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When a beam of light hits a curved mirror, it does not bounce straight back along the curve. Instead, it reflects across an invisible reference axis that stands perfectly perpendicular to the surface at the point of contact.
In geometry and calculus, this perpendicular line is called the normal line.
While a tangent line grazes a curve and shares its exact instantaneous steepness, the normal line cuts through that same point at a strict 90∘90^\circ angle.
To find the slope of the normal line, we rely on the perpendicular gradient rule: two non-vertical lines are perpendicular if and only if the product of their slopes equals βˆ’1-1 (m1β‹…m2=βˆ’1m_1 \cdot m_2 = -1).
Since rotating a slope of Ξ”yΞ”x\frac{\Delta y}{\Delta x} by 90∘90^\circ swaps the horizontal run and vertical rise while flipping the direction of change, the normal slope is the negative reciprocal of the tangent slope:
mnormal=βˆ’1fβ€²(x)m_{\text{normal}} = -\frac{1}{f'(x)}
In the 1630s, RenΓ© Descartes actually solved normal lines before tangent lines existed in modern calculus, using intersecting circles to find normals to define curve curvature.
So how do we turn this slope relationship into a full linear equation for any curve?

Constructing the Normal Equation

Every line can be determined with a single point (x0,y0)(x_0, y_0) and a slope mm using the point-slope form: yβˆ’y0=m(xβˆ’x0)y - y_0 = m(x - x_0).
Let us walk through the four-step workflow to find the equation of the normal to the curve f(x)=x2βˆ’3x+1f(x) = x^2 - 3x + 1 at x=3x = 3.