Y-Intercept
AlgebraThe 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 where a line crosses the up-down axis (the y-axis) on a graph. It is the value of $y$ when $x$ equals zero.
Example
In $y = 3x + 5$, when $x = 0$ we get $y = 5$. The y-intercept is $5$, and the line crosses the y-axis at the point $(0, 5)$.
Key Insight
The y-intercept is like the "starting value." If a graph shows a plant growing over days, the y-intercept is the plant's height on day zero.
Definition
The y-intercept of a line is the y-coordinate of the point where the line intersects the y-axis. It occurs when $x = 0$. In slope-intercept form $y = mx + b$, the constant $b$ is the y-intercept.
Example
For $4x - 2y = 8$, set $x = 0$: $-2y = 8$, so $y = -4$. The y-intercept is $-4$, and the line crosses at $(0, -4)$.
Key Insight
Every non-vertical line has exactly one y-intercept. A vertical line $x = c$ has no y-intercept unless $c = 0$, in which case the entire line is the y-axis.
Definition
The y-intercept is the value of a function $f$ at $x = 0$, written $f(0)$. For a linear function $f(x) = mx + b$ it equals $b$, the additive constant in the affine map. In regression analysis, the y-intercept represents the predicted value of the response variable when all predictors equal zero, and carries interpretive meaning only when $x = 0$ is within the observed data range.
Example
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
In Taylor series expansions, the constant term $f(0)$ plays the same role as the y-intercept. Understanding intercepts as initial conditions connects algebra to differential equations and modeling.