Algebra - Introduction to Linear Graphs and the Cartesian Plane

The Cartesian Plane is a two-dimensional surface formed by the intersection of a horizontal number line (the $x$-axis) and a vertical number line (the $y$-axis). These axes intersect at a point called the origin, denoted as $(0, 0)$. The plane is divided into four regions called quadrants, numbered I, II, III, and IV in a counter-clockwise direction starting from the top-right.

Coordinates are written as ordered pairs $(x, y)$. The first number, the $x$-coordinate (abscissa), indicates the horizontal distance from the origin (right is positive, left is negative). The second number, the $y$-coordinate (ordinate), indicates the vertical distance (up is positive, down is negative). Visually, a point like $(2, -3)$ is located by moving 2 units right and 3 units down from the origin.

A linear equation produces a straight-line graph. The most common form is the gradient-intercept form, $y = mx + c$. Visually, this line represents a constant rate of change where every step across results in a consistent step up or down.

The gradient (or slope), represented by $m$, measures the steepness of a line. Visually, a positive gradient $(m > 0)$ slopes upwards from left to right, while a negative gradient $(m < 0)$ slopes downwards. A gradient of zero $(m = 0)$ results in a perfectly horizontal line, and an undefined gradient represents a vertical line.

The $y$-intercept is the point where the graph crosses the vertical $y$-axis. At this point, the $x$-coordinate is always $0$. In the equation $y = mx + c$, the constant $c$ gives the $y$-value of this intercept. For example, if $c = 4$, the line crosses the $y$-axis at $(0, 4)$.

The $x$-intercept is the point where the graph crosses the horizontal $x$-axis. At this point, the $y$-coordinate is always $0$. To find it visually, look for where the line touches the $x$-axis; algebraically, substitute $y = 0$ into the equation and solve for $x$.

To plot a linear graph from an equation, you can use a table of values. Choose at least three values for $x$, calculate the corresponding $y$ values using the equation, plot these points as dots on the Cartesian plane, and connect them with a straight edge to show the line extends infinitely in both directions.

Horizontal lines are written in the form $y = k$, where $k$ is a constant. These lines are parallel to the $x$-axis. Vertical lines are written in the form $x = h$, where $h$ is a constant. These lines are parallel to the $y$-axis and do not have a defined gradient.