Sections 9A,9B

Functions, Linear Functions.

...

Generalities

The notion of function is designed to express the idea of dependency.

Specifically, a quantity depends on another one.

The change of one of them yields a change of the other.

Example: bacteria in a bottle.

The amount of bacteria changes with time.

Terminology

The two quantities are called variables.

A function describes how does dependent variable changes

when/if independent variable chages takes place.

Example: bacteria in a bottle.

Elapsed time since 11am is independent variable

Amount of bacterias is dependent variable.

Ways to present a function.

• mathematical formula
• graph
• data table

Formula in the example with bacteria in a bottle

$$N=2^t$$

Independent variable -- t -- time in minutes after 11:00am

Dependent variable -- N -- number of bacterias

Advantage of formula: easy to calculate

Disadvantage of formula: difficult to imagine

Graph in the example with bacteria in a bottle

Advantage of graph: easy to imagine

Data Table in the example with bacteria in a bottle

may be produced out of the formula $$N=2^t$$,
same applies to the graph:
Elapsed time, t Amount of bacterias, N
0 1
1 2
2 4
3 8
4 16
5 32
6 64

Modeling our world

VERY ROUGHLY

One starts with an idea that there may be a dependency between two quantities.

One collects raw data, possibly organized into a data table

One draws a graph from this data table in order to visualize it

One tries to figure out a formula for the function whose graph is close to this one

This function is a model for the dependency

Simplest class of functions -- Linear Functions

Linear Modeling -- modeling with linear functions

For a function, one may ask how does the dependent variable change when the independent variable changes by a unit

$$\text{rate of change} = \frac{\text{change in dependent variable}}{\text{change in independent variable}}$$

Linear function has constant rate of change

Linear function has constant rate of change.

A car is traveling along the road with a speed of 60 mph. The distance of this car to its destination point depends on time. This distance is a linear function in time. It decreases by 60 miles every hour. That is its constant rate of change of -60.

Formula:
distance to destination = initial distance -60 $$\times$$ time.

Linear Functions. General Formula

$$y=mx + b$$

was in example:
distance to destination = initial distance -60 $$\times$$ time.

$$x$$ - independent variable (was time)

$$y$$ - dependent variable (was distance to destination)

$$m$$ - rate of change aka slope of the graph (was speed of -60 mph)

$$b$$ - initial value aka y-intercept on the graph (was 400 miles)

Concluding remarks

The graph of every linear function is a straight line

The slope of this line $$slope = \frac{\text{change in y}}{\text{change in x}}$$

indicates how much the dependent variable $$y$$ changes when the independent variable $$x$$ changes by 1.

However, not every straight line can be a graph of a linear function. Exception: vertical lines are not!
They represent no dependency at all. Instead of showing how $$y$$ depends on $$x$$ this graph indicates that $$x$$ does not depend on $$y$$.