Linear Function f(x) = mx + b: Graph, Slope, Derivative and Integral

A linear function has a constant rate of change, so its graph is a straight line. Written f(x) = mx + b, the number m is the slope (how much f rises per unit of x) and b is the y-intercept (the output at x = 0).

Formula and forms

The same line can be written in other ways, depending on what you know:

Point-slope form is the one calculus uses most, because a tangent line is built from a point (a, f(a)) and a slope f'(a).

Graph of a linear function

• m > 0: increasing. m < 0: decreasing. m = 0: a horizontal line, the constant function.
• The larger |m| is, the steeper the line.
• The x-intercept is where mx + b = 0, that is x = -\frac{b}{m} when m \ne 0.

Derivative of a linear function

The derivative of mx is m by the power rule and constant multiple rule, and the derivative of b is 0. This matches the geometry: the tangent line to a line is the line itself, and its slope never changes. The second derivative is 0, so a line has no concavity.

Integral of a linear function

Integrating a line gives a quadratic function. For a definite integral over [a, b], the region under a line is a trapezoid (or two triangles if it crosses the axis), so you can check any answer with geometry.

Worked examples

Example 1: slope from two points. Find the line through (1, 3) and (4, 9).

Example 2: derivative. If f(x) = -4x + 7, then f'(x) = -4 for every x.

Example 3: definite integral. Evaluate \int_0^2 (3x + 1)\,dx.

Check: the region is a trapezoid with parallel sides f(0) = 1 and f(2) = 7 and width 2, so its area is \frac{1 + 7}{2} \cdot 2 = 8.

Example 4: a tangent line. The tangent line to y = x^2 at x = 3 has slope 2(3) = 6 and passes through (3, 9), so it is the linear function y = 6x - 9.

Common mistakes

1. Swapping the order in the slope formula. Keep the same point first on top and bottom: \frac{y_2 - y_1}{x_2 - x_1}, not \frac{y_2 - y_1}{x_1 - x_2}.
2. Calling f(x) = x^2 + 1 linear. Linear means the variable appears only to the first power.
3. Integrating mx as mx^2. The reverse power rule divides by the new exponent: \int mx\,dx = \frac{m}{2}x^2 + C.
4. Confusing "linear function" with "linear equation in two unknowns". A vertical line x = 2 is a linear equation but not a function.

How it connects

Lines are the reason derivatives exist: the derivative at a point is the slope of the best line through it. That idea powers tangent lines, linear approximation and Newton's method. Lines are degree-one polynomial functions, and integrating them leads to quadratics.