Tangent line to a parabola

A tangent line to a parabola is a line that just touches the curve, or equivalently a line that has the same slope as the curve at the point of contact¹. Here we find the tangent to $y = x^2$ at the point $(1, 1)$.

From the derivative

Differentiating $y = x^2$ gives $y' = 2x$, so the slope at $x = 1$ is $2$. The tangent line is $y - 1 = 2(x - 1)$, that is $y = 2x - 1$.

From a repeated root

Tangency can be confirmed without calculus. Solving the parabola and the line together gives $x^2 - 2x + 1 = (x - 1)^2 = 0$. The solution is a repeated root, so there is exactly one shared point. Tangency is what happens when two intersection points merge into one.

Finding the slope without differentiating

Write the line through $(1, 1)$ with slope $m$ as $y = m(x - 1) + 1$ and combine it with the parabola to get $x^2 - mx + m - 1 = 0$. Tangency requires the discriminant to vanish.

So $m = 2$, agreeing with the slope from calculus. It is satisfying that the discriminant turns out to be a perfect square, pinning the tangent slope to a single value.

The general tangent

The tangent to $y = x^2$ at $(a, a^2)$ is $y = 2ax - a^2$. Move the point of contact and the slope $2a$ moves with it, and the way it moves is nothing other than the derivative $y' = 2x$. The tangent line is where the whole idea of differentiation begins.

Two tangents from an outside point

Take $(0, -1)$: a line $y = mx - 1$ through it meets the parabola where $x^2 - mx + 1 = 0$, and $D = m^2 - 4 = 0$ gives $m = \pm 2$.

The pair of tangents embraces the curve from either side.

A tangent is an approximation

Near the point of contact, the curve may be treated as the line.

Replacing a curve locally by a straight line is the idea behind every application of the derivative. The large dot on the graph is the point of tangency.

1. Tangent, Wikipedia