X-Intercept
AlgebraThe x-intercept is the point where a line or curve crosses the x-axis, found by setting y equal to zero in the equation.
Formula
\text{Set } y = 0, \text{ solve for } x
Definition
The x-intercept is where a line crosses the left-right axis (the x-axis) on a graph. It is the value of $x$ when $y$ equals zero.
Example
In $y = 2x - 6$, set $y = 0$: $0 = 2x - 6$, so $2x = 6$, $x = 3$. The x-intercept is $3$, and the line crosses the x-axis at $(3, 0)$.
Key Insight
The x-intercept is where something "runs out" or "breaks even." If a graph shows money over days, the x-intercept is the day the money hits zero.
Definition
The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. It is found by substituting $y = 0$ into the equation and solving for $x$. A line may have zero, one, or (if it is the x-axis itself) infinitely many x-intercepts.
Example
For $3x + 6y = 12$, set $y = 0$: $3x = 12$, $x = 4$. The x-intercept is $4$, at the point $(4, 0)$.
Key Insight
For quadratic and higher-degree equations, x-intercepts are the real roots or zeros of the function. Finding x-intercepts is one of the most important skills in all of algebra.
Definition
The x-intercept is a zero of the function $f$, meaning $f(x) = 0$. For polynomials, the Fundamental Theorem of Algebra guarantees $n$ zeros (counting multiplicity) in the complex numbers for a degree-$n$ polynomial, but real x-intercepts correspond only to real zeros. The x-intercept and the root/zero concept unify across polynomial, exponential, and transcendental equations.
Example
$f(x) = x^3 - 6x^2 + 11x - 6$ factors as $(x-1)(x-2)(x-3)$, giving x-intercepts at $x = 1, 2, 3$.
Key Insight
In numerical analysis, root-finding algorithms (Newton's method, bisection) are dedicated entirely to locating x-intercepts. The concept extends to finding equilibria in differential equations.