The Cartesian coordinate system consists of a horizontal line, known as the x-axis, and a vertical line, known as the y-axis, which intersect each other at a central point called the origin. On the Cartesian plane, students graph equations, which may be either straight lines or curves. The point or points at which one of these graphs crosses or touches the x-axis is known as its x-intercept, and the point or points where such a graph crosses or touches the y-axis is known as the y-intercept.
An x-intercept is written in the form (x, 0), where x is any rational number but y is always zero. A y-intercept is written in the form (0, y), where y is any number and x is always zero. For example, (-3, 0) is an x-intercept, and (0, 1.5) is a y-intercept. An equation's x-intercepts are also known as its roots, solutions or zeroes. For instance, finding the roots of an equation is akin to finding its x-intercepts. However, y-intercepts go by no name other than y-intercepts.
The x- and y- intercepts are always two separate points; the only exception is if a graph crosses through the origin, where both intercepts equal zero. The number of intercepts an equation has depends on the nature of the particular equation. For instance, a straight line, such as y = x + 1, has one x-intercept, while a sine wave has an infinite number of x-intercepts.
It is possible to determine intercepts from an equation without graphing it. Because the x-intercept is the point where the y-value is zero, you can substitute zero in for "y" in an equation and solve that equation for "x," and the solution will be the x-intercept. Likewise, since the y-intercept is the point where the x-value is zero, you can plug zero into an equation for "x" and solve for "y" to find the y-intercept. For instance, suppose you want to find the x-intercept of 3x + 2y = 12 without graphing. Replace the "y" with a "0," rendering 3x = 12, and divide by three, yielding an x-intercept of four.