Rectangular Coordinates, Circles, and Relations in College Algebra

Rectangular Coordinates and Graphs

Ordered Pairs and the Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, is formed by the intersection of the x-axis (horizontal) and y-axis (vertical), dividing the plane into four quadrants. Each point in the plane is represented by an ordered pair (x, y).

• Quadrants: The plane is divided into four regions called quadrants, labeled I, II, III, and IV.

• Ordered Pair: An ordered pair (x, y) specifies the location of a point, where x is the horizontal coordinate and y is the vertical coordinate.

The Distance Formula

The distance between two points in the coordinate plane can be found using the distance formula, which is derived from the Pythagorean Theorem.

• Formula:

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

• Application: Used to find the length between two points, useful in geometry and graphing.

• Example: Find the distance between P(3, 5) and Q(-1, 2): $d = \sqrt{(3 - (-1))^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Right Triangle Condition

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

• Pythagorean Theorem: Used to verify right triangles in coordinate geometry.

Graphing Equations in Two Variables

Graphing by Point Plotting

To graph an equation in two variables, find ordered pairs that satisfy the equation and plot them.

• Step 1: Find the intercepts (where the graph crosses the axes).

• Step 2: Find additional ordered pairs as needed.

• Step 3: Plot the ordered pairs and connect them appropriately.

• Example: For $y = 2x + 5$, find three ordered pairs: (x, y): (-2, 1), (0, 5), (2, 9)

Graphing Nonlinear Equations

Equations such as $x = \sqrt{y + 1}$ and $y = -x^2 + 1$ produce curves rather than straight lines.

• Example: For $y = -x^2 + 1$, ordered pairs include (0, 1), (1, 0), (2, -3).

Circles in the Coordinate Plane

Center-Radius Form of a Circle

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:

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

• Center: (h, k)

• Radius: r

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

General Form of a Circle

The general form of the equation of a circle is:

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

• Completing the Square: Used to convert the general form to center-radius form and find the center and radius.

• Example: $x^2 + y^2 + 4x + 8y = 44$ Complete the square to find center (-2, -4), radius 8.

Determining Existence of a Circle

If the radius squared is negative or zero, the graph is a point or nonexistent.

• Example: $x^2 + y^2 - 6x + 2y + 12 = 0$ Completing the square yields a negative radius squared, so the graph is nonexistent (imaginary).

Relations and Functions

Definitions

• Relation: A set of ordered pairs.

• Function: A relation in which each input (x-value) corresponds to exactly one output (y-value).

• Dependent Variable: The output variable, usually y.

• Independent Variable: The input variable, usually x.

Determining Functions

To determine if a relation is a function, check if any x-value repeats with a different y-value.

• Example: M = {(-4, 0), (-2, 1), (-3, 1), (3, 1)} is a function (no x repeats).

• Example: P = {(-4, 3), (2, 1), (4, 3), (2, -3)} is not a function (x = 2 repeats).

Domain and Range

The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Domain: Set of all x-values.

• Range: Set of all y-values.

• Example: For relation {(-4, -2), (-1, 0), (1, 2), (3, 5)}, Domain: {-4, -1, 1, 3}, Range: {-2, 0, 2, 5}

Finding Domain and Range from Graphs

Analyze the graph to list all x-values (domain) and y-values (range) represented by points on the graph.

• Example: If the graph passes through (0, 2), (1, 4), (2, 6), Domain: {0, 1, 2}, Range: {2, 4, 6}

Increasing, Decreasing, and Constant Functions

Definitions

• Increasing Function: As x increases, y increases.

• Decreasing Function: As x increases, y decreases.

• Constant Function: y remains the same as x changes.

Summary Table: Forms of the Equation of a Circle

Form	Equation	Center	Radius	Notes
Center-Radius Form	$(x - h)^2 + (y - k)^2 = r^2$	(h, k)	r	Standard form for graphing
General Form	$x^2 + y^2 + Dx + Ey + F = 0$	Complete the square to find	Complete the square to find	May represent a circle, point, or nothing

Additional info: These notes expand on the provided materials with definitions, examples, and formulas for clarity and completeness, suitable for College Algebra students.