My name is Professor Aburto!

Today, I will be teaching the class about functions, including linear functions

First of all, you guys will have to know the key points about functions

• Every input has an output
• Pocess the VLT (Vertical Line test)
• Functions CAN NOT have multiple outputs for one input
• A mathematical model is a function where the outputs are dependent on the input

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Example:
Function
(y) yards : output
(t) time : input
y = f(t)
                                            Peyton Manning passing yards over 5 years

Time	Yards
2009	3500
2010	4216
2011	5857
2012	6510
2013	5218

Is this example a Mathematical model? NO

Functions can be represented in (graphs, tables, formulas, words)
T = f(h)
If q is a function of t, Q= f(t)
Q: Output, dependent
f(t): Input, Independent

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Example: Regular Sea Level VS. Sea Level Depth


This function IS linear because the rate of change is CONSTANT (look below for explanation)

This is how check for linearity


(30-16) / (34-0)= 7/17, (37-30) / (51-34)=7/17, (44-37) / (68-51)=7/17

7/17 is the constant rate of change.

Rate of Change (ROC)
Is the ROC constant at every interval?

General formula for the family of linear functions
Output = intitial + ROC * input
example:
Pressure = initial (y int) + ROC * depth pressure

Slope Intercept Fomula: y = b + mx OR mx + b

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example:
We can estimate temperature by counting the number of times a snowy tree cricket chirps in 15 sec. and adding 40. T is the output, R is the input value. Evaluate in minutes (1/4 of a minute)

T= 1/4R + 40

(T is the output, 1/4 is the rate of change, R is the input, and 40 is the y int/initial value)

R (Chirp rate)	T (Temperature)
0	40
20	45
40	50
60	55
80	60
100	65
120	70
140	80

Check for linearity, use points from the table.


(45-40) / (20-0) = 1/4 , (50-45) / (40/20) = 1/4

So, this IS a linear function because the ROC is 1/4 a constant value.

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Point Slope Formula: y-y1 = m(x-x1)
example: (40,50) point from chart
y-y1 = 1/4(x-x1)
y - 50 = 1/4 (x - 40)
y = 1/4x + 40