Section 2.1 - Rectangular Coordinates and Graphs

Section 2.1 - Rectangular Coordinates and Graphs#

Definition 4 (Major Definitions:)

An ordered pair consists of two components, written inside parentheses.
Given two real number lines intersecting perpendicularly at $(0, 0)$ makes up the rectangular coordinate system.
The intersection of the two number lines at $(0, 0)$ is called the origin.
The number lines are called axis, the horizontal axis is the $x$ -axis and the vertical axis is the $y$ -axis.
The plane into which the coordinate system is introduced is the coordinate plane, or $x y$ -plane.
The $x$ -axis and $y$ -axis divide the plane into four regions, or quadrants. Points on the $x$ -axis or $y$ -axis belong to no quadrant.
Let $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ be two points on the $x y$ -plane
$\begin{array}{r} \begin{aligned} Δ x & = x_{2} - x_{1} \\ Δ y & = y_{2} - y_{1} \\ d & = \sqrt{{(Δ x)}^{2} + {(Δ y)}^{2}} \\ = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} \\ M & = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) \end{aligned} \end{array}$
where $Δ x$ is the change in $x$ , $Δ y$ is the change in $y$ , $d$ is the distance between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , and $M$ is the midpoint between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .
The solution to a one variable equation can be represented as $x = a$ on the $x$ -axis.
The solution to a two variable equation can be represented as an ordered on the $x y$ -plane.
The graph of an equation is found by plotting ordered pairs that are solutions to the equation, sometimes called constructing a table of values. The intercepts of the graph are good points to plot first. An $x$ -intercept is a point where the graph intersects the $x$ -axis. A $y$ -intercept is a point where the graph intersects the $y$ -axis.

Example 24

Determine whether the three points are collinear.

$(0, - 7)$ , $(- 3, 5)$ , and $(2, - 15)$

$(- 1, - 3)$ , $(- 5, 12)$ , and $(1, - 11)$

Example 25

Plot the following from a table of values:

$x$	-7	-5	4	6
$y$	-5	4	-2	5

$x$	-3	-2	-1	0
$y$	9	4	1	0

Given the two variable equation construct a table of values and plot the ordered pairs on the $x y$ -plane.

$2 y - x = - 2$
$y - \sqrt{x} = - 3$
$y = x^{2}$