#
Exponential Growth

**[y_=_x_+_1]**
**[y = x * x]**
**[Phi_=_1.618...]**
**[Phi_=_-0.618?]**

**[Derive_Phi_Graphically]**

## y = x * x

This function represents a process of **growth** which always **multiplies** by the
same relative amount. In this, the most simple case, it
**multiplies by itself**. When you connect the dots, you get a particular type of
curved line called a **parabola**.
We have a shorthand for expressions like '**x * x**':

2
x * x ==> x

Because the '**2**' is called an *exponent*, this type of function is said
to be *exponential*.

With an **exponential function**, when you change **x** by some amount,
**y** always changes at a rate that depends directly on the value you're changing
**from**. The larger **x** is to start with, the more **y** will increase
relative to the increase in **x**.
This is the sort of growth seen in nearly '**mathematically perfect**' systems,
such as large **populations** of lifeforms. Exponential functions can
grow to very large numbers very quickly.

**Geometric proportion** is always represented by an **exponential function**.

**All the values** that will ever appear on the '**x * x**' side
of our equation will be somewhere on this **red line**.

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