Rectangular Coordinates, Graphs, and Introduction to Functions

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, where x is the horizontal coordinate and y is the vertical coordinate.

• Example: The point (transportation, 8796) represents the amount spent on transportation.

• Example: The point (healthcare, 9728) represents the amount spent on healthcare.

The Rectangular Coordinate System

The rectangular coordinate system (also called the Cartesian coordinate system) is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four regions called quadrants.

• The plane is called the coordinate plane.

• The axes divide the plane into four quadrants.

Distance Formula

The distance formula gives the distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ in the coordinate plane:

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

• Example: The distance between $P(3, -5)$ and $Q(-2, 8)$ is $\sqrt{(3 - (-2))^2 + ((-5) - 8)^2} = \sqrt{25 + 169} = \sqrt{194}$.

If the sides a, b, and c of a triangle satisfy $a^2 + b^2 = c^2$, then the triangle is a right triangle with legs a and b and hypotenuse c.

The Midpoint Formula

The midpoint of a line segment with endpoints $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by:

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

• 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

An equation in two variables can be represented by a set of ordered pairs (x, y) that satisfy the equation. These pairs can be plotted to form the graph of the equation.

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

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

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

Graphing Equations

To graph an equation:

1. Find the intercepts.

2. Find additional ordered pairs as needed.

3. Plot the ordered pairs.

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

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

• Example: The graph of $x = y^2 + 1$ is a parabola opening to the right.

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

Circles in the Coordinate Plane

Center-Radius Form

A circle is the set of all points in a plane that are a given distance (radius) from a fixed point (center). The center-radius form of the equation of a circle with center $(h, k)$ and radius $r$ is:

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

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

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

General Form of the Equation of a Circle

The general form of the equation of a circle is:

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

To find the center and radius from the general form, complete the square for both x and y.

• Example: $x^2 + 4x + y^2 - 8y = 45$ can be rewritten as $(x + 2)^2 + (y - 4)^2 = 65$, so the center is $(-2, 4)$ and the radius is $\sqrt{65}$.

Application: Locating the Epicenter of an Earthquake

Given distances from three stations to the epicenter, the intersection point of three circles (each representing a fixed distance from a station) gives the location of the epicenter.

Relations and Functions

Relations and Functions: Definitions

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).

• Example: $M = \{(-4, 0), (-3, 1), (3, 1)\}$ is a function.

• Example: $N = \{(-2, 4), (2, 4), (5, -4)\}$ is a function.

• Example: $P = \{(-4, 3), (0, 6), (2, 8), (-4, -3)\}$ is not a function because the input -4 corresponds to two different outputs (3 and -3).

Domain and Range

The domain of a relation or function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

Function Notation

Functions are often written as $f(x)$, where $x$ is the input and $f(x)$ is the output.

Increasing, Decreasing, and Constant Functions

• A function is increasing on an interval if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.

• A function is decreasing on an interval if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.

• A function is constant on an interval if $f(x_1) = f(x_2)$ for all $x_1, x_2$ in the interval.

Summary Table: Forms of the Equation of a Circle

Form	Equation	Center	Radius
Center-Radius Form	$(x - h)^2 + (y - k)^2 = r^2$	$(h, k)$	$r$
General Form	$x^2 + y^2 + Dx + Ey + F = 0$	Complete the square to find	Complete the square to find

Additional info: Some context and explanations have been expanded for clarity and completeness, including definitions, step-by-step examples, and the summary table.