y = sin(x – c)
π
2
Sketch the graph of y  cos( x  )
y  cos( x 
y
3

2



1

2
2
1

3
2
2
5
2
π
)
2
x
y = cosx
Math 30-1
7
Graphing y = sin(x – c) + d
Sketch the graph of y  3sin2( x  2)  2 .using transformations.
1. Sketch the graph of y = sin x. (radians)
y = sin x
Math 30-1
8
Graphing y  3sin2( x  2)  2 .
2. Sketch the graph of y = 3sin x.
y = 3sin x
y = sin x
Math 30-1
9
Graphing y  3sin2( x  2)  2 .
2. Sketch the graph of y = 3sin x.
3. Sketch the graph of y = 3sin x – 2.
y = 3sin x
y = sin x
y = 3sin x - 2
Math 30-1
10
Graphing y  3sin2( x  2)  2 .
4. Sketch the graph of y = 3sin 2x – 2.
y = 3sin x – 2
y = 3sin 2x – 2
Math 30-1
11
Graphing y  3sin2( x  2)  2 .
5. Sketch the graph of y = 3sin 2(x + 2) – 2.
y = 3sin 2(x + 2) – 2
y = 3sin 2x – 2
Math 30-1
12
Graphing y  3sin2( x  2)  2 .
y = sin x
y = 3sin 2(x + 2) – 2
Domain: the set of all real numbers
Range: {-5 ≤ y ≤ 1}
Amplitude: 3
Vertical Displacement: 2 units down
Period: 
Phase Shift: 2 units to the left
Math 30-1
13
Analyzing a Sine Function
y  3sin2(x 


4
) 2
2
y- intercept:
Domain: the set of all real numbers
Range: -5 ≤ y ≤ 1
Amplitude: 3
Vertical Displacement: 2 units down
Period: 

units to the left
Phase Shift:
Math 30-1
4
y  3sin2(x 

4
x=0
) 2

y  3sin2(0  )  2
4

y  3sin( )  2
2
y  3(1)  2
y1
14
Determining an Equation From a Graph
A partial graph of a sine function is shown.
Determine the equation as a function of sine.
a=2
d=1
c

6
2
period =
b
2
 =
b
Therefore, the equation is y  2 sin2(x 
Math 30-1
b=2

6
)  1.
15
Determining an Equation From a Graph
A partial graph of a cosine function is shown.
Determine the equation as a function of cosine.
a=2
d = -1
c

4
2
period =
b
2
 =
b
b=2

 230-1cos 2(x  )  1.
Therefore, the equation is yMath
4
16
Determining an Equation From a Graph
A partial graph of a sine function is shown.
Determine the equation as a function of sine.
Amplitude:
3
Vertical Displacement:
2
Period:
 Math 30-1
The equation as a
function of sine is
y  3 sin2x  2.
17
Textbook p. 250 – 255
Low: (Basic Drill and Practice)
1 – 7, 12,
Medium: (Problem Solving and Word Problems)
8, 9, 10, 13, 14, 17, 19, 23, 24, 27, 28
High: (Extension and Higher Level)
11, 15, 16, 20, 21, 22, 26, C3, C4
Math 30-1
18
The End
Math 30-1
19
Identify the key points of your basic graph
1. Find the new period (2π/b)
2. Find the new beginning (bx - c = 0)
3. Find the new end (bx - c = 2π)
4. Find the new interval (new period / 4) to divide
the new reference period into 4 equal parts to
create new x values for the key points
5. Adjust the y values of the key points by applying
the change in height (a) and the vertical shift
(d)
6. Graph key points and connect the dots
Math 30-1
20