Introduction to Graphs - Linear Graphs

The Cartesian Plane and Origin: A graph is drawn on a flat surface called the Cartesian Plane, formed by two perpendicular number lines. The horizontal line is the $x$-axis and the vertical line is the $y$-axis. Their point of intersection is called the Origin, represented as $(0, 0)$. Visually, the origin is the central 'starting point' from which all distances are measured.

Coordinates and Ordered Pairs: Every point on a graph is identified by an ordered pair $(x, y)$. The $x$-coordinate (abscissa) indicates the horizontal distance from the $y$-axis, and the $y$-coordinate (ordinate) indicates the vertical distance from the $x$-axis. For example, to locate $(3, -4)$, you move $3$ units to the right and $4$ units down on the grid.

The Four Quadrants: The two axes divide the plane into four regions called quadrants, numbered I to IV in a counter-clockwise direction. In Quadrant I (top-right), coordinates are $(+, +)$; in Quadrant II (top-left), they are $(-, +)$; in Quadrant III (bottom-left), they are $(-, -)$; and in Quadrant IV (bottom-right), they are $(+, -)$.

Linear Equations and Straight Lines: A linear equation in two variables, such as $y = 2x + 3$, represents a relationship that forms a straight line when plotted. Visually, this means that for every constant increase in $x$, there is a constant increase or decrease in $y$, creating a perfectly straight path on the graph paper.

Independent and Dependent Variables: In the equation $y = mx + c$, $x$ is typically the independent variable (values we choose) and $y$ is the dependent variable (the result). On a graph, the independent variable is always plotted on the horizontal $x$-axis, while the dependent variable is plotted on the vertical $y$-axis.

Horizontal and Vertical Lines: Not all lines are slanted. An equation like $x = a$ (where $a$ is a constant) results in a vertical line parallel to the $y$-axis. An equation like $y = b$ results in a horizontal line parallel to the $x$-axis. For example, $y = 5$ is a flat line passing through the point $5$ on the vertical axis.

Intercepts on the Axes: The $x$-intercept is the point where the line crosses the $x$-axis (the $y$-value is $0$). The $y$-intercept is the point where the line crosses the $y$-axis (the $x$-value is $0$). Visually, these are the 'anchor points' where the line touches the bold axis lines of your grid.