Rectangular Coordinates, Graphs, and Circles: Precalculus Study Notes

Rectangular Coordinates and Graphs

Ordered Pairs

In mathematics, an ordered pair is a pair of elements written in a specific order, usually as (x, y). Ordered pairs are used to represent points in the plane.

• Definition: An ordered pair (x, y) consists of two elements, where x is the first component and y is the second.

• Application: Used to express relationships between two quantities, such as category and amount spent.

• Example: (Transportation, 8576) or (Healthcare, 9728)

The Rectangular Coordinate System

The rectangular coordinate system (also called the Cartesian coordinate system) is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis).

• The plane is divided into four regions called quadrants.

• Each point in the plane is identified by an ordered pair (x, y).

• The axes intersect at the origin (0, 0).

Distance Formula

The distance formula calculates the distance between two points in the coordinate plane.

• Formula:

$d(P, R) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: The distance between P(3, -5) and Q(-2, 8) is:

$\sqrt{(-2-3)^2 + (8-(-5))^2} = \sqrt{25 + 169} = \sqrt{194}$

• Application: Used to determine lengths, such as the sides of a triangle.

The Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points.

• Formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: The midpoint of (-7, -5) and (2, 13) is:

$\left( \frac{-7+2}{2}, \frac{-5+13}{2} \right) = (-2.5, 4)$

Equations in Two Variables

Equations involving two variables can be represented as sets of ordered pairs that satisfy the equation.

• Example: For $y = -2x + 5$, some solutions are (0, 5), (1, 3), (2, 1).

• For $x = \sqrt{y+1}$, solutions include (1, 0), (0, -1), (-2, -9).

Graphing Equations

To graph an equation in two variables:

1. Find the intercepts.

2. Find additional ordered pairs as needed.

3. Plot the points on the coordinate plane.

4. Join the points with a smooth line or curve.

• Example: The graph of $y = -2x + 5$ is a straight line.

• The graph of $x = \sqrt{y+1}$ is a sideways parabola.

• The graph of $y = -x^2 + 1$ is a downward-opening parabola.

Circles in the Coordinate Plane

Center-Radius Form

A circle is the set of all points in a plane that are a fixed distance (radius) from a given point (center).

• Center-radius form:

$(x - h)^2 + (y - k)^2 = r^2$

• where (h, k) is the center and r is the radius.

• Example: Center (1, -2), radius 3: $(x-1)^2 + (y+2)^2 = 9$

• Center (0, 0), radius 2: $x^2 + y^2 = 4$

General Form of the Equation of a Circle

The general form of a circle's equation is:

$x^2 + y^2 + Dx + Ey + F = 0$

• Can be converted to center-radius form by completing the square.

• Example: $x^2 + 4x + y^2 - 8y = 45$ can be rewritten as $(x+2)^2 + (y-4)^2 = 65$

Determining Existence of a Circle

If the radius squared is negative after completing the square, the graph does not represent a real circle.

• Example: $(x-3)^2 + (y+1)^2 = -2$ has no real solution, so the graph is nonexistent.

Applications

Equations of circles can be used to solve real-world problems, such as locating the epicenter of an earthquake using distances from known points.

Relations and Functions

Relations and Functions: Domain and Range

A relation is a set of ordered pairs. A function is a relation in which each input (first component) corresponds to exactly one output (second component).

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Example: N = {(-4, 0), (-3, 1), (4, 5), (5, -4)} is a function because each x-value is unique.

• P = {(4, -3), (0, 6), (2, 8), (-4, -3)} is not a function if any x-value repeats with a different y-value.

Identifying Functions

• To determine if a relation is a function, check that no x-value is paired with more than one y-value.

Summary Table: Key Formulas

Concept	Formula	Description
Distance between (x1, y1) and (x2, y2)	$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$	Distance Formula
Midpoint between (x1, y1) and (x2, y2)	$\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)$	Midpoint Formula
Circle with center (h, k) and radius r	$(x-h)^2 + (y-k)^2 = r^2$	Center-Radius Form
General form of a circle	$x^2 + y^2 + Dx + Ey + F = 0$	General Form

Additional info: These notes cover foundational concepts in analytic geometry, including coordinate systems, equations of lines and circles, and the basics of functions and relations, all of which are essential for Precalculus students.