Introduction to Graphs - The Cartesian Plane and Coordinates

The Cartesian Plane is a two-dimensional surface formed by the intersection of two perpendicular number lines: the horizontal 'x-axis' and the vertical 'y-axis'. These axes intersect at a point called the 'Origin', represented by the letter $O$ and coordinates $(0, 0)$. Visualizing this, the plane looks like a grid where the x-axis goes from left to right and the y-axis goes from bottom to top.

A point in the Cartesian plane is identified by an 'Ordered Pair' written as $(x, y)$. The first number $x$ is the 'abscissa' (horizontal distance from the y-axis), and the second number $y$ is the 'ordinate' (vertical distance from the x-axis). For example, to find $(2, 3)$, you move $2$ units right from the origin and $3$ units up.

The x-axis and y-axis divide the plane into four regions called 'Quadrants', numbered I, II, III, and IV in a counter-clockwise direction starting from the top right. In Quadrant I, coordinates are $(+, +)$; in Quadrant II, they are $(-, +)$; in Quadrant III, they are $(-, -)$; and in Quadrant IV, they are $(+, -)$.

Points that lie directly on the axes have one coordinate equal to zero. Any point on the x-axis has a y-coordinate of $0$, written as $(x, 0)$. Similarly, any point on the y-axis has an x-coordinate of $0$, written as $(0, y)$. For instance, $(5, 0)$ is a point on the x-axis, while $(0, -4)$ is on the y-axis.

A 'Linear Graph' is a graph that is a whole unbroken line. It is formed by joining points that have a constant relationship. When you plot points from a table and they all lie on a single straight path, the resulting figure is a line graph representing a linear relationship between the variables.

The independent variable (a value that changes on its own, like time) is usually plotted on the horizontal x-axis, while the dependent variable (a value that changes based on the independent variable, like distance) is plotted on the vertical y-axis.