Rectangular Coordinates, Distance, Midpoint, Circles, and Functions: Precalculus Study Notes

Rectangular Coordinates and Graphs

Ordered Pairs

In mathematics, an ordered pair is a pair of numbers used to locate a point in a plane, typically written as $(x, y)$. The first number represents the horizontal position (x-coordinate), and the second number represents the vertical position (y-coordinate).

• Example: The ordered pair (transportation, 8796) expresses the relationship between a category and the amount spent.

• Application: Ordered pairs are used to represent data, such as (healthcare, 9728).

The Rectangular Coordinate System

The rectangular coordinate system (or Cartesian coordinate system) consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four regions called quadrants.

• Origin: The point (0, 0) where the axes intersect.

• Quadrants: Numbered I, II, III, IV, starting from the upper right and moving counterclockwise.

• Coordinate Plane: The plane in which points are plotted using ordered pairs.

Distance and Midpoint Formulas

Distance Formula

The distance formula calculates 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 $\\sqrt{(3 - (-2))^2 + ((-5) - 8)^2} = \\sqrt{25 + 169} = \\sqrt{194}$.

Pythagorean Theorem and Right Triangles

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 hypotenuse $c$.

• Application: Use the distance formula to verify if three points form a right triangle.

Midpoint Formula

The midpoint formula finds the midpoint $M$ of a line segment with endpoints $P(x_1, y_1)$ and $Q(x_2, y_2)$:

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

• 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 and Graphing

Ordered-Pair Solutions of Equations

An equation in two variables, such as $y = -2x + 5$, has solutions that are ordered pairs $(x, y)$ satisfying the equation.

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

Graphing Equations

To graph an equation:

1. Find the intercepts.

2. Find additional ordered pairs as needed.

3. Plot the points.

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

• Example: Graphs of $y = -2x + 5$, $x = \\sqrt{y + 1}$, and $y = -x^2 + 1$ show different types of lines and curves.

Circles: Center-Radius and General Form

Center-Radius Form

A circle with center $(h, k)$ and radius $r$ has the equation:

• $(x - h)^2 + (y - k)^2 = r^2$

• 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 a Circle

The general form of a circle's equation is:

• $x^2 + y^2 + Dx + Ey + F = 0$

• Completing the square can convert this to center-radius form.

• Example: $x^2 + 4x + y^2 - 8y = 45$ becomes $(x + 2)^2 + (y - 4)^2 = 69$

Nonexistent Circles

If the radius squared is negative, the graph does not exist in the real plane.

• Example: $(x - 3)^2 + (y + 1)^2 = -2$ has no real graph.

Application: Locating the Epicenter of an Earthquake

Systems of equations involving circles can be used to solve real-world problems, such as locating an earthquake's epicenter using distances from multiple stations.

Relations and Functions

Relations and Functions

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: $M = \\{(-4, 0), (-3, 1), (3, 1)\\}$ is a function; $N = \\{(-2, 4), (3, 4), (5, -4)\\}$ is a function; $P = \\{(-4, 3), (0, 6), (2, 8), (-4, -3)\\}$ is not a function (since $-4$ maps to both $3$and$-3$).

Function Notation

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

• Example: $f(x) = x^2$ means for each $x$, the output is $x^2$.

Increasing, Decreasing, and Constant Functions

A function is increasing if its output rises as the input increases, decreasing if its output falls, and constant if its output remains the same.

• Example: $f(x) = x$ is increasing; $f(x) = -x$ is decreasing; $f(x) = 5$ is constant.

Summary Table: Key Formulas

Concept	Formula	Example
Distance between $(x_1, y_1)$ and $(x_2, y_2)$	$d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	$d((3, -5), (-2, 8)) = \\sqrt{25 + 169} = \\sqrt{194}$
Midpoint of $(x_1, y_1)$ and $(x_2, y_2)$	$M = \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)$	$M((-7, -5), (2, 13)) = (-2.5, 4)$
Circle (center $(h, k)$, radius $r$)	$(x - h)^2 + (y - k)^2 = r^2$	$(x - 1)^2 + (y + 2)^2 = 9$
General form of a circle	$x^2 + y^2 + Dx + Ey + F = 0$	$x^2 + 4x + y^2 - 8y = 45$

Additional info:

• Some context and terminology were inferred for completeness and clarity.

• Examples and applications were expanded for student understanding.