Tangent line and normal line

Once the derivative $f'(x_0)$, that is the slope, is known, the equation of the tangent line is written using the formula for the pencil of lines through a point.

Property — The tangent line

The tangent line to the graph of $f$ at the point $A(x_0, f(x_0))$ has equation: $y - f(x_0) = f'(x_0)\,(x - x_0).$ The normal line (perpendicular to the tangent) has slope $-1/f'(x_0)$.

Example — Tangent to the parabola

Find the tangent to the graph of $y = \dfrac{x^2}{2}-3x+1$ at the point with $x=2$.

Compute the ordinate: $y(2) = 2-6+1 = -3$, so $A(2;-3)$. The derivative is $y' = x-3$, whence $y'(2) = -1$, that is $m=-1$.

Tangent: $y+3 = -(x-2)$, that is $\boxed{y = -x-1}$.

Links

Topics: Derivatives
Concepts: Derivative · Line · Normal line · Tangent line
Skills: Analytic geometry · Using formulae