Fundamentals of the Cartesian Plane and Coordinate Geometry

Cartesian Plane and Its Structure

Definition and Components

The Cartesian plane is a two-dimensional surface defined by two perpendicular axis lines: the horizontal x-axis and the vertical y-axis. These axes intersect at a point called the origin.

• Origin: The point (0, 0) where the x-axis and y-axis intersect.

• Axis lines: Reference lines used to determine the position of points in relation to each other.

• Quadrants: The axes divide the plane into four regions called quadrants, each with distinct sign conventions for coordinates.

Quadrant Classification

Quadrant	x-value	y-value
I	+	+
II	-	+
III	-	-
IV	+	-

Additional info: Quadrants are numbered counterclockwise starting from the upper right.

Ordered Pairs and Plotting Points

Ordered Pair Definition

An ordered pair (x, y) consists of two numbers that specify the location of a point on the Cartesian plane. The x-coordinate gives the horizontal position, and the y-coordinate gives the vertical position.

• Example: The point (3, 5) is 3 units to the right of the origin and 5 units up.

Plotting Points

To plot a point, locate its x-coordinate on the x-axis and its y-coordinate on the y-axis, then mark the intersection.

• Example: Plot the points: A (3, 5), B (4, 3), C (-4, 2), D (0, 3), E (-3, 0).

Solutions to Equations and Graphing

Testing Solutions

A point (x, y) is a solution to an equation if substituting x and y into the equation yields a true statement.

• Example: Is (2, 8) a solution to $y = -4x + 10$? Substitute: $8 = -4(2) + 10 = -8 + 10 = 2$ (False).

• Example: Is (1, 6) a solution to $y = -4x + 10$? Substitute: $6 = -4(1) + 10 = -4 + 10 = 6$ (True).

Graphing an Equation

To graph an equation, create a table of values for x and y, plot the corresponding points, and connect them if appropriate.

Intercepts

x-intercept and y-intercept

• x-intercept: The point where the graph crosses the x-axis. Set y = 0 and solve for x.

• y-intercept: The point where the graph crosses the y-axis. Set x = 0 and solve for y.

Distance and Midpoint Formulas

Distance Formula

The distance formula calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$:

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

• Example: Find the distance between (-2, 2) and (3, 4): $d = \sqrt{(3 - (-2))^2 + (4 - 2)^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}$

Midpoint Formula

The midpoint formula finds the point exactly halfway between $(x_1, y_1)$ and $(x_2, y_2)$:

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

• Example: Find the midpoint between (-4, 2) and (2, 5): $\left( \frac{-4 + 2}{2}, \frac{2 + 5}{2} \right) = \left( \frac{-2}{2}, \frac{7}{2} \right) = (-1, 3.5)$

Equation of a Circle in Standard Form

Standard Form Equation

The standard form of a circle with center $(h, k)$ and radius $r$ is:

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

• Example: The equation $(x - 3)^2 + (y - 8)^2 = 64$ represents a circle centered at (3, 8) with radius $r = \sqrt{64} = 8$.

Additional info: To find the center and radius from a given equation, compare to the standard form and extract $h$, $k$, and $r$.