AI Tutor
If you swing a ball on a string in a circle and cut the cord, the ball does not fly outward β€” it shoots along a perfectly straight line touching the curve at that exact instant.
That trajectory is the tangent line, while the string pulling directly perpendicular to it is the normal line.
How do we write the algebraic equation for both of these straight lines given any curve y=f(x)y = f(x)?

The Core Formulas

Any straight line passing through a point (x0,y0)(x_0, y_0) with slope mm follows the point-slope form: yβˆ’y0=m(xβˆ’x0)y - y_0 = m(x - x_0).
The derivative fβ€²(x0)f'(x_0) gives the instantaneous slope of the curve at that point, which is precisely the tangent slope mT=fβ€²(x0)m_T = f'(x_0).
Because perpendicular lines meet at 90∘90^\circ, rotating a slope Ξ”yΞ”x\frac{\Delta y}{\Delta x} by a right angle inverts and negates it, giving the normal slope mN=βˆ’1mTm_N = -\frac{1}{m_T} whenever mTβ‰ 0m_T \neq 0.
In the 1630s, Pierre de Fermat first used early tangent methods to find peak points of curves, laying the groundwork for calculus decades before Newton and Leibniz formalized derivatives.
Let's put this into action with a concrete polynomial curve.

Worked Example: Polynomial Curve

Find the equations of the tangent and normal lines to the curve y=x3βˆ’3x+2y = x^3 - 3x + 2 at x=2x = 2.