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# System prepended metadata

title: The Math 4i Reference Textbook

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# The Math 4i Reference Textbook
This book is a collaborative effort that will serve as a living text reference for learners in Math 4i. For proper formatting, we need to agree on certain conventions.

#### Usage
1. Do NOT use “normal text” for titles of chapters and sections.
2.  DO use consistent styles for chapters (“Heading 2”) and sections (“Heading 3”).
3.  It is OK to reorganize the material as we cover more topics and concepts. Some chapters might become sections, etc...
4.  Let’s use “Heading 4” for the headers of definitions, examples, remarks, or even usage, etc...
5.  It is easy to include images and even interactive Desmos or GeoGebra plots by copying and pasting export codes. 
6.  Finally, make sure to use proper grammar and mathematical notation.

## Hyperbolas

### The Geometric Definition of a Hyperbola

First, let's look at an example. Say we have two given points in the $xy$-plane, $F_1 = (-2,0)$ and $F_2 = (2,0)$. We can ask: "What would the collection of all points $P$ such that $PF_1 - PF_2 = \pm 2$ look like?"  It turns out just like on the graph below.
<iframe src="https://www.geogebra.org/calculator/vzsq3quu?embed" width="800" height="600" allowfullscreen style="border: 1px solid #e4e4e4;border-radius: 4px;" frameborder="0"></iframe>

## Polar Coordinates
We are used to giving coordinates for a point $P$ in the plane in the form $(a,b)$, where $a$ represents the horizontal "signed" distance between the point $P$ and the $y$-axis.  The number $a$ is also called the _$x$-coordinate_ of the point $P$.  Similarly $b$, the _$y$-coordinate_ of $P$, is the vertical "signed" distance between $P$ and the $x$-axis.  We call $(a,b)$ the _rectangular_ or _cartesian_ coordinates of point $P$.

Polar coordinates are another way to describe the position of $P$ in the plane.
### Definition
> The polar coordinates of a point $P$ in the $xy$-plane are a pair of number $(r;\theta)$ in which $r$ is the distance from $P$ to the origin $O$, and $\theta$ is the size of an angle in _standard position_ that has $\overrightarrow{OP}$ as its terminal ray.

### Caution
There is more than one correct value for $\theta$.  (And also for $r$, but more on that later.)




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