March 24, 2021
Unit 5: Sine Functions
Lesson 27: Sinusoidal Functions
Example 1:
Graph y = 3sinx
Parent function is y = sinx
Here, we have a stretch of
0
(90 , 3)
factor 3 -> the amplitude is 3.
(0,0)
(3600, 0)
(1800, 0)
(2700, -3)
How does that compare to y = sinx?
y = 3sinx
y = sinx
March 24, 2021
Example 2:
Graph y = 2sinx
How does that compare to y = sinx?
y = 2sinx
y = sinx
March 24, 2021
Example 3:
Graph y = 0.5sinx
How does that compare to y = sinx?
y = 0.5sinx
y = sinx
March 24, 2021
Example 4:
Graph y = ­sinx
How does that compare to y = sinx?
y = ­sinx
y = sinx
March 24, 2021
The graph of y = asinx will be a periodic function
that has the same x­intercepts as y = sinx.
• The amplitude of y = asinx is |a| (don't take the
negative sign)
• If a is negative, there is a reflection in the x­axis.
March 24, 2021
Example 5:
functions?
What is the equation of each of the following
(900, 4)
y = 4sinx
a)
y=0
(3600, 0)
(1800, 0)
(0,0)
(2700, -4)
b)
(2700, 2)
y = -2sinx
(1800, 0)
(0,0)
(900, -2)
(3600, 0)
March 24, 2021
y = sinx + 1
y = sinx
Amplitude: y = max - min
2
y = sinx + 1
y= 2-0
2
y=1
y=1
March 24, 2021
Example 5:
Graph y = sinx + 1
How does that compare to y = sinx?
y = sinx + 1
y = sinx
March 24, 2021
Example 6:
Graph y = sinx ­ 2
How does that compare to y = sinx?
y = sinx
y = -2
y = sinx ­ 2
March 24, 2021
The graph of y = sinx + c will be a vertical
translation of the graph of y = sinx.
If c is positive, the graph moves up c units.
If c is negative, the graph moves down c units.
NOTE: In MathSpace, the equation is written
y = c + sinx
Also note that the axis of the curve (or midline as
Mathspace calls it) is always y = c
March 24, 2021
Example 7:
functions?
What is the equation of each of the following
Axis of Curve: y = 3 + 1 = 2
2
a)
y = sinx + 2
y=2
Axis of Curve: y =
b)
0.5 + (-1.5) = -1
2
2
y = -1
2
y = sinx - 1
2
March 24, 2021
Example 8:
Graph y = sin(x ­ 500)
This is called a horizontal
translation. When talking
about periodic functions, we
say it is a phase shift.
How does that compare to y = sinx?
y = sin(x ­ 500)
y = sinx
March 24, 2021
Example 9:
Graph y = sin(x + 100)
How does that compare to y = sinx?
y = sin(x + 100)
y = sinx
March 24, 2021
The graph of y = sin(x ­ c) will be a horizontal
translation (called a phase shift)
of the graph of y = sinx.
If there is a minus in the brackets, move right.
If there is a plus in the brackets, move left.
March 24, 2021
Example 10:
functions?
What is the equation of each of the following
Axis of Curve: y = 1 + (-1) = 0
2
a)
(300, 0)
y=0
y = sin(x - 300)
b)
Axis of Curve: y = 1 + (-1) = 0
2
y = sin(x+500)
(-500, 0)
y=0
March 24, 2021
Example 11: Given y = 2sinx. What is the
amplitude?
amplitude is 2
Example 12: Given y = sinx + 7. What is the axis
of the curve?
axis of curve is y = 7
Example 13: Given y = sin(x + 700). What is the
phase shift?
phase shift (horizontal translation) left 700