Slope intercept form
Slope intercept form expresses a linear equation as \(y = mx + b\). The parameter \(m\) sets the slope, and the parameter \(b\) sets the \(y\)-intercept.
Parameter meanings.
- Slope \(m\). The constant rate of change of \(y\) with respect to \(x\), expressed as \(\Delta y / \Delta x\).
- \(y\)-intercept \(b\). The value of \(y\) when \(x = 0\), so the line crosses the \(y\)-axis at \((0,b)\).
Geometric interpretation on the coordinate plane
In slope intercept form, the point \((0,b)\) lies on the line immediately because substituting \(x=0\) yields \(y=b\). The slope \(m\) specifies a consistent “rise over run” from any point on the line: a horizontal change of \(\Delta x\) produces a vertical change of \(\Delta y = m \cdot \Delta x\).
Positive \(m\) corresponds to an increasing line from left to right, negative \(m\) corresponds to a decreasing line, and \(m=0\) corresponds to a horizontal line \(y=b\).
Algebraic structure and equivalent forms
Slope intercept form emphasizes the linear function viewpoint: \(y\) depends on \(x\) through a constant rate \(m\) and a constant offset \(b\). Rearrangement yields standard form: \[ y = mx + b \;\Longrightarrow\; mx - y = -b \;\Longrightarrow\; Ax + By = C, \] where \(A\), \(B\), and \(C\) are constants. The representation changes, while the set of points \((x,y)\) on the line remains the same.
Recovering parameters from points
Two points \((x_1,y_1)\) and \((x_2,y_2)\) with \(x_2 \ne x_1\) determine a unique slope \[ m = \frac{y_2 - y_1}{x_2 - x_1}. \] Substituting one point into \(y = mx + b\) yields \(b = y_1 - mx_1\), which completes slope intercept form.
| Parameter | Algebraic role in \(y = mx + b\) | Graph meaning |
|---|---|---|
| \(m\) | Coefficient of \(x\) | Steepness and direction; rise per 1 unit of run |
| \(b\) | Constant term | Intersection with the \(y\)-axis at \((0,b)\) |
Worked example
A line has slope \(m = -2\) and \(y\)-intercept \(b = 7\). Slope intercept form is \[ y = -2x + 7. \] At \(x = 3\), the corresponding value is \[ y = -2(3) + 7 = 1. \] The point \((3,1)\) lies on the line because it satisfies the equation.
Common pitfalls
Confusing the intercepts is common: \(b\) is the \(y\)-value at \(x=0\), not the \(x\)-intercept. The \(x\)-intercept occurs when \(y=0\), giving \(0 = mx + b\) and \(x = -\frac{b}{m}\) when \(m \ne 0\).
A negative \(m\) indicates a decreasing line from left to right; the sign belongs to the slope, not to the intercept unless the constant term is also negative.
Vertical lines do not have slope intercept form because they have undefined slope; their equations are of the form \(x = c\).