Introduction to Graphing: Cartesian Coordinates, Distance, Midpoint, and Circles

Introduction to Graphing

Cartesian Coordinate System

The Cartesian Coordinate System is a fundamental tool in algebra for representing points, equations, and geometric figures in a two-dimensional plane. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0, 0).

• Quadrants: The plane is divided into four quadrants, labeled I, II, III, and IV, starting from the upper right and moving counterclockwise.

• Points: Each point is represented as an ordered pair (x, y), where x is the horizontal position and y is the vertical position relative to the origin.

Example: Plotting points (−2, 4), (3, 3), and (0, −3) involves locating each on the plane according to their x and y values.

Solutions of Equations

A solution of an equation is a value or set of values for the variable(s) that make the equation true. In the context of graphing, solutions often correspond to points on the graph of the equation.

• Verification: Substitute the values into the equation to check if the equality holds.

Example: For the equation $2x + 3y = 18$:

• Test (−5, 7): $2(-5) + 3(7) = -10 + 21 = 11 \neq 18$ (Not a solution)

• Test $(\frac{3}{2}, 5)$: $2(\frac{3}{2}) + 3(5) = 3 + 15 = 18$ (Solution)

Graphs of Equations

To graph an equation means to draw the set of all points (x, y) that satisfy the equation. The graph visually represents the relationship between variables.

• x-intercept: The point where the graph crosses the x-axis, found by setting y = 0.

• y-intercept: The point where the graph crosses the y-axis, found by setting x = 0.

Example: For $2x + 3y = 18$:

• x-intercept: $2x + 3(0) = 18 \Rightarrow x = 9$; point is (9, 0).

• y-intercept: $2(0) + 3y = 18 \Rightarrow y = 6$; point is (0, 6).

The Distance Formula

The Distance Formula calculates the length between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane.

• Formula:

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

Example:

• Between (−2, 2) and (3, −6): $d = \sqrt{(3 - (-2))^2 + (-6 - 2)^2} = \sqrt{5^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.4$

• Between (−1, −5) and (−1, 2): $d = \sqrt{(-1 - (-1))^2 + (2 - (-5))^2} = \sqrt{0^2 + 7^2} = \sqrt{49} = 7$

Midpoint of Segments

The Midpoint Formula finds the point exactly halfway between two endpoints $(x_1, y_1)$ and $(x_2, y_2)$.

• Formula:

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

Example: For endpoints (−4, −2) and (2, 5):

• Midpoint: $\left( \frac{-4 + 2}{2}, \frac{-2 + 5}{2} \right) = \left( \frac{-2}{2}, \frac{3}{2} \right) = (-1, 1.5)$

Circles

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

• Standard Form: $(x - h)^2 + (y - k)^2 = r^2$ where (h, k) is the center and r is the radius.

• Circle at Origin: $x^2 + y^2 = r^2$

Example:

• Circle with radius 5 and center (3, −7): $(x - 3)^2 + (y + 7)^2 = 25$

• Circle $(x - 2)^2 + (y + 1)^2 = 16$: Center is (2, −1), radius is $\sqrt{16} = 4$

Summary Table: Key Formulas

Concept	Formula	Description
Distance between points	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Length between two points
Midpoint of segment	$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$	Point halfway between endpoints
Equation of a circle	$(x - h)^2 + (y - k)^2 = r^2$	Set of points at distance r from (h, k)
x-intercept	Set y = 0, solve for x	Where graph crosses x-axis
y-intercept	Set x = 0, solve for y	Where graph crosses y-axis

Additional info: Academic context and explanations have been expanded for clarity and completeness, including definitions, formulas, and examples for each concept.