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

**[Why A Graph?]**

First off, let's **observe convention:**
In the **language of mathematics**, we will be working with a **variable** we'll call
'**x**', **the unknown**.
We want to find a value for **x** that makes our **equation true**. When
that happens (*if* it happens), the number that **x** equals at that time
will be **phi**. So let's start by restating our equation as:
**x + 1 = x * x**.

An **equation** is a statement that **shows a relationship**
between **two sets of numbers**. It states that,
**under the conditions it describes**, the expression on the **left**
side of the '**=**' sign has the **same value** as the expression
on the **right** side.

A **special** type of **equation** is the *function*.
A **function** says that if you **perform the calculations**
('*evaluate the expression*') described on the **right-hand** side
of the '**=**' sign, you will get some **result**.
By **convention**, we generally call the
**number you perform the calculations on**
'**x**' and the **result** of the calculations '**y**'.

The **simplest functions** just return a **constant value**;
'**y = 1**' is an example. This says that, **no matter**
what value **x** takes on,
**y** will always be **1**. Well, so what? **x** isn't even in the equation.

Things get **more interesting** when you **bring in x**.
For instance, '**y = x**' says that no matter what value you give **x**,
**y** will be equal to the **same number**.
Or, remembering **simple geometry**, '**y = pi * x**' tells you that,
in a **circle**, if you take the value of the **diameter** (**x**) and multiply it by **pi**,
the result you get in **y** will be the **circumference of the circle**,
**no matter** what **diameter circle** you're talking about.

There are **two ways** we usually **solve such equations**.
One is to use **algebra**, which gives **more accurate results**
but at the expense of **only yielding up a number**.

The **other** is through the use of **graphs**,
which don't give as accurate a solution but **show what a function 'looks' like**.

So let's go back to our equation. It says that there exists
**some number** ('**x**') such that when you **add 1** to it
you get the **same number** as when you **multiply it by itself**.
Wait a minute - we're **performing calculations** and **getting results** here;
isn't that what **functions do**?
The equation really says that **two functions return the same number**, **doesn't it?**
Let's **rewrite it** as two functions,

**
y = x + 1
y = x * x
**

A while back a man named **Descartes** realized that you could take a **function**
and, **instead** of **solving** it for just **one value**, solve it for a **whole range**
of values. You imagine that every **x** corresponds to a point on a **horizontal line**
(the '**x axis**'), and every **y** to a point on a **vertical line** (the '**y axis**').
Then, for every **x** on the horizontal line, you calculate the associated **y**,
move **vertically** from the **horizontal** line, and **mark a point**.
**With any luck**, when you're done the result is like a
**connect-the-dots drawing** of the function.

What happens when we **graph our functions**?

**
****[The_Golden_Mean]
[The_Graphical_Derivation_of_Phi]
[Derive_Phi_Exactly]
[Unfold_the_Golden_Rectangle]
]
**