2. Warm-up
Describe the transformation for each of the
following equations, as based on the parent equation
of y = sin x.
1. y = sin(x + 14) 2. y = sin x + 14
Shifted 14 units left Shifted 14 units up
3. y = sin x - 14 4. y = sin(x - 14)
Shifted 14 units down Shifted 14 units right
3. Phase Shift
The smallest positive or largest negative number
used in a horizontal translation for a sine or cosine
wave
4. Example 1
a. Graph two cycles of the following function
(
y = cos x − π
2 )
b. What is the phase shift of this function when
compared to the parent function y = cos x?
π
...to the right
2
5. c. What is the phase shift when compared to
y = sin x?
It IS y = sin x
6. Theorem
We can rewrite some of these phase shifts quite
easily:
sin: + + - - cos: + - - +
Just follow these patterns
7. Example 2
Find another possible equation for each function
listed below.
a. y = cos ( π
2
−x ) (
b. y = sin x − π )
++-- --++
y = sin x y = -sin x
8. Example 3
Compare the following graphs.
y = sin x y = sin x + ( 5π
6 )+2
5π
Phase shift: −
6
Vertical Shift: 2
9. Example 4
Compare the following graphs:
y = tan x y = −1 + tan x
There was a vertical shift of -1
10. In phase:
When voltage and current flow coincide with each
other
Out-of-phase:
When the current is behind the voltage
Inductance:
When current flow is out-of-phase
11. Example 5
In question 14 from section 4-7, AC current y was
modeled with the equation
(
y = 10 sin 120πx )
where x is time in seconds. Maximum inductance
π
occurs when the voltage leads the current by
2
seconds. What is an equation for the current in the
new model?
⎛ ⎡ π ⎤⎞
y = 10 sin ⎜ 120π ⎢x − ⎥ ⎟
⎝ ⎣ 2 ⎦⎠
