We use math to help describe real world phenomenon by establishing a relationship between variables. Often this is started with the simple equality:

$$ y = f(x) $$

where y, a dependent variable, is a function of x, an independent variable.

Now we can define f(x) as a polynomial, which is just an expression with many terms. A linear function which is a polynomial with degree one

$$ f(x) = ax + b $$

A polynomial with degree two can be expressed as

$$ f(x) = ax^2 + bx + c $$

and is a quadratic function.

We can graph all of these by plotting the coordinates on a cartesian plane. Then we can calculate various things. For a linear polynomial we can discover the slope of the line as well as its root or where on the x axis y is zero.

Determining the root(s) of a quadratic function become more complicated but still require simple algrebra but also the introduction to complex numbers.

For instance the root of the quadratic function when b squared is greater than 4ac is

$$ x = \frac {-b \pm \sqrt {b^2 - 4ac}} {2a} $$

which is known as the quadratic formula.

When b squared is equal to 4ac then it’s equal to

$$ x_1 = x_2 = - \frac b {2a} $$

When b squares is less than 4ac then it can be expressed in terms of complex numbers.

Beyond quadratic functions there are cubic functions and quartic functions with degree 4.

Linear factors. There are n linear factors of an nth degree polynomial.

Orders of magnitude are used often to simplify the notation of the representative data. We typically use the Power of 10 or sometimes other bases such as 2. It doesn’t really matter what base we use to express the information. One order of magnitude that is used is “e” where:

$$ e = 2.718281828 $$

This can be explained by looking at a simple growth rate starting at time zero then moving forward in time in discrete steps. We then assume there is some constant that increases y at each step and that constant can be something like a birth rate (i.e. growth rate)

$$ N(t) - N(0) = tcN(0) $$ $$ N(2t) - N(t) = tcN(t) $$ $$ N(3t) - N(2t) = tcN(2t) $$ $$ … $$ $$ N(nt) = N(0)(1 + \frac {ct} n)^2 $$

If we then increase n to infinity and take the limits

$$ N(t) = \lim_{n\rightarrow \infty} \left\lbrack N(0)(1 + \frac {ct} n)^2 \right\rbrack $$

And from that work we can generically get the exponential functions

$$ e^x = \lim_{n\rightarrow \infty} \left\lbrack \left( 1 + \frac x n \right) ^n \right\rbrack $$