Y-Intercept
The y-intercept is the point where a line or curve crosses the y-axis, found by setting x equal to zero in the equation.
Formula
\text{Set } x = 0, \text{ solve for } y
Definition
The y-intercept is the point where a line or curve crosses the y-axis, the value of $y$ when $x$ equals zero, written $f(0)$ for a function $f$. In slope-intercept form $y = mx + b$, the constant $b$ is exactly this value. Every non-vertical line has exactly one y-intercept; a vertical line $x = c$ has none unless $c = 0$, in which case the entire line is the y-axis. In regression analysis, the y-intercept represents the predicted value of the response variable when all predictors equal zero, and it carries meaningful interpretation only when $x = 0$ falls within the observed data range.
Example
In $y = 3x + 5$, setting $x = 0$ gives $y = 5$, so the line crosses the y-axis at $(0, 5)$. For $4x - 2y = 8$, setting $x = 0$ gives $-2y = 8$, so $y = -4$ and the line crosses at $(0, -4)$. In the regression model $\text{salary} = 30{,}000 + 5{,}000(\text{years\_experience})$, the y-intercept $30{,}000$ estimates starting salary with zero experience.
Key Insight
The y-intercept is like the "starting value": if a graph tracks a plant's height over days, the y-intercept is its height on day zero. In Taylor series expansions, the constant term $f(0)$ plays exactly this same role, connecting the idea of an intercept as an initial condition to differential equations and mathematical modeling.