Graphing Form of Sine and Cosine Functions. Period The length of one cycle of a graph.
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Transcript of Graphing Form of Sine and Cosine Functions. Period The length of one cycle of a graph.
Initial Trigonometric Graphing Form
siny a x h k Sine
Cosine
cosy a x h k
Do not write these
on your worksheet yet. We still need
to add one more
parameter.
Requirements for a Sine/Cosine Graph
x-intercept
At least one Period
(in other words, at least 5 consecutive critical points accurately plotted)
1
2
3
4
5
Arrows (to show that there
infinite cycles)
a > 10<a<1a < 0Amplitude:
Half of the distance between the maximum and minimum values of the range of a periodic function with a bounded
range.
The amplitude is the absolute value of a! It is a positive distance.
The Amplitude and the Effect of “a”
a = 1
1Amplitude =
3 0.5 1 0.5siny x
3siny x
siny x siny x
Example: Sine
32siny x y = 0
x = -3π/2
Transformation: Flip the parent graph and translate it 3Pi/2 units to the left.
Transformation:
22
New Equation:
Period:
2You need at least 5 consecutive critical points.
Example: Cosine
2cos 1y x
y = -1
x = -π/2
Transformation: Translate the parent graph Pi/2 units to the left and 1 unit down.
Transformation:
22
New Equation:
Period:
2You need at least 5 consecutive critical points.
Translation - 3 units up and …
Example: Sine or Cosine?
y = 3
New Equation:
Transformation:
22
Amplitude - 2
Orientation -
Graph -
Period - 2π
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.
3 units up and 3π/4 to the left
Example: Sine or Cosine?
342sin 3y x
x = -3π/4
New Equation:
Translation - 3 units up and … Transformation: Amplitude -
Orientation -
2 Graph - Sine
Positive
22
y = 3
Period - 2π
3 units up and π/4 to the left
Example: Sine or Cosine?
42cos 3y x
x = -π/4
New Equation:
Translation - 3 units up and … Transformation: Amplitude -
Orientation -
2 Graph - Cosine
Positive
22
y = 3
Period - 2π
Changing the PeriodFind the period for each graph and generalize the result.
siny x 14cosy x
cos 4y x sin 2y x
1 cycle in 2π
2 2
22
Period = 2π
1/4 cycle in 2π
Period = 8π
2 cycles in 2π
Period = π
4 cycles in 2π
Period = 0.5π
2The coefficient of Period x
21 Period 2 2
1 4 Period 2 4 8
22 Period 2 1
4 2Period
Determining the Period of Sine/Cosine Graph
If or , the period (the length of one cycle) is determined by:
Ex: What is the period of ?
siny bx cosy bx
2 Period b
7sin 3 2f x x
23 Period
Changing the Period w/o Affecting (h,k)
sin 2 3 4y x
sin 2 3 4y x or
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)?
Notation: Trigonometric Functions
sin 2 56
y x
sin 2 56
y x
is equivalent toCorrect way for
the calculator!
Example: Sine
0.5sin 2 1y x
y = -1
x = π/2
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.
New Equation:
Transformation:
22
You need at least 5 consecutive critical points.
22 Period:
20.5sin 2 1y x
Not in Graphing
form
Example: Cosine
12cos 1y x
y = 1
x = 0
Transformation: Change the period to 4π and translate the parent graph 1 unit up.
Transformation:
22
You need at least 5 consecutive critical points.
New Equation:
21 2 Period:
4
Translation - 2 units down and …
Example: Sine or Cosine?
y = -2
New Equation:
Transformation:
2
Amplitude - 1.5
Orientation -
Graph -
Period - π/2
22 b
4b
Period:
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.
2 units down
Example: Sine or Cosine?
1.5cos 4 2y x
x = 0
New Equation:
Translation - Transformation: Amplitude -
Orientation -
Graph - Cosine
Positive
1.5 2 units down and …
22 b
4b
Period:
Period - π/2
2
y = -2