1. Periodic Functions

2. These are curves based on sine and cosine
These curves fluctuate between a maximum and a minimum.
There are specific terms we must know here:
Crest: - top of wave
- this also known as the maximum
2) Trough: - bottom of wave
- this is also known as the minimum
3) Period: - from the beginning of a curve to the beginning of the next curve (one max and one min)
4) Amplitude: the height to the max/min from the axis of symmetry
5) Axis of Symmetry: a horizontal line that cuts the curve into mirror halves. Express this as y = ____. (max + min /2)

3. 6) x-intercepts (zeroes): where curve crosses the x-axis
6) x-intercepts (zeroes): where curve crosses the x-axis. Express this as x = ____. ex 1: Find the six categories for:
Max: 2
Min: -2
Axis of Symm: y = 0
Amp: 2
Period: (5 – 1) = 4
Zeros: x = 0, 2, 4, 6, 8
ex 2: y = sin 2x
Amp: 1
Period: (4.71 – 1.57) = 3.14
Zeros: x = 0, 1.57, 4.71, 6.28
Max: 1
Min: -1
Axis of Symm: y = 0

4. Finding Graphs from Data
In order to find an equation from data, we must plug in our data into L1 and L2.
From here we use “Sin Reg”
STAT → CALC → C:SinReg → Enter → L1 → , → L2 → , → VARS → Y-Vars → 1 → 1 → Enter
This will generate a periodic equation of the form (in the y= screen):
y = Asin(Bx + C) + D
where A = amplitude
B = period (= 2π/B)
C = horizontal shift ( -ve is right, +ve is left)
D = vertical shift (axis of symm/median)

5. Find period from equation and then in months Horizontal shift
From here we can graph, adjusting our window according to the question.
ex:
Month J F M A S O N D
Temp
-15
-10 -3 5 12 20 28 26 13 2
-17
Plot a graph for 2 years
Find period from equation and then in months
Horizontal shift
Vertical shift
Write the equation

6. a) Set window and graph:
b) Period (from equation) = 0.55
Period (in months) = 2π/B
= 2π/0.55
= 11.42
c) C = which means 2.25 to the right
d) D = 5.37
e) y = 20.48sin(0.55x – 2.25)

7. Formulas from Graphs
When given a graph, you must first find a series of points on the wave (at least 5).
Then you can plug them into L1 and L2 so that you can use SinReg.
Remember that L1 are your x-values (for time usually) and L2 are your y-values (for the other variable).
ex 1: Find the formula for:
(6, 3)
(2, 3)
(7, 2)
(1, 2)
(3, 0)
(5, 0)
(4, -1)
y = 2.2sin(1.36x )

8. ex 2: A paddlewheel boat has a reflector on it’s wheel
ex 2: A paddlewheel boat has a reflector on it’s wheel. After 3 seconds, the reflector is at it’s max. height. The height varies sinusoidally with time and completes it’s cycle every 12 seconds.
a) Draw a picture and find the equation
Time
Height
10 ft
reflector
(3, 8)
3 ft
water
height
time
(9, -2)
y = 5sin(0.524x) + 3
b) How long (in seconds) is the reflector under water?
2nd Calc → zero to find the two zeroes
x = x = 10.76
10.76 – 7.22
= 3.54 seconds under water

9. When Asked… When given a formula and asked to find:
1) Something on the y-axis (gives you an x-value):
- 2nd trace → 1:value (enter the number from question)
**give the y-value for the answer**
2) Something on the x-axis (gives you a y-value):
- go into y= and plug that value into y2
- graph
- 2nd trace → 5:intersect
**give the x-value for the answer**

10. Window Settings for Formulas
x-min: 0 x-max: read question x-scl: set this according to the x-max (usually 1 unless dealing with hundreds/thousands) y-min: D - A y-max: D + A