Download Graphing Form of Sine and Cosine Functions

Graphing Form of Sine and
Cosine Functions
Period
The length of one cycle of a graph.
The Amplitude and the Effect of “a”
Amplitude:
Half of the
distance between
the maximum and
minimum values
of the range of a
periodic function
with a bounded
range.
a
0<a<1
=<
a
>10
1
y  sin
 sin
0.5sin
 xxx 
y  3sin  x 
Amplitude =
0.5
1
3
The amplitude is the absolute value of a! It is a
positive distance.
Initial Trigonometric Graphing Form
Do not
write these
on your
worksheet
yet. We
still need
to add one
more
parameter.
Sine
y  a sin  x  h   k
Cosine
y  a cos  x  h   k
Requirements for a Sine/Cosine Graph
x-intercept
2
Arrows
(to show
that there
infinite
cycles)
1
3
5
4
At least one Period
(in other words, at least 5 consecutive
critical points accurately plotted)
Example: Sine
Transformation: Flip the parent graph and
translate it 3Pi/2 units to the left.
The Key Point is on the Sine Graph... Use the period to plot the point where it repeats…
Equally space out the remaining 3 critical points…
2
Plot more cycles by continuing the pattern.
New Equation:


y=0
2
y   sin  x 
3
2

Period:
Sine typically goes up first, but this graph
has been flipped so go down first.
x = -3π/2
2
You need at least 5 consecutive critical points.
Example: Cosine
Transformation: Translate the parent graph Pi/2
units to the left and 1 unit down.
The Key Point is not on the Cosine Graph...
Use the period to plot the point where it repeats…
Equally space out the remaining 3 critical points…
Plot more cycles by continuing the pattern.
Cosine starts
one above.
2
New Equation:
y  cos  x  2   1



2
Period:
y = -1
x = -π/2
Cosine goes down first.
2
You need at least 5 consecutive critical points.
Sine v Cosine
Sine
Cosine
The cosine
graph is a
horizontal
translation of
the sine graph
(and vice
versa)
Example: Sine or Cosine?
Transformation: Amplitude - 2 Graph Translation - 3 units up and …
Period - 2π
Orientation Find the
length of half
of the height.
Find where
the graph
can be cut in
half
vertically.
Since the Sine and Cosine
graphs are periodic and
translations of each other, there
are infinite equations that
represent the same curve.
2
Here are two examples.
y=3
2

Find how long does it take
to repeat itself.

New Equation:
Example: Sine or Cosine?
Transformation: Amplitude - 2 Graph - Sine
Translation - 3 units up and 3π/4
… to the left
Period - 2π
Orientation - Positive
The Key
Point is
only on
the Sine
Graph.
The parent Sine graph goes up first.
y=3
x = -3π/4
2

New Equation:
3
y  2sin  x 
4
3
Pick any critical point to draw the vertical axis to find the horizontal shift.

2
OR
Example: Sine or Cosine?
Transformation: Amplitude - 2 Graph - Cosine
Translation - 3 units up and π/4
… to the left
Period - 2π
Orientation - Positive
The Key
Point is
only on
the Sine
Graph.
The parent Cosine graph goes down first.
y=3
2
x = -π/4

New Equation:

y  2cos  x  4   3
Pick any critical point to draw the vertical axis to find the horizontal shift.

2
Changing the Period
Find the period for each graph and generalize the result.
y  sin  x 
1 cycle in 2π
2
Period = 2π
y  cos  14 x  1/4 cycle in 2π
2
Period = 8π
Period  21  2
Period  124  2  4  8
y  sin  2 x  2 cycles in 2π
y  cos  4 x  4 cycles in 2π
2
Period  22  
Period = π
2
Period  24  12 
2
Period  The coefficient
of x
Period = 0.5π
Determining the Period of Sine/Cosine Graph
If y  sin  bx  or y  cos  bx  , the period (the
length of one cycle) is determined by:
Period 
2
b
Ex: What is the period of f  x   7sin  3x   2?
Period 
2
3
Changing the Period w/o Affecting (h,k)
The key point (h,k) is a point on the sine graph.
Also, multiplying x by a constant changes the
period. Below are two different ways to write a
transformation. In order for the equation to be
useful, it must directly change the graph in a
specific manner. Which equation changes the
period and contains the point (-3,4)?
y  sin  2  x  3   4
or
y  sin  2 x  3  4
Graphing Form for Trigonometric
Functions
Sine
y  a sin b  x  h   k
2

Period:
b
Amplitude:
a
Height:
2 a
Cosine
y  a cos b  x  h   k
Period:
2
b
Amplitude:
a
Height:
2 a
Notation: Trigonometric Functions
Correct
way for
the
calculator!


y  sin 2  x    5
6

is equivalent to
 
 
y  sin  2  x     5
6 
 
Example: Sine Again
Transformation: Change the amplitude to 0.5
and the period to π. Then translate it π/2 units
to the right and 1 unit down.
Not in
The Key Point is on the Sine Graph...
Equally space out
the remaining 3
critical points…
2

New Equation:
Plot more cycles by
continuing the
pattern.

2
y = -1
x = π/2
Use the period to plot the
point where it repeats…
Graphing
form
y  0.5sin  2 x     1
y  0.5sin 2  x  2   1

Period:
Use the b
value to find
the period.
2
2
 
You need at least 5 consecutive critical points.
Example: Cosine Again
Transformation: Change the period to 4π and
translate the parent graph 1 unit up.
The Key Point is not on the Cosine Graph...
2
Use the period to plot the point where it repeats…
Equally space out the remaining 3 critical points…
y=1


2
New Equation:
y  cos  x   1
1
2
Period:
x=0
Use the b
value to find
the period.
2
12
 4
You need at least 5 consecutive critical points.
Example: Sine or Cosine?
Transformation: Amplitude -1.5 Graph Translation - 2 units down and …
Period - π/2
Orientation Find how long does it take
to repeat itself.

2

New Equation:
Since the Sine and Cosine graphs are periodic and
translations of each other, there are infinite equations
that represent the same curve. Here are two
examples.
Find the
length of
half of the
height.
y = -2
Find where
the graph
can be cut in
half
vertically.
Period:

b4

2
2
b
Example: Sine or Cosine?
Transformation: Amplitude -1.5 Graph - Cosine
Translation - 2 units down and …
Period - π/2
Orientation - Positive
x=0
The Key
Point is NOT
on the
Cosine
Graph.

2

New Equation:
y  1.5cos 4  x   2
y = -2
Period:

b4

2
The parent Cosine graph goes down first.
2
b
OR
Example: Sine or Cosine?
Transformation: Amplitude -1.5 Graph - Sine
Translation - 2 units down and 5π/8
… to the right
Period - π/2
Orientation - Negative
x = 5π/8
The Key
Point is
only on
the Sine
Graph.

2

New Equation:
y   sin 4  x 
3
2
y = -2
Period:

b4

2
The parent Sine graph goes up first.
2
b
5
8
2