Transcript Periodic Functions- Sine Functions

What do these situations have in common? Explain.
Periodic Functions and Trigonometry
Unit Objectives:
• Determine exact values for trigonometric functions: with and without a
calculator
• Write and graph trigonometric functions
• Find amplitude, period, maximums, minimums and phase shifts for
periodic functions
• Model problems using trigonometric functions
Today’s Objective:
I can find a cycle, period and amplitude of periodic function.
What do these situations have in common? Explain
Periodic Function: A function that repeats a pattern of outputs
(y-values) at regular intervals
Cycle:
Period:
One complete pattern
Horizontal length of a cycle
– distance along x-axis
One cycle: 𝑥 = 2 to 𝑥 = 6 or 𝑥 = 8 to 𝑥 = 12
Period: 6 − 2 = 12 − 8 = 4
One cycle
𝑥 = −2 to 𝑥 = 2
𝑥 = 3 to 𝑥 = 7
Period:
2 − (−2) = 4
Determine whether function is periodic.
If so identify one cycle and determine the period.
Not Periodic
One cycle
𝑥 = 0 to 𝑥 = 3
One cycle
𝑥 = −2 to 𝑥 = 8
Period:
3−0= 3
Period:
8 − (−2) = 10
Not Periodic
Maximum
Midline 𝑦 = 1
Minimum
amp. = 3
Midline: Horizontal line midway
between maximum and
minimum values
Amplitude: Half the difference
between maximum and
minimum
𝑦=
1
(maximum
2
1
2
+ minimum)
amp. = (max. – min.)
What is the period, the amplitude
and the equation of the midline for
each sound wave displayed below.
One cycle: 𝑥 = 0.004 to 𝑥 = 0.008
Period: 0.008 − 0.004 = 0.004
Midline: 𝑦 = 2
1
Amplitude: 2 (2.5 − 1.5) = 0.5
One cycle: 𝑥 = 0.006 to 𝑥 = 0.012
Period: 0.012 − 0.006 = 0.006
Midline: 𝑦 = −0.75
Pg. 832
#7-25 odd, 35, 36
Amplitude:
1
((−0.5) − (−1)) = 0.25
2
The Sine Function
𝑦 = sin 𝑥
Today’s Objective:
I can graph the sine function.
𝑓 𝑥 = sin 𝑥
x
0
𝜋
6
𝜋
2
5𝜋
6
𝜋
y
𝑦 = sin 𝑥
OR
𝑦 = sin 𝜃
Domain:
All real numbers
Range:
–1≤y≤1
Period:
2π
Amplitude:
1
Period of a Sine Curve
Calculator
[MODE]: Radians
[WINDOW]
Xmin = 0
Xmax = 2π
Xscl = π/2
Ymin = – 2
Ymax = 2
Yscl = 0.5
𝑌1 = sin 𝑥
𝑌3 = sin .5𝑥
𝑌2 = sin 4𝑥
𝑦 = sin 𝑏𝑥
b = # of cycles from 0 to 2π
2𝜋
Period of function =
𝑏
Amplitude of a Sine Curve
Calculator
[MODE]: Radians
𝑌2 = 2 sin 𝑥
[WINDOW]
Xmin = 0
Xmax = 2π
Xscl = π/2
Ymin = – 2
Ymax = 2
Yscl = 0.5
𝑦 = 𝑎 sin 𝑥
𝑌1 = sin 𝑥
𝑌3 = −2 sin 𝑥
𝑎 = amplitude (stretch)
– a = reflects graph
across x-axis
Find amplitude and period for each equation.
1. 𝑦 = 2 sin 3𝑥
Amplitude = 2
3..
Period = 2𝜋
3
1
2
2. 𝑦 = sin
𝑥
4
Amplitude
1
=
2
or 0.5
2𝜋
= 8𝜋
Period =
0.25
Amplitude = 3
4𝜋
2𝜋
=
Period =
3
𝑏
Write an equation for the
3
graph.
𝑦 = 3 sin 𝑥
2
Sketching a Sine Curve Graph (2 cycles) Amplitude =
2
1. Find amplitude and
period.
2. Plot 5 points:
Midline points
Beginning,
End
Middle
Amplitude points
Max
Min
3. Sketch curve.
𝑦 = 2 sin 2𝑥
2𝜋
=𝜋
Period =
2
5 points: midline – max– midline – min – midline
Sketching a Sine Curve Graph (2 cycles)
1. Find amplitude and
period.
2. Plot 5 points
3. Sketch curve.
𝜋
𝑦 = 1.5 sin 𝑥
2
Amplitude =1.5
2𝜋
Period = 𝜋 = 4
2
Pg. 856 #13-35 odd
5 points: midline – max– midline – min – midline